MIT OpenCourseWare: All Courses in MathematicsAll courses in Mathematics from MIT OpenCourseWare, provider of free and open MIT course materials.
https://ocw.mit.edu/courses/mathematics
2021-09-17T15:10:16+05:00MIT OpenCourseWare https://ocw.mit.eduen-USContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.085 Computational Science and Engineering I (MIT)This course provides the fundamental computational toolbox for solving science and engineering problems. Topics include review of linear algebra, applications to networks, structures, estimation, finite difference and finite element solutions of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, the discrete Fourier transform, and convolution. We will also explore many topics in AI and machine learning throughout the course.
https://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-summer-2020
Summer2020Zhang, Chengzhao “Richard”2021-07-28T18:21:20+05:0018.08518.0851en-USlinear algebranetworksstructuresfinite differencefinite elementLaplace's equationpotential flowboundary-value problemsFourier seriesdiscrete Fourier transformconvolutionMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S191 Introduction to Computational Thinking (MIT)This is an introductory course on computational thinking. We use the Julia programming language to approach real-world problems in varied areas, applying data analysis and computational and mathematical modeling. In this class you will learn computer science, software, algorithms, applications, and mathematics as an integrated whole. Topics include image analysis, particle dynamics and ray tracing, epidemic propagation, and climate modeling.
https://ocw.mit.edu/courses/mathematics/18-s191-introduction-to-computational-thinking-fall-2020
Fall2020Edelman, AlanSanders, David P.Sanderson, GrantSchloss, JamesDrake, Henri2021-04-05T18:59:30+05:0018.S1916.S08322.S092en-UScomputational modelingmathematical modelingcomputational scienceartificial intelligenceJulia programmingdata sciencealgorithmsstatistical modelingimage analysisparticle dynamicsray tracingepidemic propagationclimate modelingMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.031 System Functions and the Laplace Transform (MIT)This half-semester course studies basic continuous control theory as well as representation of functions in the complex frequency domain. It covers generalized functions, unit impulse response, and convolution. Also covered are the Laplace transform, system (or transfer) functions, and the pole diagram. Examples from mechanical and electrical engineering are provided. Go to OCW’s Open Learning Library site for 18.031: System Functions and the Laplace Transform. The site is free to use, just like all OCW sites. You have the option to sign up and enroll in the course if you want to track your progress, or you can view and use all the materials without enrolling.
https://ocw.mit.edu/courses/mathematics/18-031-system-functions-and-the-laplace-transform-spring-2019
Spring2019Pearce, Philip2021-02-12T19:10:59+05:0018.031en-USLaplace transformunit impulse responsestep functiondelta functionconvolutionsystem functionpole diagramMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S190 Introduction to Computational Thinking with Julia, with Applications to Modeling the COVID-19 Pandemic (MIT)This half-semester course introduces computational thinking through applications of data science, artificial intelligence, and mathematical models using the Julia programming language. This Spring 2020 version is a fast-tracked curriculum adaptation to focus on applications to COVID-19 responses. See the MIT News article Computational Thinking Class Enables Students to Engage in Covid-19 Response
https://ocw.mit.edu/courses/mathematics/18-s190-introduction-to-computational-thinking-with-julia-with-applications-to-modeling-the-covid-19-pandemic-spring-2020
Spring2020Edelman, AlanSanders, David P.2020-09-14T12:42:09+05:0018.S1906.S083en-UScomputational modelingmathematical modelingCovid-19computational scienceartificial intelligenceJulia programmingdata sciencelanguagestatistical modelingepidemiologymachine learningdrug developmentdisease modelsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.906 Algebraic Topology II (MIT)This is the second part of the two-course series on algebraic topology. Topics include basic homotopy theory, obstruction theory, classifying spaces, spectral sequences, characteristic classes, and Steenrod operations.
https://ocw.mit.edu/courses/mathematics/18-906-algebraic-topology-ii-spring-2020
Spring2020Miller, Haynes2020-08-11T15:45:49+05:0018.906en-UShomotopycohomologyclassifying spacesspectral sequences cofibrationsserre fibrationsSteenrod operationscohomology operationscobordismcobordismMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.217 Graph Theory and Additive Combinatorics (MIT)This course examines classical and modern developments in graph theory and additive combinatorics, with a focus on topics and themes that connect the two subjects. The course also introduces students to current research topics and open problems.
https://ocw.mit.edu/courses/mathematics/18-217-graph-theory-and-additive-combinatorics-fall-2019
Fall2019Zhao, Yufei2020-05-12T17:36:18+05:0018.217en-USgraph theoryadditive combinatoricsRamsey theorySchur’s theoremMantel’s theoremTurán’s theoremErdős-Stone-Simonovits theoremKővári-Sós-Turán theoremSzemerédi’s graph regularity lemmatriangle counting lemmatriangle removal lemmaRoth’s theoremhypergraph removal lemmaGreen-Tao theoremmartingale convergence theoremFreiman’s theoremRuzsa triangle inequalityRuzsa covering lemmaBalog-Szémeredi-Gowers theoremSzemerédi-Trotter theoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.785 Number Theory I (MIT)This is the first semester of a one-year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.
https://ocw.mit.edu/courses/mathematics/18-785-number-theory-i-fall-2019
Fall2019Sutherland, Andrew2020-04-23T14:42:04+05:0018.785en-USAbsolute valuesDiscrete valuationslocalizationDedekind domainsEtale algebrasDedekind extensionsIdeal NormDedekind-Kummer TheoremGalois extensionsArtin mapcomplete fieldsValuation ringsHensel's lemmasKrasner's lemmaMinkowski boundDirichlet's unit theormZeta functionRay ClassRing of AdelesIdele groupChebotarev density theoremGlobal fieldsTate cohomologyArtin reciprocityMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.600 Probability and Random Variables (MIT)This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.
https://ocw.mit.edu/courses/mathematics/18-600-probability-and-random-variables-fall-2019
Fall2019Sheffield, Scott2020-04-06T16:49:43+05:0018.600en-USProbability spacesrandom variablesdistribution functionsBinomialgeometrichypergeometricPoisson distributionsUniformexponentialnormalgamma and beta distributionsConditional probabilityBayes theoremjoint distributionsChebyshev inequalitylaw of large numberscentral limit theoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.212 Algebraic Combinatorics (MIT)This course covers the applications of algebra to combinatorics. Topics include enumeration methods, permutations, partitions, partially ordered sets and lattices, Young tableaux, graph theory, matrix tree theorem, electrical networks, convex polytopes, and more.
https://ocw.mit.edu/courses/mathematics/18-212-algebraic-combinatorics-spring-2019
Spring2019Postnikov, Alexander2019-12-19T16:21:22+05:0018.212en-USenumeration methodspermutationspartitionspartially ordered sets and latticesYoung tableauxgraph theorymatrix tree theoremelectrical networksconvex polytopesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.218 Probabilistic Method in Combinatorics (MIT)This course is a graduate-level introduction to the probabilistic method, a fundamental and powerful technique in combinatorics and theoretical computer science. The essence of the approach is to show that some combinatorial object exists and prove that a certain random construction works with positive probability. The course focuses on methodology as well as combinatorial applications.
https://ocw.mit.edu/courses/mathematics/18-218-probabilistic-method-in-combinatorics-spring-2019
Spring2019Zhao, Yufei2019-12-17T15:42:51+05:0018.218en-USprobabilistic methodRamsey numbersLovász Local Lemmahypergraph coloringsbalancing vectorssum-free setssecond Moment MethodChernoff boundMoser-Tardos algorithmJanson’s inequalitiesHarris-FKG inequalityMartingale convergenceAzuma’s inequalityentropy methodsoccupancy methodMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.783 Elliptic Curves (MIT)This graduate-level course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography.
https://ocw.mit.edu/courses/mathematics/18-783-elliptic-curves-spring-2019
Spring2019Sutherland, Andrew2019-11-05T18:49:52+05:0018.783en-USelliptic curvesnumber theorycryptographypoint-countingisogeniespairingstheory of complex multiplicationinteger factorizationprimality provingelliptic curve cryptographymodular curvesFermat's Last TheoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.04 Complex Variables with Applications (MIT)Complex analysis is a basic tool with a great many practical applications to the solution of physical problems. It revolves around complex analytic functions—functions that have a complex derivative. Unlike calculus using real variables, the mere existence of a complex derivative has strong implications for the properties of the function. Applications reviewed in this class include harmonic functions, two dimensional fluid flow, easy methods for computing (seemingly) hard integrals, Laplace transforms, and Fourier transforms with applications to engineering and physics.
https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018
Spring2018Orloff, Jeremy2019-10-30T18:49:34+05:0018.04en-USComplex algebra and functionsanalyticitycontour integrationCauchy's theoremsingularitiesTaylor and Laurent seriesresiduesevaluation of integralsmultivalued functionspotential theory in 2DFourier analysis and Laplace transformsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.335J Introduction to Numerical Methods (MIT)This course offers an advanced introduction to numerical analysis, with a focus on accuracy and efficiency of numerical algorithms. Topics include sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems), floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) are also surveyed.
https://ocw.mit.edu/courses/mathematics/18-335j-introduction-to-numerical-methods-spring-2019
Spring2019Johnson, Steven G.2019-07-10T12:59:56+05:0018.335J6.337Jen-USnumerical linear algebralinear systemseigenvalue decompositionQR/SVD factorizationnumerical algorithmsIEEE floating point standardsparse matricesstructured matricespreconditioninglinear algebra softwareMatlabMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (MIT)Linear algebra concepts are key for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization–and above all a full explanation of deep learning.
https://ocw.mit.edu/courses/mathematics/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018
Spring2018Strang, Gilbert2019-05-16T15:05:00+05:0018.06518.0651en-USdata analysissignal processingimage processingmachine learninglinear algebracomputationsingular value decompositionleast squaresweighted least squarescovariance matricescorrelation matricesdirected graphsundirected graphsmatrix factorizationsneural netsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.A34 Mathematical Problem Solving (Putnam Seminar) (MIT)This course is a seminar intended for undergraduate students who enjoy solving challenging mathematical problems, and to prepare them for the Putnam Competition. All students officially registered in the class are required to participate in the William Lowell Putnam Mathematical Competition.
https://ocw.mit.edu/courses/mathematics/18-a34-mathematical-problem-solving-putnam-seminar-fall-2018
Fall2018Zhao, Yufei2019-03-27T18:53:09+05:0018.A34en-UShidden independenceprobabilitycongruences and divisibilityrecurrenceslimitsgreatest integer functioninequalitiesroots of polynomialsPigeonhole PrincipleMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S097 Applied Category Theory (MIT)Category theory is a relatively new branch of mathematics that has transformed much of pure math research. The technical advance is that category theory provides a framework in which to organize formal systems and by which to translate between them, allowing one to transfer knowledge from one field to another. But this same organizational framework also has many compelling examples outside of pure math. In this course, we will give seven sketches on real-world applications of category theory.
https://ocw.mit.edu/courses/mathematics/18-s097-applied-category-theory-january-iap-2019
January IAP2019Spivak, David I.Fong, Brendan2019-03-25T19:18:33+05:0018.S097en-USorderadjunctionsetGalois connectionmonoidal preorderwiring diagramV-categoriesBool-categoriescategoriesfunctorslimitscolimitsmonoidal categorieshypergraph categoriessheavestoposesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.905 Algebraic Topology I (MIT)This is a course on the singular homology of topological spaces. Topics include: Singular homology, CW complexes, Homological algebra, Cohomology, and Poincare duality.
https://ocw.mit.edu/courses/mathematics/18-905-algebraic-topology-i-fall-2016
Fall2016Miller, Haynes2018-04-20T16:23:06+05:0018.905en-USAlgebraic TopologyhomologyCW complexesHomological algebraCohomologyPoincare dualityHomotopy InvarianceEilenberg-Steenrod AxiomsTopological GenealogyKünneth theoremTor functorstensor productČech" CohomologyMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.325 Topics in Applied Mathematics: Waves and Imaging (MIT)This class covers the mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography.
https://ocw.mit.edu/courses/mathematics/18-325-topics-in-applied-mathematics-waves-and-imaging-fall-2015
Fall2015Demanet, Laurent2018-03-06T16:46:19+05:0018.325en-USwavesimagingradar imagingseismic imagingRadon transformbackprojectionreflection seismologycomputerized tomographysynthetic aperture radarMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.786 Number Theory II: Class Field Theory (MIT)This course is the continuation of 18.785 Number Theory I. It begins with an analysis of the quadratic case of Class Field Theory via Hilbert symbols, in order to give a more hands-on introduction to the ideas of Class Field Theory. More advanced topics in number theory are discussed in this course, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, and quadratic forms.
https://ocw.mit.edu/courses/mathematics/18-786-number-theory-ii-class-field-theory-spring-2016
Spring2016Raskin, Sam2017-10-12T20:01:26+05:0018.786en-USClass Field Theory (CFT)Hilbert SymbolsHilbert's Theorynorm grouptame ramificationtame cohomologyHerbrand quotientsHomotopyVanishing TheoryKummer TheoryBrauer groupArtin and Brauer ReciprocityMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.650 Statistics for Applications (MIT)This course offers an in-depth the theoretical foundations for statistical methods that are useful in many applications. The goal is to understand the role of mathematics in the research and development of efficient statistical methods.
https://ocw.mit.edu/courses/mathematics/18-650-statistics-for-applications-fall-2016
Fall2016Rigollet, Philippe2017-07-31T19:59:37+05:0018.65018.6501en-USstatisticsregressionparametric inferenceparametric hypothesisBayesian statisticsprincipal component analysisMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.405J Advanced Complexity Theory (MIT)This graduate-level course focuses on current research topics in computational complexity theory. Topics include: Nondeterministic, alternating, probabilistic, and parallel computation models; Boolean circuits; Complexity classes and complete sets; The polynomial-time hierarchy; Interactive proof systems; Relativization; Definitions of randomness; Pseudo-randomness and derandomizations;Interactive proof systems and probabilistically checkable proofs.
https://ocw.mit.edu/courses/mathematics/18-405j-advanced-complexity-theory-spring-2016
Spring2016Moshkovitz, DanaBavarian, Mohammad2017-01-17T20:02:58+05:0018.405J6.841Jen-US18.405J18.4056.841J6.841Polynomial hierarchytime-space lower boundsapproximate countingToda’s TheoremRelativizationBaker-Gill-Solovayswitching lemmaRazborov-SmolenskyNEXP vs. ACC0Communication complexityPCP theoremPCP theoremHadamard codeGap amplificationNatural proofsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.156 Differential Analysis II: Partial Differential Equations and Fourier Analysis (MIT)In this course, we study elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation. Then we study Fourier and harmonic analysis, emphasizing applications of Fourier analysis. We will see some applications in combinatorics / number theory, like the Gauss circle problem, but mostly focus on applications in PDE, like the Calderon-Zygmund inequality for the Laplacian, and the Strichartz inequality for the Schrodinger equation. In the last part of the course, we study solutions to the linear and the non-linear Schrodinger equation. All through the course, we work on the craft of proving estimates.
https://ocw.mit.edu/courses/mathematics/18-156-differential-analysis-ii-partial-differential-equations-and-fourier-analysis-spring-2016
Spring2016Guth, Lawrence2016-08-15T16:18:27+05:0018.156en-USelliptic PDEdispersive PDEFourier analysisFourier transformFourier inversion theoremPlancherel theoremSchauder inequalityStrichartz inequalitySobolev spacesGauss circle problemMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.725 Algebraic Geometry (MIT)This is the first semester of a two-semester sequence on Algebraic Geometry. The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. This course is an introduction to the language of schemes and properties of morphisms.
https://ocw.mit.edu/courses/mathematics/18-725-algebraic-geometry-fall-2015
Fall2015Bezrukavnikov, Roman2016-06-16T18:37:19+05:0018.725en-USalgebraic geometryZariski topologyProduct TopologyAffine VarietiesProjective VarietiesNoether NormalizationAffine MorphismsFinite MorphismsSheavesBezout’s TheoremKahler DifferentialsCanonical BundlesRiemann-Hurwitz FormulaChevalley’s TheoremBertini’s TheoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.655 Mathematical Statistics (MIT)This course provides students with decision theory, estimation, confidence intervals, and hypothesis testing. It introduces large sample theory, asymptotic efficiency of estimates, exponential families, and sequential analysis.
https://ocw.mit.edu/courses/mathematics/18-655-mathematical-statistics-spring-2016
Spring2016Kempthorne, Peter2016-05-27T20:01:38+05:0018.655en-USLeast SquaresRoot-findingCoordinate AscentNewton-RaphsonBayes ProceduresRobustness CriteriaNeyman-Pearson LemmaConfidence BoundsConfidence IntervalsAsymptotic NormalityGaussian Linear ModelsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.177 Universal Random Structures in 2D (MIT)This graduate-level course introduces students to some fundamental 2D random objects, explains how they are related to each other, and explores some open problems in the field.
https://ocw.mit.edu/courses/mathematics/18-177-universal-random-structures-in-2d-fall-2015
Fall2015Sheffield, Scott2016-04-26T16:51:16+05:0018.177en-UScontinuum random treestable Levy treestable looptreeGaussian free fieldSchramm-Loewner evolutionpercolationuniform spanning treeloop-erased random walkIsing modelFK cluster modelconformal loop ensembleBrownian loop souprandom planar mapLiouvillequantum gravityBrownian mapBrownian snakediffusion limited aggregationfirst passage percolationand dielectric breakdown modelimaginary geometryquantum zipperpeanospherequantum Loewner evolutionMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S096 Topics in Mathematics of Data Science (MIT)This is a mostly self-contained research-oriented course designed for undergraduate students (but also extremely welcoming to graduate students) with an interest in doing research in theoretical aspects of algorithms that aim to extract information from data. These often lie in overlaps of two or more of the following: Mathematics, Applied Mathematics, Computer Science, Electrical Engineering, Statistics, and / or Operations Research.
https://ocw.mit.edu/courses/mathematics/18-s096-topics-in-mathematics-of-data-science-fall-2015
Fall2015Bandeira, Afonso2016-03-08T21:40:36+05:0018.S096en-USPrincipal Component Analysis (PCA)random matrix theoryspike modelmanifold learningDiffusion MapsSobolev Embedding TheoremSpectral ClusteringCheeger’s inequalityMesh TheoremNumber TheoryApproximation algorithmsMax-Cut problemStochastic Block ModelSynchronizationinverse problemsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.103 Fourier Analysis (MIT)This course continues the content covered in 18.100 Analysis I. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals.
https://ocw.mit.edu/courses/mathematics/18-103-fourier-analysis-fall-2013
Fall2013Jerison, David2016-03-01T22:46:38+05:0018.103en-USFourier seriesFourier analysispartial sumswavesBoolean ringsHilbert SpaceOrthonormal basesLp theoryFourier integralsmeasurecentral limit theorembrownian motionLebesgue integralperiodic functionsFourier coefficientsParseval's formulaBernoulli sequencerandom walksprobability theoryLebesgue measureMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.657 Mathematics of Machine Learning (MIT)Broadly speaking, Machine Learning refers to the automated identification of patterns in data. As such it has been a fertile ground for new statistical and algorithmic developments. The purpose of this course is to provide a mathematically rigorous introduction to these developments with emphasis on methods and their analysis.You can read more about Prof. Rigollet's work and courses on his website.
https://ocw.mit.edu/courses/mathematics/18-657-mathematics-of-machine-learning-fall-2015
Fall2015Rigollet, Philippe2016-02-08T18:50:16+05:0018.657en-USBinary ClassificationConcentration inequalitiesVC theoryVC inequalityGradient DescentMirror DescentStochastic BanditsAdversarial banditsLinear banditsBlackwell's approachabilityMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S997 The Polynomial Method (MIT)This course offers an introduction to the polynomial method as applied to solving problems in combinatorics in the last decade. The course also explores the connections between the polynomial method as used in these problems to the polynomial method in other fields, including computer science, number theory, and analysis.
https://ocw.mit.edu/courses/mathematics/18-s997-the-polynomial-method-fall-2012
Fall2012Guth, Lawrence2015-12-23T02:16:21+05:0018.S997en-USpolynomial methodcombinatoricsKakeya ProblemIncidence GeometryAlgebraic StructureCell DecompositionsRuled SurfacesProjection TheoryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.304 Undergraduate Seminar in Discrete Mathematics (MIT)This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.
https://ocw.mit.edu/courses/mathematics/18-304-undergraduate-seminar-in-discrete-mathematics-spring-2015
Spring2015Tamuz, Omer2015-12-17T21:51:46+05:0018.304en-USdiscrete mathdiscrete mathematicspresentationsstudent presentationsoral communicationcombinatoricsgraph theorydiscrete mathematicsProofs from the Bookmathematics communicationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.443 Statistics for Applications (MIT)This course is a broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics include: hypothesis testing and estimation, confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics.
https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-spring-2015
Spring2015Kempthorne, Peter2015-12-14T23:32:13+05:0018.44318.650en-UShypothesis testinghypothesis estimationconfidence intervalschi-square testsnonparametric statisticsanalysis of varianceregressioncorrelationdecision theoryBayesian statisticsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.385J Nonlinear Dynamics and Chaos (MIT)This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.
https://ocw.mit.edu/courses/mathematics/18-385j-nonlinear-dynamics-and-chaos-fall-2014
Fall2014Rosales, Rodolfo2015-12-03T23:34:47+05:0018.385J2.036Jen-USchaosFloquet theoryPoincare-Bendixson theoryphase planelimit cyclestime-dependent systemsPoincare mapsstability of equilibrianear-equilibrium dynamicscenter manifoldselementary bifurcationsnormal formsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.409 Algorithmic Aspects of Machine Learning (MIT)This course is organized around algorithmic issues that arise in machine learning. Modern machine learning systems are often built on top of algorithms that do not have provable guarantees, and it is the subject of debate when and why they work. In this class, we focus on designing algorithms whose performance we can rigorously analyze for fundamental machine learning problems.
https://ocw.mit.edu/courses/mathematics/18-409-algorithmic-aspects-of-machine-learning-spring-2015
Spring2015Moitra, Ankur2015-12-03T20:15:21+05:0018.409en-USMachine learningnonnegative matrix factorizationtensor decompositiontensor rankborder ranksparse codingsparse recoverylearning mixture modelmatrix completionMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S997 High-Dimensional Statistics (MIT)This course offers an introduction to the finite sample analysis of high- dimensional statistical methods. The goal is to present various proof techniques for state-of-the-art methods in regression, matrix estimation and principal component analysis (PCA) as well as optimality guarantees. The course ends with research questions that are currently open. You can read more about Prof. Rigollet's work and courses on his website
https://ocw.mit.edu/courses/mathematics/18-s997-high-dimensional-statistics-spring-2015
Spring2015Rigollet, Philippe2015-11-10T22:50:26+05:0018.S997en-USHigh Dimensional StatisticsRandom VariablesLinear RegressionMisspecified Linear ModelsMatrix EstimationMinmax Lower BoundsSub-GaussianMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.354J Nonlinear Dynamics II: Continuum Systems (MIT)This course introduces the basic ideas for understanding the dynamics of continuum systems, by studying specific examples from a range of different fields. Our goal will be to explain the general principles, and also to illustrate them via important physical effects. A parallel goal of this course is to give you an introduction to mathematical modeling.
https://ocw.mit.edu/courses/mathematics/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015
Spring2015Dunkel, Jörn2015-10-30T21:15:40+05:0018.354J1.062J12.207Jen-UScontinuum systemsmathematical modelingdiffusion equationequations of motionnonlinear partial differential equationscalculus of variationsBrachistochrone curvesoap filmshydrodynamicsNavier-Stokessolitonssurface tensionwavesconformal mapsairfoilsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.445 Introduction to Stochastic Processes (MIT)This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix.
https://ocw.mit.edu/courses/mathematics/18-445-introduction-to-stochastic-processes-spring-2015
Spring2015Wu, Hao2015-08-20T20:53:50+05:0018.445en-USprobabilityStochastic ProcessesMarkov chainsrandom walksmartingalesGalton-Watsom treeprobabilitylinear algebraMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.310 Principles of Discrete Applied Mathematics (MIT)This course is an introduction to discrete applied mathematics. Topics include probability, counting, linear programming, number-theoretic algorithms, sorting, data compression, and error-correcting codes. This is a Communication Intensive in the Major (CI-M) course, and thus includes a writing component.
https://ocw.mit.edu/courses/mathematics/18-310-principles-of-discrete-applied-mathematics-fall-2013
Fall2013Goemans, MichelRuff, SusanOrecchia, LorenzoPeng, Richard2015-07-27T20:48:49+05:0018.310en-USprobabilityprobability theory countingpigeonhole principleVan der Waerden's theoremChernoff boundscountingcodingsamplingrandom samplingCatalan familiesgenerating functionschord diagramslinear programmingsimplex methodZero-Sum matrixnetwork flowsmaximum flow problemsorting algorithmsQUICKSORTmedian findingsorting networksBatcher's algorithmEuclid's algorithmChinese Remainder TheoremcryptographyRSA codeprimaility testingFFTFast Fourier TransformShannon's coding theoremsLempel-Ziv codeslinear codeshamming codeMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.311 Principles of Applied Mathematics (MIT)18.311 Principles of Continuum Applied Mathematics covers fundamental concepts in continuous applied mathematics, including applications from traffic flow, fluids, elasticity, granular flows, etc. The class also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion, and group velocity.
https://ocw.mit.edu/courses/mathematics/18-311-principles-of-applied-mathematics-spring-2014
Spring2014Rosales, Rodolfo2015-07-08T20:40:32+05:0018.311en-USpartial differential equationhyperbolic equationsdimensional analysisperturbation methodshyperbolic systemsdiffusion and reaction processescontinuum modelsequilibrium modelscontinuous applied mathematicstraffic flowfluidselasticitygranular flowscontinuum limitconservation lawsquasi-equilibriumkinematic wavescharacteristicssimple wavesshocksdiffusion (linear and nonlinear)numerical solution of wave equationsfinite differencesconsistencystabilitydiscrete and fast Fourier transformsspectral methodstransforms and series (Fourier, Laplace)sonic boomsMach conecausticslatticesdispersiongroup velocityMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.303 Linear Partial Differential Equations: Analysis and Numerics (MIT)This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples.
https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-analysis-and-numerics-fall-2014
Fall2014Johnson, Steven G.2015-06-25T17:46:19+05:0018.303en-USdiffusionLaplace equationsPoissonwave equationsseparation of variablesFourier seriesFourier transformseigenvalue problemsGreen's functionHeat EquationSturm-Liouville Eigenvalue problemsquasilinear PDEsBessel functionsORDSMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.440 Probability and Random Variables (MIT)This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.
https://ocw.mit.edu/courses/mathematics/18-440-probability-and-random-variables-spring-2014
Spring2014Sheffield, Scott2015-05-14T17:12:51+05:0018.440en-USProbability spacesrandom variablesdistribution functionsBinomialgeometrichypergeometricPoisson distributionsUniformexponentialnormalgamma and beta distributionsConditional probabilityBayes theoremjoint distributionsChebyshev inequalitylaw of large numbersAnd central limit theoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.915 Graduate Topology Seminar: Kan Seminar (MIT)This is a literature seminar with a focus on classic papers in Algebraic Topology. It is named after the late MIT professor Daniel Kan. Each student gives one or two talks on each of three papers, chosen in consultation with the instructor, reads all the papers presented by other students, and writes reactions to the papers. This course is useful not only to students pursuing algebraic topology as a field of study, but also to those interested in symplectic geometry, representation theory, and combinatorics.
https://ocw.mit.edu/courses/mathematics/18-915-graduate-topology-seminar-kan-seminar-fall-2014
Fall2014Miller, Haynes2015-04-15T23:38:37+05:0018.915en-USmathematicstopologyKan SeminarDan KancommunicationpresentingMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.314 Combinatorial Analysis (MIT)This course analyzes combinatorial problems and methods for their solution. Topics include: enumeration, generating functions, recurrence relations, construction of bijections, introduction to graph theory, network algorithms, and extremal combinatorics.
https://ocw.mit.edu/courses/mathematics/18-314-combinatorial-analysis-fall-2014
Fall2014Stanley, Richard2015-03-17T17:30:01+05:0018.314en-USEnumerationGenerating functionsRecurrence relationsConstruction of BijectionsGraph TheoryNetwork AlgorithmsExtremal CombinatoricsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S096 Topics in Mathematics with Applications in Finance (MIT)The purpose of the class is to expose undergraduate and graduate students to the mathematical concepts and techniques used in the financial industry. Mathematics lectures are mixed with lectures illustrating the corresponding application in the financial industry. MIT mathematicians teach the mathematics part while industry professionals give the lectures on applications in finance.
https://ocw.mit.edu/courses/mathematics/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013
Fall2013Kempthorne, PeterLee, ChoongbumStrela, VasilyXia, Jake2015-01-05T21:47:35+05:0018.S096en-USFinancial termsValue at Risk ModelsVolatility ModelingRegularized pricingRisk ModelsRisk analysiscommodity modelsportfolio theoryIto calculusBlack-Scholes formularisk neutral valuationoption pricingQuanto credit hedgingRoss recovery theoremcounterparty credit riskMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.05 Introduction to Probability and Statistics (MIT)This course provides an elementary introduction to probability and statistics with applications. Topics include: basic combinatorics, random variables, probability distributions, Bayesian inference, hypothesis testing, confidence intervals, and linear regression. The Spring 2014 version of this subject employed the residential MITx system, which enables on-campus subjects to provide MIT students with learning and assessment tools such as online problem sets, lecture videos, reading questions, pre-lecture questions, problem set assistance, tutorial videos, exam review content, and even online exams.
https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014
Spring2014Orloff, JeremyBloom, Jonathan2014-12-19T18:39:32+05:0018.05en-USprobabilitystatisticsmodelscombinatoricsexpectationvariancerandom variablediscrete probability distributioncontinuous probability distributionBayesdistributionstatistical estimationstatistical testingconfidence intervallinear regressionnormalsignificance testingbootstrappingMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.175 Theory of Probability (MIT)This course covers topics such as sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales.
https://ocw.mit.edu/courses/mathematics/18-175-theory-of-probability-spring-2014
Spring2014Sheffield, Scott2014-12-12T22:28:19+05:0018.175en-USLaws of large numberscentral limit theoremsindependent random variablesconditioningmartingalesBrownian motionelements of diffusion theoryfunctional limit theoremsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.700 Linear Algebra (MIT)This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with 18.06 Linear Algebra, more emphasis is placed on theory and proofs.
https://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-fall-2013
Fall2013Vogan, David2014-10-22T01:03:34+05:0018.700en-USlinear algebravector spacesystem of linear equationsbaseslinear independencematricesdeterminanteigenvalueinner productquadratic formSpectral TheoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.782 Introduction to Arithmetic Geometry (MIT)This course is an introduction to arithmetic geometry, a subject that lies at the intersection of algebraic geometry and number theory. Its primary motivation is the study of classical Diophantine problems from the modern perspective of algebraic geometry.
https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013
Fall2013Sutherland, Andrew2014-05-23T17:35:46+05:0018.782en-USalgebranumber theorydiophantine equationsalgebraic geometryplane conicselliptic curveshyperelliptic curvesabelian varietiesTate-Shafarevich groupMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.821 Project Laboratory in Mathematics (MIT)Project Laboratory in Mathematics is a course designed to give students a sense of what it's like to do mathematical research. In teams, students explore puzzling and complex mathematical situations, search for regularities, and attempt to explain them mathematically. Students share their results through professional-style papers and presentations.
This course site was created specifically for educators interested in offering students a taste of mathematical research. This site features extensive description and commentary from the instructors about why the course was created and how it operates.
https://ocw.mit.edu/courses/mathematics/18-821-project-laboratory-in-mathematics-spring-2013
Spring2013Miller, HaynesStapleton, NatGlasman, SaulRuff, Susan2014-01-30T19:42:35+05:0018.821en-USmathematicsresearchcommunicationwritingpresentingLaTeXteamworkMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.353J Nonlinear Dynamics I: Chaos (MIT)This course provides an introduction to nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in engineering and science.
https://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012
Fall2012Chumakova, Lyubov2014-01-08T21:40:41+05:0018.353J2.050J12.006Jen-USnonlinear dynamicschaosdissipative systemsfree oscillatorsforced oscillatorsnonlinear phenomenabifurcation theoryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.357 Interfacial Phenomena (MIT)This graduate-level course covers fluid systems dominated by the influence of interfacial tension. The roles of curvature pressure and Marangoni stress are elucidated in a variety of fluid systems. Particular attention is given to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics.
https://ocw.mit.edu/courses/mathematics/18-357-interfacial-phenomena-fall-2010
Fall2010Bush, John W. M.2013-08-06T19:47:35+05:0018.357en-USfluid dynamicsfluid mechanicsinterfacial phenomenawater-repellencysurfactantsMarangoni flowscapillary origamicontact line dynamicstears of wineMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S996 Category Theory for Scientists (MIT)The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common scientific models. The power of the language will be tested by its ability to penetrate into taken-for-granted ideas, either by exposing existing weaknesses or flaws in our understanding, or by highlighting hidden commonalities across scientific fields.
https://ocw.mit.edu/courses/mathematics/18-s996-category-theory-for-scientists-spring-2013
Spring2013Spivak, David I.2013-07-02T15:29:17+05:0018.S996en-USSetsfunctionscommutative diagramsproductscoproductsfinite limitsmonoidsgroupsgraphsordersschemasinstancesdatabasescategoriesfunctorsmathematicsnatural transformationslimitscolimitsadjoint functorsmonadsoperadsisomorphismmolecular dynamicsologMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.703 Modern Algebra (MIT)This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics.
https://ocw.mit.edu/courses/mathematics/18-703-modern-algebra-spring-2013
Spring2013McKernan, James2013-06-26T05:17:39+05:0018.703en-USalgebragroup theoryfinite groupsring theoryunique factorizationEuclidean ringsfield theoryfinite fieldsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.100C Real Analysis (MIT)This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. MIT students may choose to take one of three versions of Real Analysis; this version offers three additional units of credit for instruction and practice in written and oral presentation.
The three options for 18.100:
Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible.
Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology.
Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication. This fulfills the MIT CI requirement.
https://ocw.mit.edu/courses/mathematics/18-100c-real-analysis-fall-2012
Fall2012Seidel, Paul2013-04-11T17:35:39+05:0018.100Cen-USmathematical analysisArchimedean principledecimal expansionCauchy-Schwarzmetric spacesopen subsetsEuclidean spaceconvergent sequencessubsequential limitsinverse functionsStone-Weierstrass theoremtheory of integrationRiemann-Stjeltjes integralFourier seriesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S997 Introduction To MATLAB Programming (MIT)This course is intended to assist undergraduates with learning the basics of programming in general and programming MATLAB® in particular.
https://ocw.mit.edu/courses/mathematics/18-s997-introduction-to-matlab-programming-fall-2011
Fall2011Farjoun, Yossi2013-01-28T21:12:18+05:0018.S997en-USMATLAB,programmingvariablesplottingscriptsfunctionsflow controlstatisticsdata structuresimagesvectorsmatricesroot-findingNewton's MethodSecant MethodBasins of AttractionConway Game of LifeGame of Lifevectorizationdebuggingscopefunction blockMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.781 Theory of Numbers (MIT)This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions.
https://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-spring-2012
Spring2012Kumar, Abhinav2013-01-22T21:45:18+05:0018.781en-USprimesdivisibilityfundamental theorem of arithmeticgcdEuclidean algorithmcongruencesChinese remainder theoremHensel's lemmaprimitive rootsquadratic residuesreciprocityarithmetic functionsDiophantine equationscontinued fractionsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.100A Introduction to Analysis (MIT)Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology. Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication.
https://ocw.mit.edu/courses/mathematics/18-100a-introduction-to-analysis-fall-2012
Fall2012Mattuck, Arthur2013-01-16T15:12:26+05:0018.100Aen-USmathematical analysisestimationslimit of a sequencelimit theoremssubsequencescluster pointsinfinite seriespower serieslocal and global propertiescontinuityintermediate-value theoremconvexityintegrabilityRiemann integralcalculusconvergenceGamma functionStirlingquantifiers and negationLeibnizFubiniimproper integralsLebesgue integralmathematical proofsdifferentiationintegrationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.337J Parallel Computing (MIT)This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language, a free, open-source, high-performance dynamic programming language for technical computing.
https://ocw.mit.edu/courses/mathematics/18-337j-parallel-computing-fall-2011
Fall2011Edelman, Alan2012-12-21T19:42:47+05:0018.337J6.338Jen-UScloud computingdense linear algebrasparse linear algebraN-body problemsmultigridfast-multipolewaveletsFourier transformspartitioningmesh generationapplications oriented architectureparallel programming paradigmsMPIdata parallel systemsStar-Pparallel Pythonparallel Matlabgraphics processorsvirtualizationcachesvector processorsVHLLsVery High Level LanguagesJulia programming languagedistributed parallel executionMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.330 Introduction to Numerical Analysis (MIT)This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra.
https://ocw.mit.edu/courses/mathematics/18-330-introduction-to-numerical-analysis-spring-2012
Spring2012Demanet, Laurent2012-12-14T21:24:02+05:0018.330en-USseries expansionsroot findinginterpolationFourier transformapproximation functionsleast-squares approximationprincipal component analysisMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.904 Seminar in Topology (MIT)This course is a seminar in topology. The main mathematical goal is to learn about the fundamental group, homology and cohomology. The main non-mathematical goal is to obtain experience giving math talks.
https://ocw.mit.edu/courses/mathematics/18-904-seminar-in-topology-spring-2011
Spring2011Snowden, Andrew2012-12-13T14:13:53+05:0018.904en-USstudent lecturesmath writingtopologyfundamental groupcovering spacescommunicationoral communicationmathematical writingMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.152 Introduction to Partial Differential Equations (MIT)This course introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. It includes mathematical tools, real-world examples and applications.
https://ocw.mit.edu/courses/mathematics/18-152-introduction-to-partial-differential-equations-fall-2011
Fall2011Speck, Jared 2012-06-28T12:14:02+05:0018.152en-USdiffusionelliptichyperbolicpartial differential equationInitial and boundary value problems for ordinary differential equationsSturm-Liouville theory and eigenfunction expansionsinitial value problemswave equation;heat equationDirichlet problemLaplace operator and potential theoryBlack-Scholes equationwater wavesscalar conservation lawsfirst order equationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.03SC Differential Equations (MIT)The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.
https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011
Fall2011Mattuck, ArthurMiller, HaynesOrloff, JeremyLewis, John2012-02-08T19:08:08+05:0018.03SCen-USOrdinary Differential EquationsODEmodeling physical systemsfirst-order ODE'sLinear ODE'ssecond order ODE'ssecond order ODE's with constant coefficientsUndetermined coefficientsvariation of parametersSinusoidal signalsexponential signalsoscillationsdampingresonanceComplex numbers and exponentialsFourier seriesperiodic solutionsDelta functionsconvolutionLaplace transform methodsMatrix systemsfirst order linear systemseigenvalues and eigenvectorsNon-linear autonomous systemscritical point analysisphase plane diagramsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.06SC Linear Algebra (MIT)This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines such as physics, economics and social sciences, natural sciences, and engineering. It parallels the combination of theory and applications in Professor Strang’s textbook Introduction to Linear Algebra.
https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011
Fall2011Strang, Gilbert2012-01-24T21:18:50+05:0018.06SCen-USmatrix theorylinear algebrasystems of equationsvector spacesdeterminantseigenvaluessimilaritypositive definite matricesleast-squares approximationsstability of differential equationsnetworksFourier transformsMarkov processesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.024 Multivariable Calculus with Theory (MIT)This course is a continuation of 18.014. It covers the same material as 18.02 (Multivariable Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.
https://ocw.mit.edu/courses/mathematics/18-024-multivariable-calculus-with-theory-spring-2011
Spring2011Breiner, Christine2011-12-08T15:51:15+05:0018.024en-USlinear algebravector integral calculusCalculus of several variablesVector algebra in 3-spacedeterminantsmatricesVector-valued functions of one variablespace motionScalar functions of several variablespartial differentiationgradientoptimization techniquesDouble integrals and line integrals in the planeexact differentials and conservative fieldsGreen's theorem and applicationstriple integralsline and surface integrals in spaceDivergence theoremStokes' theoremapplicationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.702 Algebra II (MIT)This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.
https://ocw.mit.edu/courses/mathematics/18-702-algebra-ii-spring-2011
Spring2011Artin, Michael2011-10-28T16:57:09+05:0018.702en-USSylow theoremsGroup Representationsdefinitionsunitary representationscharactersSchur's LemmaRings: Basic DefinitionshomomorphismsfractionsFactorizationunique factorizationGauss' Lemmaexplicit factorizationmaximal idealsQuadratic Imaginary IntegersGauss Primesquadratic integersideal factorizationideal classesLinear Algebra over a Ringfree modulesinteger matricesgenerators and relationsstructure of abelian groupsRings: Abstract Constructionsrelations in a ringadjoining elementsFields: Field Extensionsalgebraic elementsdegree of field extensionruler and compasssymbolic adjunctionfinite fieldsFields: Galois Theorythe main theoremcubic equationssymmetric functionsprimitive elementsquartic equationsquintic equationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.701 Algebra I (MIT)This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.
https://ocw.mit.edu/courses/mathematics/18-701-algebra-i-fall-2010
Fall2010Artin, Michael2011-10-28T16:56:15+05:0018.701en-USGroup TheoryLinear AlgebraGeometrygroupsvector spaceslinear transformationssymmetry groupsbilinear formslinear groupsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.022 Calculus of Several Variables (MIT)This is a variation on 18.02 Multivariable Calculus. It covers the same topics as in 18.02, but with more focus on mathematical concepts.
https://ocw.mit.edu/courses/mathematics/18-022-calculus-of-several-variables-fall-2010
Fall2010McKernan, James2011-06-08T17:16:40+05:0018.022en-USvector algebradeterminantmatrixmatricesvector-valued functionsspace motionscalar functionspartial differentiationgradientoptimization techniquesdouble integralsline integralsexact differentialsconservative fieldsGreen's theoremtriple integralssurface integralsdivergence theoremStokes' theoremgeometryvector fieldslinear algebraMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.100B Analysis I (MIT)Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.
https://ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010
Fall2010Wehrheim, Katrin2011-06-02T16:47:08+05:0018.100Ben-USmathematical analysisconvergence of sequencesconvergence of seriescontinuitydifferentiabilityRiemann integralsequences and series of functionsuniformityinterchange of limit operationsutility of abstract conceptsconstruction of proofspoint-set topologyn-spaceMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.014 Calculus with Theory (MIT)18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.
https://ocw.mit.edu/courses/mathematics/18-014-calculus-with-theory-fall-2010
Fall2010Breiner, Christine2011-05-18T17:00:27+05:0018.014en-USaxioms for the real numbersthe Riemann integrallimitstheorems on continuous functionsderivatives of functions of one variablethe fundamental theorems of calculusTaylor's theoreminfinite seriespower seriesrigorous treatment of the elementary functionsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.03 Differential Equations (MIT)Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.
https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010
Spring2010Miller, HaynesMattuck, Arthur2011-03-16T18:26:50+05:0018.03en-USOrdinary Differential EquationsODEmodeling physical systemsfirst-order ODE'sLinear ODE'ssecond order ODE'ssecond order ODE's with constant coefficientsUndetermined coefficientsvariation of parametersSinusoidal signalsexponential signalsoscillationsdampingresonanceComplex numbers and exponentialsFourier seriesperiodic solutionsDelta functionsconvolutionLaplace transform methodsMatrix systemsfirst order linear systemseigenvalues and eigenvectorsNon-linear autonomous systemscritical point analysisphase plane diagramsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.01SC Single Variable Calculus (MIT)This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics.
https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010
Fall2010Jerison, David2011-01-12T17:16:43+05:0018.01SCen-USdifferentiation of functionsintegration of functionslimitscontinuitydifferentiation rulesextremum problemsdefinite integrationindefinite integrationfundamental theorem of calculustechniques of integrationapproximation of definite integralsimproper integralsl'Hôpital's ruleMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.02SC Multivariable Calculus (MIT)This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics.
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010
Fall2010Auroux, Denis2010-12-20T16:04:13+05:0018.02SCen-UScalculuscalculus of several variablesvector algebradeterminantsmatrixmatricesvector-valued functionspace motionscalar functionpartial differentiationgradientoptimization techniquesdouble integralsline integralsexact differentialconservative fieldsGreen's theoremtriple integralssurface integralsdivergence theorem Stokes' theoremapplicationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.786 Topics in Algebraic Number Theory (MIT)This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants.
https://ocw.mit.edu/courses/mathematics/18-786-topics-in-algebraic-number-theory-spring-2010
Spring2010Kumar, Abhinav2010-12-16T12:58:58+05:0018.786en-USnumber fieldsdedekind domainprime idealclass grouplattice methodMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.06 Linear Algebra (MIT)This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.
https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010
Spring2010Strang, Gilbert2010-09-10T14:23:13+05:0018.06en-USmatrix theorylinear algebrasystems of equationsvector spacesdeterminantseigenvaluessimilaritypositive definite matricesleast-squares approximationsstability of differential equationsnetworksFourier transformsMarkov processesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.712 Introduction to Representation Theory (MIT)The goal of this course is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces.
https://ocw.mit.edu/courses/mathematics/18-712-introduction-to-representation-theory-fall-2010
Fall2010Etingof, Pavel2010-09-07T14:34:12+05:0018.712en-USfinite dimensional algebrasQuiver Representationsseries Representationsfinite groupsrepresentation theoryLie algebrasTensor productsdensity theoremJordan-H?older theoremKrull-Schmidt theoremMaschke?s TheoremFrobenius-Schur indicatorFrobenius divisibilityBurnside?s TheoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.409 Topics in Theoretical Computer Science: An Algorithmist's Toolkit (MIT)
This course covers a collection of geometric techniques that apply broadly in modern algorithm design.
https://ocw.mit.edu/courses/mathematics/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009
Fall2009Kelner, Jonathan2010-05-10T19:48:35+05:0018.409en-USSpectral graph theoryIterative methods for linear algebraConvex geometryLattices and basis reductionLPs and SDPs for approximating NP-hard problemsGraph LaplaciansCheeger inequalitiesFritz John?s theoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.735 Double Affine Hecke Algebras in Representation Theory, Combinatorics, Geometry, and Mathematical Physics (MIT)
Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems.
https://ocw.mit.edu/courses/mathematics/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009
Fall2009Etingof, Pavel2010-04-30T19:38:50+05:0018.735en-USdunkl operatorscherednikaffine algebrarepresentation theoryheckeknizknik-zamoldchikovorbifoldscalogero-moser spacehilbert schemealgebramacdonald-mehta integralintegrable systemMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.769 Topics in Lie Theory: Tensor Categories (MIT)
This course will give a detailed introduction to the theory of tensor categories and review some of its connections to other subjects (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.
https://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009
Spring2009Etingof, Pavel2010-04-30T18:17:00+05:0018.769en-USmonoidal functorstensorpivotalsphericalMacLane'sGrthendieckmodule categoriesbraided tensorMuger centralizersymmetric categoriesdeligne's theoremradford formulasquared normsglobal dimensionscohomologyoceanu ridigityrobenius-perronlifting theoryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.306 Advanced Partial Differential Equations with Applications (MIT)
The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc.
https://ocw.mit.edu/courses/mathematics/18-306-advanced-partial-differential-equations-with-applications-fall-2009
Fall2009Rosales, Rodolfo2009-12-16T21:40:27+05:0018.306en-USpartial differential equations (pde)nonlinear pde. DiffusiondispersionInitial and boundary value problemsCharacteristics and shocksSeparation of variablestransform methodsGreen's functionsAsymptoticsgeometrical theoryDimensional analysisself-similaritytraveling wavesSingular perturbation and boundary layersSolitonsVariational methodsFree-boundary problemsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.034 Honors Differential Equations (MIT)This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems.
https://ocw.mit.edu/courses/mathematics/18-034-honors-differential-equations-spring-2009
Spring2009Hur, Vera Mikyoung2009-12-07T19:11:00+05:0018.034en-USQuadratureMaximum PrincipleLaplace TransformExistence TheoryAutonomous SystemLyapunovLimit CyclesFourier SeriesBoundary Value ProblemsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.102 Introduction to Functional Analysis (MIT)This is a undergraduate course. It will cover normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of L-p spaces; Hilbert space; compact, Hilbert-Schmidt and trace class operators; as well as spectral theorem.
https://ocw.mit.edu/courses/mathematics/18-102-introduction-to-functional-analysis-spring-2009
Spring2009Melrose, Richard2009-10-21T16:40:59+05:0018.102en-USlinear spacesmetric spacesnormed spacesBanach spacesLebesgue integrabilityLebesgue integrable functionsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.336 Numerical Methods for Partial Differential Equations (MIT)
This graduate-level course is an advanced introduction to applications and theory of numerical methods for solution of differential equations. In particular, the course focuses on physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods.
https://ocw.mit.edu/courses/mathematics/18-336-numerical-methods-for-partial-differential-equations-spring-2009
Spring2009Seibold, Benjamin2009-10-07T00:15:22+05:0018.336en-USadvection equationheat equationwave equationAiry equationconvection-diffusion problemsKdV equationhyperbolic conservation lawsPoisson equationStokes problemNavier-Stokes equationsinterface problemsconsistencystabilityconvergenceLax equivalence theoremerror analysisFourier approachesstaggered gridsshocksfront propagationpreconditioningmultigridKrylov spacessaddle point problemsfinite differencesfinite volumesfinite elementsENO/WENOspectral methodsprojection approaches for incompressible owslevel set methodsparticle methodsdirect and iterative methodsmultigridMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.726 Algebraic Geometry (MIT)This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.
https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009
Spring2009Kedlaya, Kiran2009-10-02T16:28:34+05:0018.726en-UScategory theorysheavesabelian sheavesshcemesmorphismsprojective morphismsdifferentialsdivisorshomological algebraalgebraic geometrycohomologyquasicoherent sheavesprojective spaceshilbert polynomialsgagaserre dualitycohen-macaulay schemesriemann-rochetale cohomologyMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.969 Topics in Geometry: Mirror Symmetry (MIT)
This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor.
https://ocw.mit.edu/courses/mathematics/18-969-topics-in-geometry-mirror-symmetry-spring-2009
Spring2009Auroux, Denis2009-08-20T15:32:17+05:0018.969en-USmirror symmetrydeformationhodge theorypseudoholomorphicgromov-wittencohomologyyukawamonodromypicard-fuchslagrangian floer theoryhomologySYZ conjecturesubmanifoldsK3 surfacesmatricesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.727 Topics in Algebraic Geometry: Algebraic Surfaces (MIT)The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.
https://ocw.mit.edu/courses/mathematics/18-727-topics-in-algebraic-geometry-algebraic-surfaces-spring-2008
Spring2008Kumar, Abhinav2009-04-20T19:24:01+05:0018.727en-USnear equivalencealgebraic equivalencenumerical equivalencebirationalrationalmapssurfacesruled surfacesrational surfaceslinear systemscastelnuovo's criterionrationalitypicardalbaneseclassificationK3ellipticKodaira dimensionbiellipticMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.085 Computational Science and Engineering I (MIT)This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications. Note: This course was previously called "Mathematical Methods for Engineers I."
https://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008
Fall2008Strang, Gilbert2009-03-31T14:33:33+05:0018.085en-USlinear algebranetworksLagrange multipliersdifferential equations of equilibriumLaplace's equationpotential flowboundary-value problemsFourier seriesdiscrete Fourier transformconvolutionlinear algebranetworksLagrange multipliersdifferential equations of equilibriumLaplace's equationpotential flowboundary-value problemsFourier seriesdiscrete Fourier transformconvolutionMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.705 Commutative Algebra (MIT)
In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.
https://ocw.mit.edu/courses/mathematics/18-705-commutative-algebra-fall-2008
Fall2008Kleiman, Steven2009-02-25T20:38:22+05:0018.705en-USringsidealsmoduleschain conditionsintegrallocalizationdecompositiondedekind domaintensordimension theoryZorn's lemmahilbert theoremDVRnormalizationartin ringnakayama's lemmazerodivisorsnoethernullsetellensatzMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.704 Seminar in Algebra and Number Theory: Computational Commutative Algebra and Algebraic Geometry (MIT)
In this undergraduate level seminar series, topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.
https://ocw.mit.edu/courses/mathematics/18-704-seminar-in-algebra-and-number-theory-computational-commutative-algebra-and-algebraic-geometry-fall-2008
Fall2008Kleiman, Steven2009-02-17T21:38:10+05:0018.704en-USComputational algebraalgebraic geometryGeometryAlgebraAlgorithmsGroebner BasesElimination TheoryAlgebra-Geometry DictionaryPolynomial FunctionsRational FunctionsGeometric Theorem ProvingInvariant Theory of Finite GroupsProjective Algebraic GeometryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.112 Functions of a Complex Variable (MIT)
This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.
This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.
https://ocw.mit.edu/courses/mathematics/18-112-functions-of-a-complex-variable-fall-2008
Fall2008Helgason, Sigurdur2009-02-06T20:02:10+05:0018.112en-USfunctions of one complex variableCauchy's theoremholomorphic functionsmeromorphic functionsresiduescontour integralsconformal mappingInfinite series and productsthe gamma functionthe Mittag-Leffler theoremHarmonic functionsDirichlet's problemThe Riemann mapping theoremThe Riemann Zeta functionMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.950 Differential Geometry (MIT)
This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.
https://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008
Fall2008Seidel, Paul2009-02-04T06:38:09+05:0018.950en-USdifferential geometrygeometry of plane curveshypersurfacesgeometry of lengths and distancesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.369 Mathematical Methods in Nanophotonics (MIT)
Find out what solid-state physics has brought to Electromagnetism in the last 20 years. This course surveys the physics and mathematics of nanophotonics—electromagnetic waves in media structured on the scale of the wavelength.
Topics include computational methods combined with high-level algebraic techniques borrowed from solid-state quantum mechanics: linear algebra and eigensystems, group theory, Bloch's theorem and conservation laws, perturbation methods, and coupled-mode theories, to understand surprising optical phenomena from band gaps to slow light to nonlinear filters.
Note: An earlier version of this course was published on OCW as 18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics, Fall 2005.
https://ocw.mit.edu/courses/mathematics/18-369-mathematical-methods-in-nanophotonics-spring-2008
Spring2008Johnson, Steven G.2008-12-22T20:10:33+05:0018.369en-USlinear algebraeigensystems for Maxwell's equationssymmetry groupsrepresentation theoryBloch's theoremnumerical eigensolver methodstime and frequency-domain computationperturbation theorycoupled-mode theorieswaveguide theoryadiabatic transitionsOptical phenomenaphotonic crystalsband gapsanomalous diffractionmechanisms for optical confinementoptical fibersintegrated optical devicesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.917 Topics in Algebraic Topology: The Sullivan Conjecture (MIT)
The goal of this course is to describe some of the tools which enter into the proof of Sullivan's conjecture.
https://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007
Fall2007Lurie, Jacob2008-10-03T20:54:14+05:0018.917en-USThe Sullivan ConjectureSteenrod OperationsAdem RelationsAdmissible MonomialsFree Unstable ModulesGabriel-Kuhn-PopescoInjectivity of the cohomology of BVGenerating Analytic FunctorsTensor products and algebrasThe Dual Steenrod AlgebraThe FrobeniusFiniteness ConditionsLannes' T-functorFree E-infinity Algebrasp-adic Homotopy TheoryAtomicityThe Arithmetic SquareQuaternionic Projective SpaceThe Nil-FiltrationThe Krull FiltrationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.02 Multivariable Calculus (MIT)This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space. MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the Fall and Spring terms at MIT, and is a required subject for all MIT undergraduates.
https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007
Fall2007Auroux, Denis2008-06-13T16:02:35+05:0018.02en-UScalculuscalculus of several variablesvector algebradeterminantsmatrixmatricesvector-valued functionspace motionscalar functionpartial differentiationgradientoptimization techniquesdouble integralsline integralsexact differentialconservative fieldsGreen's theoremtriple integralssurface integralsdivergence theorem Stokes' theoremapplicationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.098 Street-Fighting Mathematics (MIT)This course teaches the art of guessing results and solving problems without doing a proof or an exact calculation. Techniques include extreme-cases reasoning, dimensional analysis, successive approximation, discretization, generalization, and pictorial analysis. Applications include mental calculation, solid geometry, musical intervals, logarithms, integration, infinite series, solitaire, and differential equations. (No epsilons or deltas are harmed by taking this course.) This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT that runs from the first week of January until the end of the month.
https://ocw.mit.edu/courses/mathematics/18-098-street-fighting-mathematics-january-iap-2008
January IAP2008Mahajan, Sanjoy2008-05-27T14:03:51+05:0018.0986.099en-USextreme-cases reasoningdimensional analysisdiscretizationdragfluid mechanicspendulumpictorial proofsanalogyoperatorssummationsquare rootslogarithmsmusical intervalstaking out the big partintegrationdifferentiationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.966 Geometry of Manifolds (MIT)
This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.
https://ocw.mit.edu/courses/mathematics/18-966-geometry-of-manifolds-spring-2007
Spring2007Auroux, Denis2007-10-16T04:53:25+05:0018.966en-USDifferential formsLie groupsDeRhamRiemannian manifoldscurvatureHodgeHodge theorymanifoldsRiemannian geometryholonomysymplectic geometrycomplex geometryHodge-Kahler theorysmooth manifold topologyMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.465 Topics in Statistics: Statistical Learning Theory (MIT)
The main goal of this course is to study the generalization ability of a number of popular machine learning algorithms such as boosting, support vector machines and neural networks. Topics include Vapnik-Chervonenkis theory, concentration inequalities in product spaces, and other elements of empirical process theory.
https://ocw.mit.edu/courses/mathematics/18-465-topics-in-statistics-statistical-learning-theory-spring-2007
Spring2007Panchenko, Dmitry2007-08-30T12:39:03+05:0018.465en-USmachine learning algorithmsboostingsupportmachine learning algorithmsboostingsupport vector machinesneural networksVapnik- Chervonenkis theoryconcentration inequalities in product spacesempirical process theoryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.01 Single Variable Calculus (MIT)This introductory calculus course covers differentiation and integration of functions of one variable, with applications.
https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006
Fall2006Jerison, David2007-07-24T10:30:40+05:0018.01en-USdifferentiation and integration of functions of one variablelimitscontinuitydifferentiation rulesextremum problemsdefinite and indefinite integrationfundamental theorem of calculuselementarytechniques of integrationapproximation of definite integralsimproper integralsl'H?pital's ruleMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.404J Theory of Computation (MIT)This graduate level course is more extensive and theoretical treatment of the material in Computability, and Complexity (6.045J / 18.400J). Topics include Automata and Language Theory, Computability Theory, and Complexity Theory.
https://ocw.mit.edu/courses/mathematics/18-404j-theory-of-computation-fall-2006
Fall2006Sipser, Michael2007-05-25T04:27:42+05:0018.404J6.840Jen-USComputability, computational complexity theoryRegular and context-free languagesDecidable and undecidable problems, reducibility, recursive function theoryTime and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systemsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.366 Random Walks and Diffusion (MIT)This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.
https://ocw.mit.edu/courses/mathematics/18-366-random-walks-and-diffusion-fall-2006
Fall2006Bazant, Martin2007-05-17T05:44:29+05:0018.366en-USDiscrete and continuum modeling of diffusion processes in physics, chemistry, and economicscentral limit theoremscontinuous-time random walksLevy flightscorrelationsextreme eventsmixingrenormalizationand percolationpercolationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.969 Topics in Geometry: Dirac Geometry (MIT)
This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.
https://ocw.mit.edu/courses/mathematics/18-969-topics-in-geometry-dirac-geometry-fall-2006
Fall2006Gualtieri, Marco2007-05-17T05:34:36+05:0018.969en-USgeneralized geometryDirac geometryGerbesB-fieldsCourant algebroidssigma modelsbaby String theorylinear algebrapure spinorsRiemannian structuresHodge starintegrabilityDirac structuresLie algebroids and bialgebroidsholomorphic bundlesPicard groupKodaira-Spencer-Kuranishi deformation theoryKahler geometryHermitian geometryCalabi-Yau structuresD-branesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.104 Seminar in Analysis: Applications to Number Theory (MIT)18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.
https://ocw.mit.edu/courses/mathematics/18-104-seminar-in-analysis-applications-to-number-theory-fall-2006
Fall2006Ciubotaru, Dan2007-03-16T03:25:07+05:0018.104en-USInfinitude of the primesSumming powers of integersBernoulli polynomialssine product formula$\zeta(2n)$Fermat's Little TheoremFermat's Great TheoremAverages of arithmetic functionsarithmetic-geometric meanGauss' theoremWallis's formulaStirling's formulaprime number theoremRiemann's hypothesisEuler's proof of infinitude of primesDensity of prime numbersEuclidean algorithmGolden RatioMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.303 Linear Partial Differential Equations (MIT)
This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.
https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006
Fall2006Hancock, Matthew2007-02-23T03:41:11+05:0018.303en-USdiffusionLaplace equationsPoissonwave equationsseparation of variablesFourier seriesFourier transformseigenvalue problemsGreen's functionHeat EquationSturm-Liouville Eigenvalue problemsquasilinear PDEsBessel functionsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.443 Statistics for Applications (MIT)
This course offers a broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics include: hypothesis testing and estimation, confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, and correlation.
OCW offers an earlier version of this course, from Fall 2003. This newer version focuses less on estimation theory and more on multiple linear regression models. In addition, a number of Matlab examples are included here.
https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2006
Fall2006Panchenko, Dmitry2007-02-23T03:36:48+05:0018.44318.650en-UShypothesis testing and estimationconfidence intervalschi-square testsnonparametric statisticsanalysis of varianceregressioncorrelationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.02 Multivariable Calculus (MIT)This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include Vectors and Matrices, Partial Derivatives, Double and Triple Integrals, and Vector Calculus in 2 and 3-space.
https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-spring-2006
Spring2006Jerison, DavidMattuck, Arthur2006-11-07T15:31:42+05:0018.02en-USCalculuscalculus of several variablesvector algebradeterminantsmatrixmatricesvector-valued functionspace motionscalar functionpartial differentiationgradientoptimization techniquesdouble integralsline integralsexact differentialconservative fieldsGreen's theoremtriple integralssurface integralsdivergence theorem Stokes' theoremapplicationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.086 Mathematical Methods for Engineers II (MIT)This graduate-level course is a continuation of Mathematical Methods for Engineers I (18.085). Topics include numerical methods; initial-value problems; network flows; and optimization.
https://ocw.mit.edu/courses/mathematics/18-086-mathematical-methods-for-engineers-ii-spring-2006
Spring2006Strang, Gilbert2006-10-27T15:33:44+05:0018.086en-USScientific computing: Fast Fourier Transformfinite differencesfinite elementsspectral methodnumerical linear algebraComplex variables and applicationsInitial-value problems: stability or chaos in ordinary differential equationswave equation versus heat equationconservation laws and shocksdissipation and dispersionOptimization: network flowslinear programmingScientific computing: Fast Fourier Transform, finite differences, finite elements, spectral method, numerical linear algebraComplex variables and applicationsInitial-value problems: stability or chaos in ordinary differential equations, wave equation versus heat equation, conservation laws and shocks, dissipation and dispersionOptimization: network flows, linear programmingMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.152 Introduction to Partial Differential Equations (MIT)
This course provides a solid introduction to Partial Differential Equations for advanced undergraduate students. The focus is on linear second order uniformly elliptic and parabolic equations.
https://ocw.mit.edu/courses/mathematics/18-152-introduction-to-partial-differential-equations-fall-2005
Fall2005Colding, Tobias2006-10-20T10:50:54+05:0018.152en-USHarmonic functionsHarnack inequalitygradient estimateHopf Maximum PrinciplePoincare InequalitiesCacciopolli InequalityDirichlet problemCampanato's lemmaMorrey's lemmaMoser's ApproachMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.01 Single Variable Calculus (MIT)This introductory calculus course covers differentiation and integration of functions of one variable, with applications.
https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2005
Fall2005Starr, Jason2006-10-11T19:12:03+05:0018.01en-USdifferentiation and integration of functions of one variablelimitscontinuitydifferentiation rulesextremum problemsdefinite and indefinite integrationfundamental theorem of calculuselementarytechniques of integrationapproximation of definite integralsimproper integralsl'H?pital's ruleMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.786 Topics in Algebraic Number Theory (MIT)
This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets.
https://ocw.mit.edu/courses/mathematics/18-786-topics-in-algebraic-number-theory-spring-2006
Spring2006Kedlaya, Kiran2006-08-08T20:40:33+05:0018.786en-USalgebraic number theorynumber fieldsclass numbersDirichlet's units theoremcyclotomic fieldslocal fieldsvaluationsdecomposition and inertia groupsramificationbasic analytic methodsbasic class field theoryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.727 Topics in Algebraic Geometry: Intersection Theory on Moduli Spaces (MIT)
The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.
https://ocw.mit.edu/courses/mathematics/18-727-topics-in-algebraic-geometry-intersection-theory-on-moduli-spaces-spring-2006
Spring2006Coskun, Izzet2006-08-02T19:50:41+05:0018.727en-USintersection theorymoduli spacesgeometry of homogeneous varietiesDeligne-Mumford moduli spacesstable curvesKontsevich moduli spacesstable mapsLittlewood-Richardson rulesGrassmanniansdivisor theorycohomologyBrill-Noether theorylimit linear seriesample coneseffective conesGromov-Witten invariantssimple homogeneous varietiesKontsevich moduli spacesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.318 Topics in Algebraic Combinatorics (MIT)
The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.
https://ocw.mit.edu/courses/mathematics/18-318-topics-in-algebraic-combinatorics-spring-2006
Spring2006Stanley, Richard2006-07-10T20:40:31+05:0018.318en-USalgebraic combinatoricsmatrix-tree theoremlinear algebracommutative algebraexterior algebracounting faces of simplicial complexestilingsYoung's latticeShannon capacityFisher inequalityHadamard matricesf-vectorsSperner Property-Binomial CoeffcientsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.307 Integral Equations (MIT)
This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.
https://ocw.mit.edu/courses/mathematics/18-307-integral-equations-spring-2006
Spring2006Margetis, Dionisios2006-07-06T17:52:09+05:0018.307en-USintegral equationsapplied mathematicsVolterra equationFredholm equationFredholm theoryHilbert-Schmidt theoremWiener-Hopf Methodpartial differential equationsHilbert Problemingular integral equationsCauchy typeinverse scattering transformgroup theoryfluid mechanicssolid mechanicsacousticsquantum mechanicsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.101 Analysis II (MIT)This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis. The first half of the course covers multivariable calculus. The rest of the course covers the theory of differential forms in n-dimensional vector spaces and manifolds.
https://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005
Fall2005Guillemin, Victor2006-04-24T22:14:03+05:0018.101en-USDifferentiable mapsinverse and implicit function theoremsn-dimensional Riemann integralchange of variables in multiple integralsmanifoldsdifferential formsand n-dimensional version of Stokes' theoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.315 Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative Combinatorics (MIT)
This course serves as an introduction to major topics of modern enumerative and algebraic combinatorics with emphasis on partition identities, young tableaux bijections, spanning trees in graphs, and random generation of combinatorial objects. There is some discussion of various applications and connections to other fields.
https://ocw.mit.edu/courses/mathematics/18-315-combinatorial-theory-introduction-to-graph-theory-extremal-and-enumerative-combinatorics-spring-2005
Spring2005Pak, Igor2006-04-11T14:32:04+05:0018.315en-USenumerative combinatoricsalgebraic combinatoricspartition identitiesyoung tableaux bijectionsspanning treesrandom generation of combinatorial objectsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.319 Geometric Combinatorics (MIT)
This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems.
https://ocw.mit.edu/courses/mathematics/18-319-geometric-combinatorics-fall-2005
Fall2005Toth, Csaba2006-03-24T15:00:01+05:0018.319en-USdiscrete geometrycomputational geometryconvex partitionsbinary space partitionsart gallery problemsPlanar graphspseudo-triangulationsencompassing graphsgeometric graphscrossing numbersextremal graph theoryGallai-Sylvester problemsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.091 Mathematical Exposition (MIT)
This course provides techniques of effective presentation of mathematical material. Each section of this course is associated with a regular mathematics subject, and uses the material of that subject as a basis for written and oral presentations. The section presented here is on chaotic dynamical systems.
https://ocw.mit.edu/courses/mathematics/18-091-mathematical-exposition-spring-2005
Spring2005Carberry, Emma2005-11-02T22:09:48+05:0018.091en-USoral presentationmathematics writingmathematics presentation17.88117.882MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.155 Differential Analysis (MIT)This is the first semester of a two-semester sequence on Differential Analysis. Topics include fundamental solutions for elliptic; hyperbolic and parabolic differential operators; method of characteristics; review of Lebesgue integration; distributions; fourier transform; homogeneous distributions; asymptotic methods.
https://ocw.mit.edu/courses/mathematics/18-155-differential-analysis-fall-2004
Fall2004Melrose, Richard2005-10-26T06:27:25+05:0018.155en-USelliptichyperbolicparabolic differential operatorsLebesgue integrationDistributionsFourier transformHomogeneous distributionsAsymptotic methodsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.117 Topics in Several Complex Variables (MIT)
This course covers harmonic theory on complex manifolds, the Hodge decomposition theorem, the Hard Lefschetz theorem, and Vanishing theorems. Some results and tools on deformation and uniformization of complex manifolds are also discussed.
https://ocw.mit.edu/courses/mathematics/18-117-topics-in-several-complex-variables-spring-2005
Spring2005Guillemin, Victor2005-10-18T06:26:28+05:0018.117en-USHarmonic theorycomplex manifoldsHodge decomposition theoremHard Lefschetz theoremVanishing theoremsdeformation of complex manifoldsuniformization of complex manifoldsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.465 Topics in Statistics: Nonparametrics and Robustness (MIT)This graduate-level course focuses on one-dimensional nonparametric statistics developed mainly from around 1945 and deals with order statistics and ranks, allowing very general distributions. For multidimensional nonparametric statistics, an early approach was to choose a fixed coordinate system and work with order statistics and ranks in each coordinate. A more modern method, to be followed in this course, is to look for rotationally or affine invariant procedures. These can be based on empirical processes as in computer learning theory. Robustness, which developed mainly from around 1964, provides methods that are resistant to errors or outliers in the data, which can be arbitrarily large. Nonparametric methods tend to be robust.
https://ocw.mit.edu/courses/mathematics/18-465-topics-in-statistics-nonparametrics-and-robustness-spring-2005
Spring2005Dudley, Richard2005-10-14T07:03:10+05:0018.465en-USRank TestsRobustnessM-estimationMultivariate robustnessVC combinatoricsNonparametric classificationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.385J Nonlinear Dynamics and Chaos (MIT)This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.
https://ocw.mit.edu/courses/mathematics/18-385j-nonlinear-dynamics-and-chaos-fall-2004
Fall2004Rosales, Rodolfo2005-10-14T06:05:14+05:0018.385J2.036Jen-USPhase planelimit cyclesPoincare-Bendixson theoryTime-dependent systemsFloquet theoryPoincare mapsaveragingStability of equilibrianear-equilibrium dynamicsCenter manifoldselementary bifurcationsnormal formschaos18.385J18.3852.036J2.036MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.013A Calculus with Applications (MIT)This is an undergraduate course on differential calculus in one and several dimensions. It is intended as a one and a half term course in calculus for students who have studied calculus in high school. The format allows it to be entirely self contained, so that it is possible to follow it without any background in calculus.
https://ocw.mit.edu/courses/mathematics/18-013a-calculus-with-applications-spring-2005
Spring2005Kleitman, Daniel2005-10-07T06:58:34+05:0018.013Aen-USvector algebrataylor seriesnumerical methodsdifferential calculus18.013A18.013MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.435J Quantum Computation (MIT)
This course provides an introduction to the theory and practice of quantum computation. Topics covered include: physics of information processing, quantum logic, quantum algorithms including Shor's factoring algorithm and Grover's search algorithm, quantum error correction, quantum communication, and cryptography.
https://ocw.mit.edu/courses/mathematics/18-435j-quantum-computation-fall-2003
Fall2003Shor, Peter2005-04-27T01:37:31+05:0018.435J2.111JESD.79Jen-USquantum computationphysics of information processingquantum logicquantum algorithms including Shor's factoring algorithm and Grover's search algorithmquantum error correctionquantum communicationcryptography18.435J18.3452.111J2.111ESD.79JESD.79MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.755 Introduction to Lie Groups (MIT)
This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry. The text for this class is Differential Geometry, Lie Groups and Symmetric Spaces by Sigurdur Helgason (American Mathematical Society, 2001).
Much of the course material is based on Chapter I (first half) and Chapter II of the text. The text however develops basic Riemannian Geometry, Complex Manifolds, as well as a detailed theory of Semisimple Lie Groups and Symmetric Spaces.
https://ocw.mit.edu/courses/mathematics/18-755-introduction-to-lie-groups-fall-2004
Fall2004Helgason, Sigurdur2005-04-22T23:41:56+05:0018.755en-USManifoldsLie groupsexponential mappingLie algebrasHomogeneous spacestransformation groupsAdjoint representationCovering groupsAutomorphism groupsInvariant differential formscohomology of Lie groupshomogeneous spaces.ManifoldsLie GroupsExponential MappingLie AlgebrasHomogeneous SpacesTransformation GroupsAdjoint representationCovering GroupsAutomorphism GroupsInvariant Differential FormsCohomology of Lie GroupsHomogeneous Spaces.MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.330 Introduction to Numerical Analysis (MIT)
This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations and direct and iterative methods in linear algebra.
https://ocw.mit.edu/courses/mathematics/18-330-introduction-to-numerical-analysis-spring-2004
Spring2004Toomre, Alar2005-04-20T17:05:29+05:0018.330en-USRoot findinginterpolationapproximation functionintegrationdifferential equationsdirect iterative methodslinear algebra.MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.994 Seminar in Geometry (MIT)
In this course, students take turns in giving lectures. For the most part, the lectures are based on Robert Osserman's classic book A Survey of Minimal Surfaces, Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.
https://ocw.mit.edu/courses/mathematics/18-994-seminar-in-geometry-fall-2004
Fall2004Carberry, Emma2005-03-31T23:16:58+05:0018.994en-USMinimal SurfacesDifferential Geometry of Curves and SurfacesCalculus on ManifoldsComplex AnalysisScientific GraphicsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.901 Introduction to Topology (MIT)
This course introduces topology, covering topics fundamental to modern analysis and geometry. It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group.
https://ocw.mit.edu/courses/mathematics/18-901-introduction-to-topology-fall-2004
Fall2004Munkres, James2005-03-31T23:12:17+05:0018.901en-UStopologytopological spacescontinuous functionsconnectednesscompactnessseparation axiomsfunction spacesmetrization theoremsembedding theoremsthe fundamental groupMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.704 Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves (MIT)
This is a seminar for mathematics majors, where the students present the lectures. No prior experience giving lectures is necessary.
https://ocw.mit.edu/courses/mathematics/18-704-seminar-in-algebra-and-number-theory-rational-points-on-elliptic-curves-fall-2004
Fall2004Rogalski, Daniel2005-03-31T23:10:06+05:0018.704en-USRational points on elliptic curvesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.417 Introduction to Computational Molecular Biology (MIT)This course introduces the basic computational methods used to understand the cell on a molecular level. It covers subjects such as the sequence alignment algorithms: dynamic programming, hashing, suffix trees, and Gibbs sampling. Furthermore, it focuses on computational approaches to: genetic and physical mapping; genome sequencing, assembly, and annotation; RNA expression and secondary structure; protein structure and folding; and molecular interactions and dynamics.
https://ocw.mit.edu/courses/mathematics/18-417-introduction-to-computational-molecular-biology-fall-2004
Fall2004Lippert, Ross2005-03-24T22:28:08+05:0018.417en-USbasic computational methods cell on a molecular levelsequence alignment algorithmsdynamic programminghashingsuffix treesGibbs samplinggenetic and physical mappinggenome sequencingassemblyand annotationRNA expression and secondary structureprotein structure and foldingand molecular interactions and dynamicsannotationmolecular interactions and dynamicsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.075 Advanced Calculus for Engineers (MIT)This course analyzes the functions of a complex variable and the calculus of residues. It also covers subjects such as ordinary differential equations, partial differential equations, Bessel and Legendre functions, and the Sturm-Liouville theory.
https://ocw.mit.edu/courses/mathematics/18-075-advanced-calculus-for-engineers-fall-2004
Fall2004Margetis, DionisiosBush, John W. M.2005-03-24T22:23:55+05:0018.075en-USFunctions of complex variablecalculus of residuesOrdinary differential equationsBessel and Legendre functionsSturm-Liouville theorypartial differential equationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.338J Infinite Random Matrix Theory (MIT)
In this course on the mathematics of infinite random matrices, students will learn about the tools such as the Stieltjes transform and Free Probability used to characterize infinite random matrices.
https://ocw.mit.edu/courses/mathematics/18-338j-infinite-random-matrix-theory-fall-2004
Fall2004Win, MoeEdelman, Alan2005-03-18T03:03:24+05:0018.338J16.394Jen-USInfinite Random MatricesThe Hermite EnsembleWigner's Semi-Circle Law;The Laguerre EnsembleMarcenko-Pastur TheoremThe Jacobi EnsembleMcKay's Random Graph TheoremThe ?Semi-Circular? ElementCentral Limit TheoremFree Cumulants in Free ProbabilityNon-Crossing PartitionsmFree CumulantsThe Semi-Circular and ?Free Poisson? distributionsAdditive Free ConvolutionThe R-Transform and the Marcenko-Pastur TheoremMultiplicative Free ConvolutionThe S-TransformInfinite Random MatricesInfinite Random MatricesThe Hermite EnsembleWigner's Semi-Circle Law;The Laguerre EnsembleMarcenko-Pastur TheoremThe Jacobi EnsembleMcKay's Random Graph TheoremThe ?Semi-Circular? ElementCentral Limit TheoremFree Cumulants in Free ProbabilityNon-Crossing PartitionsFree CumulantsThe Semi-Circular and ?Free Poisson? distributionsAdditive Free ConvolutionThe R-Transform and the Marcenko-Pastur TheoremMultiplicative Free ConvolutionThe S-TransformOrthogonal Polynomials and the Classical Matrix EnsemblesTracy Widom DistributionEigenvalue Spectrum FluctuationsFree Probability and FluctuationsZonal Polynomials and Random MatricesSymmetric Group Representations and Free Probability18.338J16.394J18.33816.394Wigner's Semi-Circle Law18.338J18.33816.394J16.394MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.965 Geometry of Manifolds (MIT)
Geometry of Manifolds analyzes topics such as the differentiable manifolds and vector fields and forms. It also makes an introduction to Lie groups, the de Rham theorem, and Riemannian manifolds.
https://ocw.mit.edu/courses/mathematics/18-965-geometry-of-manifolds-fall-2004
Fall2004Mrowka, Tomasz2005-03-18T02:17:37+05:0018.965en-USDifferentiable manifoldsvector fields formsLie groupsDeRham theoremRiemannian manifoldsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.315 Combinatorial Theory: Hyperplane Arrangements (MIT)
This is a graduate-level course in combinatorial theory. The content varies year to year, according to the interests of the instructor and the students. The topic of this course is hyperplane arrangements, including background material from the theory of posets and matroids.
https://ocw.mit.edu/courses/mathematics/18-315-combinatorial-theory-hyperplane-arrangements-fall-2004
Fall2004Stanley, Richard2005-03-18T02:15:45+05:0018.315en-USCombinatorial TheoryHyperplane ArrangementsIntersection PosetMatroidsGeometric LatticesBroken CircuitsModular ElementsSupersolvabilityFinite FieldsCombinatorial TheoryHyperplaneArrangementsintersection posetMatroidsgeometric latticesBroken circuitsmodular elementssupersolvabilityFinite fieldsHyperplane ArrangementsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.305 Advanced Analytic Methods in Science and Engineering (MIT)
Advanced Analytic Methods in Science and Engineering is a comprehensive treatment of the advanced methods of applied mathematics. It was designed to strengthen the mathematical abilities of graduate students and train them to think on their own.
https://ocw.mit.edu/courses/mathematics/18-305-advanced-analytic-methods-in-science-and-engineering-fall-2004
Fall2004Cheng, Hung2005-03-18T01:27:34+05:0018.305en-USelementary methods complex analysisordinary differential equationspartial differential equationsexpansions around regular irregular singular pointsasymptotic evaluation integralsregular perturbationsWKB methodmultiple scale methodboundary-layer techniques.asymptotic evaluation integrals, regular perturbationsasymptotic evaluation integrals, regular perturbationsregular perturbationsboundary-layer techniquesasymptotic evaluation integrals, regular perturbationsasymptotic evaluation integralsregular perturbationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.409 Behavior of Algorithms (MIT)
This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.
https://ocw.mit.edu/courses/mathematics/18-409-behavior-of-algorithms-spring-2002
Spring2002Spielman, Daniel2004-09-17T01:31:15+05:0018.409en-USCondition numberlargest singluar value of a matrixSmoothed analysisGaussian eliminationGrowth factors of partial and complete pivotingGE of graphs with low bandwidth or small separatorsSpectral Partitioning of planar graphsspectral paritioning of well-shaped meshesspectral paritioning of nearest neighbor graphsTurner's theorembandwidth of semi-random graphs.McSherry's spectral bisection algorithmLinear Programmingvon Neumann's algorithmprimal and dual simplex methods, and duality Strong duality theoremRenegar's condition numbersMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.413 Error-Correcting Codes Laboratory (MIT)
This course introduces students to iterative decoding algorithms and the codes to which they are applied, including Turbo Codes, Low-Density Parity-Check Codes, and Serially-Concatenated Codes. The course will begin with an introduction to the fundamental problems of Coding Theory and their mathematical formulations. This will be followed by a study of Belief Propagation--the probabilistic heuristic which underlies iterative decoding algorithms. Belief Propagation will then be applied to the decoding of Turbo, LDPC, and Serially-Concatenated codes. The technical portion of the course will conclude with a study of tools for explaining and predicting the behavior of iterative decoding algorithms, including EXIT charts and Density Evolution.
https://ocw.mit.edu/courses/mathematics/18-413-error-correcting-codes-laboratory-spring-2004
Spring2004Spielman, Daniel2004-09-16T19:55:33+05:0018.413en-USiterative decodingerror-correcting codesTurbo CodesLow-Density Parity-Check Codesserially concatenated codesaid code designiterative decoding algorithmsBelief Propagation Serially-Concatenated codesEXIT chartsDensity EvolutionMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.996A Simplicity Theory (MIT)
This is an advanced topics course in model theory whose main theme is simple theories. We treat simple theories in the framework of compact abstract theories, which is more general than that of first order theories. We cover the basic properties of independence (i.e., non-dividing) in simple theories, the characterization of simple theories by the existence of a notion of independence, and hyperimaginary canonical bases.
https://ocw.mit.edu/courses/mathematics/18-996a-simplicity-theory-spring-2004
Spring2004Ben-Yaacov, Itay2004-09-16T18:01:08+05:0018.996Aen-USuniversal domainscompact abstract theoriesindiscernibilityindiscernible sequencesdividingsimplicityindependenceLascar strong typesindependence theoremhyperimaginariescanonical basessupersimplicityLascar inequalitiesstabilitystable theoriesgeneric automorphismtype-definable groupslovely pairsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.125 Measure and Integration (MIT)This graduate-level course covers Lebesgue's integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.
https://ocw.mit.edu/courses/mathematics/18-125-measure-and-integration-fall-2003
Fall2003Viaclovsky, Jeff2004-09-15T23:10:49+05:0018.125en-USLebesgue integralconvergence theoremsLebesgue measure in RnLpspacesRadon-Nikodym TheoremLebesgue Differentiation TheoremFubini TheoremHausdorff measureArea and Coarea Formulasmeasure theoryconvolutionFourier transformLebesque Integration TheoryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.156 Differential Analysis (MIT)The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.
https://ocw.mit.edu/courses/mathematics/18-156-differential-analysis-spring-2004
Spring2004Viaclovsky, Jeff2004-09-11T23:51:27+05:0018.156en-USSobolev spacesFredholm alternativeVariable coefficient elliptic, parabolic and hyperbolic linear partial differential equationsVariational methodsViscosity solutions of fully nonlinear partial differential equationsSchauder theoryHolder estimateslinear equationssecond derivativesellipticparabolicnonlinear partial differential equationslinear partial differential equationsharmonic functionselliptic equationsparabolic equationsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.034 Honors Differential Equations (MIT)This course covers the same material as 18.03 with more emphasis on theory. Topics include first order equations, separation, initial value problems, systems, linear equations, independence of solutions, undetermined coefficients, and singular points and periodic orbits for planar systems.
https://ocw.mit.edu/courses/mathematics/18-034-honors-differential-equations-spring-2004
Spring2004Starr, Jason2004-09-11T04:20:08+05:0018.034en-USFirst order equationsSeparationinitial value problemsSystemslinear equationsindependence of solutionsundetermined coefficientsSingular pointsperiodic orbits for planar systemsfirst order ode'ssecond order ode'sfourier serieslaplace transformlinear systemsnonlinear systemsconstant coefficientsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.725 Algebraic Geometry (MIT)This course covers the fundamental notions and results about algebraic varieties over an algebraically closed field. It also analyzes the relations between complex algebraic varieties and complex analytic varieties.
https://ocw.mit.edu/courses/mathematics/18-725-algebraic-geometry-fall-2003
Fall2003Olsson, Martin2004-09-11T04:11:22+05:0018.725en-USalgebraic varieties over algebraically closed fieldcomplex algebraic varietiescomplex analytic varietiescurves and surfacesirreducible componentsprojective spacetopological diversionsheavespresheavesalgebraic geometryfibersmorphismsvarietiesprojective varietiesapplicationsdimensionkrull dimensioncompletenesscomplex topologyChow's lemmaanalytic spacescurvesMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.996 Random Matrix Theory and Its Applications (MIT)
This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications.
https://ocw.mit.edu/courses/mathematics/18-996-random-matrix-theory-and-its-applications-spring-2004
Spring2004Win, MoeEdelman, Alan2004-09-11T03:58:31+05:0018.99616.399en-USRandom matrix theoryMatrix JacobiansWishart MatricesWigner's Semi-Circular lawsMatrix beta ensemblesfree probabilityspherical coordinateswedgingPlucker coordinatesmatrix factorizationshouseholder transformationsStiefel manifoldCauchey-Binet theoremTelatar's paperlevel densitiesorthogonal polynomialsmatrix integralshypergeometric functionswireless communictionseigenvalue densitysample covariance matricesMarcenko-Pastur theoremwireless communications18.99616.399MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.997 Topics in Combinatorial Optimization (MIT)
In this graduate-level course, we will be covering advanced topics in combinatorial optimization. We will start with non-bipartite matchings and cover many results extending the fundamental results of matchings, flows and matroids. The emphasis is on the derivation of purely combinatorial results, including min-max relations, and not so much on the corresponding algorithmic questions of how to find such objects. The intended audience consists of Ph.D. students interested in optimization, combinatorics, or combinatorial algorithms.
https://ocw.mit.edu/courses/mathematics/18-997-topics-in-combinatorial-optimization-spring-2004
Spring2004Goemans, Michel2004-09-09T20:40:14+05:0018.997en-UScombinatorial optimizationEar decompositionsNonbipartite matchingGallai-Milgram and Bessy-Thomasse theorems on partitioning/covering graphs by directed paths/cyclesMinimization of submodular functionsMatroid intersectionPolymatroid intersectionJump systemsMatroid unionMatroid matching, path matchingsPacking trees and arborescencesPacking directed cuts and the Lucchesi-Younger theoremSubmodular flows and the Edmonds-Giles theoremGraph orientationConnectivity tree and connectivity augmentationMulticommodity flowsConnectivity treeconnectivity augmentationGallai-Milgram TheoremBessy-Thomasse Theoremparitioning graphscovering graphsdirected pathsdirected cyclesmatroid matchingpath matchingpacking directed cutsLuchessi-Younger Theorempacking treesarborescencessubmodular flowsEdmonds-Giles TheoremMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.06CI Linear Algebra - Communications Intensive (MIT)
This is a communication intensive supplement to Linear Algebra (18.06). The main emphasis is on the methods of creating rigorous and elegant proofs and presenting them clearly in writing. The course starts with the standard linear algebra syllabus and eventually develops the techniques to approach a more advanced topic: abstract root systems in a Euclidean space.
https://ocw.mit.edu/courses/mathematics/18-06ci-linear-algebra-communications-intensive-spring-2004
Spring2004Brooke-Taylor, AndrewLachowska, Anna2004-09-09T19:35:45+05:0018.06CIen-USLinear AlegebraLatexLaTeX2emathematical writinglinear spacesbasisdimensionlinear mappingsmatricessubspacesdirect sumsreflectionsEuclidean spaceabstract root systemssimple rootspositive rootsCartan matrixDynkin diagramsclassification18.06CI18.06MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.443 Statistics for Applications (MIT)
This course provides a broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. The course topics include hypothesis testing and estimation. It also includes confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, and correlation.
https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003
Fall2003Panchenko, Dmitry2004-08-31T10:55:46+05:0018.44318.650en-UShypothesis testing and estimation; confidence intervals; chi-square tests; nonparametric statistics; analysis of variance; regression; correlationhypothesis testing and estimationconfidence intervalschi-square testsnonparametric statisticsanalysis of varianceregressioncorrelationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.327 Wavelets, Filter Banks and Applications (MIT)
Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.
https://ocw.mit.edu/courses/mathematics/18-327-wavelets-filter-banks-and-applications-spring-2003
Spring2003Strang, GilbertAmaratunga, Kevin2004-06-03T17:56:36+05:0018.3271.130en-USDiscrete-time filtersconvolutionFourier transformowpass and highpass filtersSampling rate change operationsupsampling and downsamplingractional samplinginterpolationFilter Bankstime domain (Haar example) and frequency domainconditions for alias cancellation and no distortionperfect reconstructionhalfband filters and possible factorizationsModulation and polyphase representationsNoble identitiesblock Toeplitz matrices and block z-transformspolyphase examplesMatlab wavelet toolboxOrthogonal filter banksparaunitary matricesorthogonality condition (Condition O) in the time domainmodulation domain and polyphase domainMaxflat filtersDaubechies and Meyer formulasSpectral factorizationMultiresolution Analysis (MRA)requirements for MRAnested spaces and complementary spaces; scaling functions and waveletsRefinement equationiterative and recursive solution techniquesinfinite product formulafilter bank approach for computing scaling functions and waveletsOrthogonal wavelet basesconnection to orthogonal filtersorthogonality in the frequency domainBiorthogonal wavelet basesMallat pyramid algorithmAccuracy of wavelet approximations (Condition A)vanishing momentspolynomial cancellation in filter banksSmoothness of wavelet basesconvergence of the cascade algorithm (Condition E)splinesBases vs. framesSignal and image processingfinite length signalsboundary filters and boundary waveletswavelet compression algorithmsLiftingladder structure for filter banksfactorization of polyphase matrix into lifting stepslifting form of refinement equationSecWavelets and subdivisionnonuniform gridsmultiresolution for triangular meshesrepresentation and compression of surfacesNumerical solution of PDEsGalerkin approximationwavelet integrals (projection coefficients, moments and connection coefficients)convergenceconvergenceSubdivision wavelets for integral equationsCompression and convergence estimatesCompression and convergence estimatesM-band waveletsDFT filter banks and cosine modulated filter banksMultiwavelets18.3271.130MIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.433 Combinatorial Optimization (MIT)
Combinatorial Optimization provides a thorough treatment of linear programming and combinatorial optimization. Topics include network flow, matching theory, matroid optimization, and approximation algorithms for NP-hard problems.
https://ocw.mit.edu/courses/mathematics/18-433-combinatorial-optimization-fall-2003
Fall2003Vempala, Santosh2004-03-19T07:17:26+05:0018.433en-USlinear programmingcombinatorial optimizationnetwork flowmatching theorymatroid optimizationapproximation algorithms for NP-hard problemsapproximation algorithmsNP-hard problemsdiscrete mathematicsfundamental algorithmic techniquesconvex programmingflow theoryrandomizationMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.238 Geometry and Quantum Field Theory (MIT)
Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory.
https://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002
Fall2002Etingof, Pavel2004-02-20T21:12:50+05:0018.238en-USperturbative quantum field theoryclassical field theoryfree quantum theoriesFeynman diagramsRenormalization theoryLocal operatorsOperator product expansionRenormalization group equationclassicalfieldtheoryFeynmandiagramsfreequantumtheorieslocaloperatorsproductexpansionperturbativerenormalizationgroupequationsfunctionalfunctionintergralsoperatorQFTstringphysicsmathematicsgeometrygeometricalgebraictopologynumber0-dimensional1-dimensionald-dimensionalsupergeometrysupersymmetryconformalstationaryphaseformulacalculuscombinatoricsmatrixmechanicslagrangianshamiltonsleastactionprinciplelimitsformalismFeynman-KaccurrentchargesNoether?stheorempathintegralapproachdivergencesperturbative quantum field theoryfunctional integralsclassical field theoryfee quantum theoriesFeynman diagramsrenormalization theorylocal operatorsoperator product expansionrenormalization group equationmathematical languagestring theory0-dimensional QFTStationary Phase FormulaMatrix ModelsLarge N Limits1-dimensional QFTClassical MechanicsLeast Action PrinciplePath Integral ApproachQuantum MechanicsPerturbative Expansion using Feynman DiagramsOperator FormalismFeynman-Kac Formulad-dimensional QFTFormalism of Classical Field TheoryCurrentsNoether?s TheoremPath Integral Approach to QFTPerturbative ExpansionRenormalization TheoryConformal Field Theoryalgebraic topologyalgebraic geometrynumber theoryclassical field theoryMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.S66 The Art of Counting (MIT)
The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment.
https://ocw.mit.edu/courses/mathematics/18-s66-the-art-of-counting-spring-2003
Spring2003Stanley, Richard2003-09-04T08:01:23+05:0018.S66en-USenumerative combinatoricsfinite setsum of positive integersbijective proofsbijection (one-to-one correspondence)permutationspartitionsCatalan numbersYoung tableauxlattice paths and tilingsMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.04 Complex Variables with Applications (MIT)
The following topics are covered in the course: complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis and Laplace transforms.
https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-fall-1999
Fall1999Rosales, R.2003-06-11T22:11:45+05:0018.04en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm18.996 Topics in Theoretical Computer Science : Internet Research Problems (MIT)
We will discuss numerous research problems that are related to the internet. Sample topics include: routing algorithms such as BGP, communication protocols such as TCP, algorithms for intelligently selecting a resource in the face of uncertainty, bandwidth sensing tools, load balancing algorithms, streaming protocols, determining the structure of the internet, cost optimization, DNS-related problems, visualization, and large-scale data processing. The seminar is intended for students who are ready to work on challenging research problems. Each lecture will discuss:
methods used today
issues and problems
formulation of concrete problems
potential new lines of research
A modest amount of background information will be provided so that the importance and context of the problems can be understood. No previous study of the internet is required, but experience with algorithms and/or theoretical computer science at the graduate/research level is needed.
https://ocw.mit.edu/courses/mathematics/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002
Spring2002Maggs, BruceSundaram, RaviTeng, Shang-HuaLeighton, Tom2003-06-06T22:32:25+05:0018.996en-USMIT OpenCourseWare https://ocw.mit.eduContent within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm