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
2021-10-21T14:06:13+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.htm18.404J Theory of Computation (MIT)This course emphasizes computability and computational complexity theory. Topics include regular and context-free languages, decidable and undecidable problems, reducibility, recursive function theory, time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.
https://ocw.mit.edu/courses/mathematics/18-404j-theory-of-computation-fall-2020
Fall2020Sipser, Michael2021-10-06T12:04:43+05:0018.404J18.4041J6.840Jen-UScomputabilitycomputational complexity theoryregular and context-free languagesdecidable and undecidable problemsreducibilityrecursive function theorytime and space measures on computationhierarchy theoremsinherently complex problemsprobabilistic computationinteractive 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.htm18.085 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.htm