MIT OpenCourseWare: New Courses in MathematicsNew courses in Mathematics from MIT OpenCourseWare, provider of free and open MIT course materials.
https://ocw.mit.edu/courses/mathematics
2020-02-07T15:45:31+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.htm18.212 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.htm18.785 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-2017
Fall2017Sutherland, Andrew2018-03-14T20:31:30+05:0018.785en-USAbsolute valuesDiscrete valuationslocalizationDedekind domainsEtale algebrasDedekind extensionsIdeal NormDedekind-Kummer TheoremGalois extensionsFrobeniusArtin mapcomplete fieldsValuation ringsHensel's lemmasKrasner's lemmaMinkowski boundDirichletUnit theoremRiemannZeta functionKroneckerWeberRay 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.htm18.325 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.htm