New courses in Mathematics from MIT OpenCourseWare, provider of free and open MIT course materials. https://ocw.mit.edu/courses/mathematics 2022-02-11T17:21:40+05:00 MIT OpenCourseWare https://ocw.mit.edu en-US Content 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 This course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography. While this is an introductory course, we will (gently) work our way up to some fairly advanced material, including an overview of the proof of Fermat's last theorem.  https://ocw.mit.edu/courses/mathematics/18-783-elliptic-curves-spring-2021 Spring 2021 Sutherland, Andrew 2021-12-16T13:48:12+05:00 18.783 18.7831 en-US elliptic curves number theory cryptography point-counting isogenies pairings theory of complex multiplication integer factorization primality proving elliptic curve cryptography modular curves Fermat's Last Theorem MIT OpenCourseWare https://ocw.mit.edu Content 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 Functional analysis helps to solve problems where the vector space is no longer finite-dimensional, a situation that arises very naturally in many concrete problems. Topics will include normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of Lp spaces; Hilbert spaces; compact and self-adjoint operators; and the spectral theorem. https://ocw.mit.edu/courses/mathematics/18-102-introduction-to-functional-analysis-spring-2021 Spring 2021 Rodriguez, Casey 2021-11-03T19:47:08+05:00 18.102 18.1021 en-US linear spaces metric spaces normed spaces Banach spaces Lebesgue integrability Lebesgue integrable functions null functions Monotonicity Fatou's Lemma Hilbert spaces Baire's theorem Bessel's inequality compact sets weak convergence weak compactness Fourier series Fourier transform MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2020 Sipser, Michael 2021-10-06T12:04:43+05:00 18.404J 18.4041J 6.840J en-US computability computational complexity theory regular and context-free languages decidable and undecidable problems reducibility recursive function theory time and space measures on computation hierarchy theorems inherently complex problems probabilistic computation interactive proof systems MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Summer 2020 Zhang, Chengzhao “Richard” 2021-07-28T18:21:20+05:00 18.085 18.0851 en-US linear algebra networks structures finite difference finite element Laplace's equation potential flow boundary-value problems Fourier series discrete Fourier transform convolution MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2020 Edelman, Alan Sanders, David P. Sanderson, Grant Schloss, James Drake, Henri 2021-04-05T18:59:30+05:00 18.S191 6.S083 22.S092 en-US computational modeling mathematical modeling computational science artificial intelligence Julia programming data science algorithms statistical modeling image analysis particle dynamics ray tracing epidemic propagation climate modeling MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2019 Pearce, Philip 2021-02-12T19:10:59+05:00 18.031 en-US Laplace transform unit impulse response step function delta function convolution system function pole diagram MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2020 Edelman, Alan Sanders, David P. 2020-09-14T12:42:09+05:00 18.S190 6.S083 en-US computational modeling mathematical modeling Covid-19 computational science artificial intelligence Julia programming data science language statistical modeling epidemiology machine learning drug development disease models MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2020 Miller, Haynes 2020-08-11T15:45:49+05:00 18.906 en-US homotopy cohomology classifying spaces spectral sequences cofibrations serre fibrations Steenrod operations cohomology operations cobordism cobordism MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2019 Zhao, Yufei 2020-05-12T17:36:18+05:00 18.217 en-US graph theory additive combinatorics Ramsey theory Schur’s theorem Mantel’s theorem Turán’s theorem Erdős-Stone-Simonovits theorem Kővári-Sós-Turán theorem Szemerédi’s graph regularity lemma triangle counting lemma triangle removal lemma Roth’s theorem hypergraph removal lemma Green-Tao theorem martingale convergence theorem Freiman’s theorem Ruzsa triangle inequality Ruzsa covering lemma Balog-Szémeredi-Gowers theorem Szemerédi-Trotter theorem MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2019 Sutherland, Andrew 2020-04-23T14:42:04+05:00 18.785 en-US Absolute values Discrete valuations localization Dedekind domains Etale algebras Dedekind extensions Ideal Norm Dedekind-Kummer Theorem Galois extensions Artin map complete fields Valuation rings Hensel's lemmas Krasner's lemma Minkowski bound Dirichlet's unit theorm Zeta function Ray Class Ring of Adeles Idele group Chebotarev density theorem Global fields Tate cohomology Artin reciprocity MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2019 Sheffield, Scott 2020-04-06T16:49:43+05:00 18.600 en-US Probability spaces random variables distribution functions Binomial geometric hypergeometric Poisson distributions Uniform exponential normal gamma and beta distributions Conditional probability Bayes theorem joint distributions Chebyshev inequality law of large numbers central limit theorem MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2019 Postnikov, Alexander 2019-12-19T16:21:22+05:00 18.212 en-US enumeration methods permutations partitions partially ordered sets and lattices Young tableaux graph theory matrix tree theorem electrical networks convex polytopes MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2019 Zhao, Yufei 2019-12-17T15:42:51+05:00 18.218 en-US probabilistic method Ramsey numbers Lovász Local Lemma hypergraph colorings balancing vectors sum-free sets second Moment Method Chernoff bound Moser-Tardos algorithm Janson’s inequalities Harris-FKG inequality Martingale convergence Azuma’s inequality entropy methods occupancy method MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2018 Orloff, Jeremy 2019-10-30T18:49:34+05:00 18.04 en-US Complex algebra and functions analyticity contour integration Cauchy's theorem singularities Taylor and Laurent series residues evaluation of integrals multivalued functions potential theory in 2D Fourier analysis and Laplace transforms MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2019 Johnson, Steven G. 2019-07-10T12:59:56+05:00 18.335J 6.337J en-US numerical linear algebra linear systems eigenvalue decomposition QR/SVD factorization numerical algorithms IEEE floating point standard sparse matrices structured matrices preconditioning linear algebra software Matlab MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Spring 2018 Strang, Gilbert 2019-05-16T15:05:00+05:00 18.065 18.0651 en-US data analysis signal processing image processing machine learning linear algebra computation singular value decomposition least squares weighted least squares covariance matrices correlation matrices directed graphs undirected graphs matrix factorizations neural nets MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2018 Zhao, Yufei 2019-03-27T18:53:09+05:00 18.A34 en-US hidden independence probability congruences and divisibility recurrences limits greatest integer function inequalities roots of polynomials Pigeonhole Principle MIT OpenCourseWare https://ocw.mit.edu Content 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 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 IAP 2019 Spivak, David I. Fong, Brendan 2019-03-25T19:18:33+05:00 18.S097 en-US order adjunction set Galois connection monoidal preorder wiring diagram V-categories Bool-categories categories functors limits colimits monoidal categories hypergraph categories sheaves toposes MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2016 Miller, Haynes 2018-04-20T16:23:06+05:00 18.905 en-US Algebraic Topology homology CW complexes Homological algebra Cohomology Poincare duality Homotopy Invariance Eilenberg-Steenrod Axioms Topological Genealogy Künneth theorem Tor functors tensor product Čech" Cohomology MIT OpenCourseWare https://ocw.mit.edu Content 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 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 Fall 2015 Demanet, Laurent 2018-03-06T16:46:19+05:00 18.325 en-US waves imaging radar imaging seismic imaging Radon transform backprojection reflection seismology computerized tomography synthetic aperture radar MIT OpenCourseWare https://ocw.mit.edu Content 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