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
2017-06-06T15:12:52+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.htm18.785 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-2016
Fall2016Sutherland, Andrew2017-05-02T17:52:23+05:0018.785en-USAbsolute valuesdiscrete valuationslocalizationDedekind domainsEtale algebrasDedekind extensionsIdeal NormDedekind-Kummer TheoremGalois extensionsFrobeniusArtin mapcomplete fieldsValuation ringsHensel's lemmasKrasner's lemmaHaar measuresMinkowski boundDirichletUnit theoremRiemannZeta functionKroneckerWeberRay ClassRing of AdelesIdele groupChebotarev density 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.htm18.405J 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, Mohammad2017-01-17T15:02:58+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, Lawrence2016-08-15T12:18:27+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, Roman2016-06-16T14:37:19+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, Peter2016-05-27T16:01:38+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, Scott2016-04-26T12:51:16+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, Afonso2016-03-08T16:40:36+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, David2016-03-01T17:46:38+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, Philippe2016-02-08T13:50:16+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, Lawrence2015-12-22T21:16:21+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, Omer2015-12-17T16:51:46+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, Peter2015-12-14T18:32:13+05:0018.443en-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, Rodolfo2015-12-03T18:34:47+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, Ankur2015-12-03T15:15:21+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, Philippe2015-11-10T17:50:26+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, Jörn2015-10-30T17:15:40+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, Hao2015-08-20T16:53:50+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, Richard2015-07-27T16:48:49+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, Rodolfo2015-07-08T16:40:32+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.2015-06-25T13:46:19+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.htm