Mathematics
The language every quantitative field is written in.
165 courses from MIT OpenCourseWare.
18.01 · Undergraduate · Fall 2006
This introductory calculus course covers differentiation and integration of functions of one variable, with applications.
18.01 · Undergraduate · Fall 2005
This introductory calculus course covers differentiation and integration of functions of one variable, with applications.
18.01 · Undergraduate · Fall 2020
<p>Master the calculus of derivatives, integrals, coordinate systems, and infinite series.</p> <p>In this three-part series you will learn the mathematical notation, physical meaning, and geometric interpretation of a variety of calculus concepts. Along with the fundamental computational skills required to solve these problems, you will also gain insight into real-world applications of these mathematical ideas.</p> <ul> <li>Part 1: Differentiation</li> <li>Part 2: Integration</li> <li>Part 3: C…
18.01SC · Undergraduate · Fall 2010
<p>This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics.</p> <p>Course Format</p> <p>This course has been designed for independent study. It includes all of the materials you will need to understand the concepts covered in this subject. The materials in this course include:</p> <ul> <li><strong>Lecture…
18.02 · Undergraduate · Fall 2007
<p>This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.</p> <p>MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the…
18.02 · Undergraduate · Spring 2006
This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include Vectors and Matrices, Partial Derivatives, Double and Triple Integrals, and Vector Calculus in 2 and 3-space.
18.02SC · Undergraduate · Fall 2010
<p>This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics.</p> <p>The materials have been organized to support independent study. The website includes all of the materials you will need to understand the concepts covered in this subject. The materials in this course include:</p> <ul> <li><strong>Lecture Videos</strong> r…
18.03 · Undergraduate · Spring 2010
Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE’s) deal with functions of one variable, which can often be thought of as time.
18.03SC · Undergraduate · Fall 2011
<p>The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.</p> <p>Course Format</p> <p>This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials includ…
18.04 · Undergraduate · Spring 2018
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 tra…
18.04 · Undergraduate · Fall 1999
The following topics are covered in the course: complex algebra and functions; analyticity; contour integration, Cauchy’s theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis and Laplace transforms.
18.05 · Undergraduate · Spring 2022
<p>This course provides an elementary introduction to probability and statistics with applications. Topics include basic combinatorics, random variables, probability distributions, Bayesian inference, hypothesis testing, confidence intervals, and linear regression.</p> <p>These same course materials, including interactive components (online reading questions and problem checkers) are available on MIT’s Open Learning Library, which is free to use. You have the option to enroll and track you…
18.06 · Undergraduate · Spring 2010
This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.
18.06CI · Undergraduate · Spring 2004
This is a communication intensive supplement to Linear Algebra (18.06). The main emphasis is on the methods of creating rigorous and elegant proofs and presenting them clearly in writing. The course starts with the standard linear algebra syllabus and eventually develops the techniques to approach a more advanced topic: abstract root systems in a Euclidean space.
18.06SC · Undergraduate · Fall 2011
<p>This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines such as physics, economics and social sciences, natural sciences, and engineering. It parallels the combination of theory and applications in Professor Strang’s textbook <em>Introduction to Linear Algebra</em>.</p> <p>Course Format</p> <p>This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials inc…
18.013A · Undergraduate · Spring 2005
This is an undergraduate course on differential calculus in one and several dimensions. It is intended as a one and a half term course in calculus for students who have studied calculus in high school. The format allows it to be entirely self contained, so that it is possible to follow it without any background in calculus.
18.014 · Undergraduate · Fall 2010
18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.
18.022 · Undergraduate · Fall 2010
<p>This is a variation on <em>18.02 Multivariable Calculus</em>. It covers the same topics as in 18.02, but with more focus on mathematical concepts.</p> <p>Acknowledgement</p> <p>Prof. McKernan would like to acknowledge the contributions of Lars Hesselholt to the development of this course.</p>
18.024 · Undergraduate · Spring 2011
This course is a continuation of <em>18.014 Calculus with Theory</em><em>.</em> It covers the same material as <em>18.02 Multivariable Calculus</em>, but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.
18.031 · Undergraduate · Spring 2019
<p>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.</p> <p>Go to OCW’s Open Learning Library site for <em>18.031: System Functions and the Laplace Transform</em>. The site is fr…
18.034 · Undergraduate · Spring 2009
This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems.
18.034 · Undergraduate · Spring 2004
This course covers the same material as 18.03 with more emphasis on theory. Topics include first order equations, separation, initial value problems, systems, linear equations, independence of solutions, undetermined coefficients, and singular points and periodic orbits for planar systems.
18.065 · Undergraduate · Spring 2018
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.
18.075 · Graduate · Fall 2004
This course analyzes the functions of a complex variable and the calculus of residues. It also covers subjects such as ordinary differential equations, partial differential equations, Bessel and Legendre functions, and the Sturm-Liouville theory.
18.085 · Undergraduate · Summer 2020
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.
18.085 · Graduate · Fall 2008
<p>This course provides a review of linear algebra, including applications to networks, structures, and estimation, Lagrange multipliers. Also covered are: differential equations of equilibrium; Laplace’s equation and potential flow; boundary-value problems; minimum principles and calculus of variations; Fourier series; discrete Fourier transform; convolution; and applications.</p> <p>Note: This course was previously called “Mathematical Methods for Engineers I.”</p>
18.086 · Graduate · Spring 2006
This graduate-level course is a continuation of Mathematical Methods for Engineers I (18.085). Topics include numerical methods; initial-value problems; network flows; and optimization.
18.091 · Undergraduate · Spring 2005
This course provides techniques of effective presentation of mathematical material. Each section of this course is associated with a regular mathematics subject, and uses the material of that subject as a basis for written and oral presentations. The section presented here is on chaotic dynamical systems.
18.100A · Undergraduate, Graduate · Fall 2020
This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an understanding and construction of proofs.
18.100A · Undergraduate · Fall 2012
<p>Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space.</p> <p>MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives appli…
18.100B · Undergraduate · Fall 2010
Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.
18.100B · Undergraduate, Graduate · Spring 2025
<p>This course gives an introduction to analysis, and the goal is twofold: </p> <p> 1. To learn how to prove mathematical theorems in analysis and how to write proofs. <br> 2. To prove theorems in calculus in a rigorous way.</p> <p>The course will start with real numbers, limits, convergence, series and continuity. We will continue on with metric spaces, differentiation and Riem…
18.100C · Undergraduate · Fall 2012
<p>This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. MIT students may choose to take one of three versions of Real Analysis; this version offers three additional units of credit for instruction and practice in writ…
18.101 · Undergraduate · Fall 2005
This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis. The first half of the course covers multivariable calculus. The rest of the course covers the theory of differential forms in n-dimensional vector spaces and manifolds.
18.102 · Undergraduate · Spring 2009
This is a undergraduate course. It will cover normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of L-p spaces; Hilbert space; compact, Hilbert-Schmidt and trace class operators; as well as spectral theorem.
18.102 · Undergraduate · Spring 2021
Functional analysis helps us study and solve both linear and nonlinear problems posed on a normed space that is no longer finite-dimensional, a situation that arises very naturally in many concrete problems. Topics include normed spaces, completeness, functionals, the 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.
18.103 · Undergraduate · Fall 2013
This course continues the content covered in <em>18.100 Analysis I</em>. 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.
18.104 · Undergraduate · Fall 2006
18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.
18.112 · Undergraduate · Fall 2008
<p>This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.</p> <p>This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.</p>
18.117 · Graduate · Spring 2005
This course covers harmonic theory on complex manifolds, the Hodge decomposition theorem, the Hard Lefschetz theorem, and Vanishing theorems. Some results and tools on deformation and uniformization of complex manifolds are also discussed.
18.125 · Graduate · Fall 2003
This graduate-level course covers Lebesgue’s integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.
18.152 · Undergraduate · Fall 2005
This course provides a solid introduction to Partial Differential Equations for advanced undergraduate students. The focus is on linear second order uniformly elliptic and parabolic equations.
18.152 · Undergraduate · Fall 2011
This course introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. It includes mathematical tools, real-world examples and applications.
18.155 · Graduate · Fall 2004
This is the first semester of a two-semester sequence on Differential Analysis. Topics include fundamental solutions for elliptic; hyperbolic and parabolic differential operators; method of characteristics; review of Lebesgue integration; distributions; fourier transform; homogeneous distributions; asymptotic methods.
18.156 · Graduate · Spring 2016
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 …
18.156 · Graduate · Spring 2004
The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.
18.156 · Graduate · Spring 2025
<p>The class studies projection theory, starting from the first questions and building up to recent developments. Projection theory studies how a set <em>X</em> behaves under different orthogonal projections. Questions of this type aren’t usually emphasized in the graduate analysis curriculum, but they come up in many areas of math, including harmonic analysis, analytic number theory, additive combinatorics, and homogeneous dynamics.</p> <p>We will survey several applications of projection theo…
18.175 · Graduate · Spring 2014
This course covers topics such as sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales.
18.177 · Graduate · Fall 2015
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.
18.200 · Undergraduate · Spring 2024
This course will teach you illustrative topics in discrete applied mathematics, including counting, generating functions, probability, linear optimization, algebraic structures, basic number theory, information theory, and coding theory. It is a CI-M (Communication Intensive in the Major) course and thus includes a writing component.
18.212 · Undergraduate · Spring 2019
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.
18.218 · Graduate · Spring 2021
In this course, we will mostly be studying Fourier analysis of Boolean functions, which is a useful tool in theoretical computer science, combinatorics, and more. We will start with basic concepts such as influences, noise sensitivity, and hypercontractivity and some basic results in the area.
18.225 · Graduate · Fall 2023
<p>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.</p> <p>This course was previously numbered 18.217.</p>
18.226 · Graduate · Fall 2022
This course is a graduate-level introduction to the probabilistic methods, 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.
18.238 · Graduate · Spring 2023
The main goal of this course is to present the basic ideas of quantum field theory (QFT) in a completely rigorous and mathematical way. Topics of the course include generalities on classical and quantum mechanics and field theory, 0-dimensional QFT, 1-dimensional QFT, <em>d</em>-dimensional QFT, supergeometry and field theories with fermions, and an introduction to 2-dimensional conformal field theory.
18.303 · Undergraduate · Fall 2006
This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green’s functions.
18.303 · Undergraduate · Fall 2014
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 introduce…
18.304 · Undergraduate · Spring 2015
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.
18.305 · Graduate · Fall 2004
Advanced Analytic Methods in Science and Engineering is a comprehensive treatment of the advanced methods of applied mathematics. It was designed to strengthen the mathematical abilities of graduate students and train them to think on their own.
18.306 · Graduate · Fall 2009
The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc.
18.307 · Graduate · Spring 2006
This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, …
18.310 · Undergraduate · Fall 2013
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.
18.311 · Undergraduate · Spring 2014
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 method…
18.314 · Undergraduate · Fall 2014
This course analyzes combinatorial problems and methods for their solution. Topics include: enumeration, generating functions, recurrence relations, construction of bijections, introduction to graph theory, network algorithms, and extremal combinatorics.
18.315 · Graduate · Spring 2005
This course serves as an introduction to major topics of modern enumerative and algebraic combinatorics with emphasis on partition identities, young tableaux bijections, spanning trees in graphs, and random generation of combinatorial objects. There is some discussion of various applications and connections to other fields.
18.315 · Graduate · Fall 2004
This is a graduate-level course in combinatorial theory. The content varies year to year, according to the interests of the instructor and the students. The topic of this course is hyperplane arrangements, including background material from the theory of posets and matroids.
18.318 · Graduate · Spring 2006
The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.
18.319 · Graduate · Fall 2005
This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems.
18.325 · Graduate · Fall 2015
This class covers the mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography.
18.330 · Undergraduate · Spring 2004
This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations and direct and iterative methods in linear algebra.
18.330 · Undergraduate · Spring 2012
This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra.
18.336 · Graduate · Spring 2009
This graduate-level course is an advanced introduction to applications and theory of numerical methods for solution of differential equations. In particular, the course focuses on physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods.
18.357 · Graduate · Fall 2010
This graduate-level course covers fluid systems dominated by the influence of interfacial tension. The roles of curvature pressure and Marangoni stress are elucidated in a variety of fluid systems. Particular attention is given to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics.
18.366 · Graduate · Fall 2006
This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.
18.369 · Graduate · Spring 2008
<p>Find out what solid-state physics has brought to Electromagnetism in the last 20 years. This course surveys the physics and mathematics of nanophotonics—electromagnetic waves in media structured on the scale of the wavelength.</p> <p>Topics include computational methods combined with high-level algebraic techniques borrowed from solid-state quantum mechanics: linear algebra and eigensystems, group theory, Bloch’s theorem and conservation laws, perturbation methods, and coupled-mode theories,…
18.385J · Graduate · Fall 2014
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.
18.385J · Graduate · Fall 2004
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.
18.408 · Graduate · Fall 2022
In this course, we will present the theory of Probabilistically Checkable Proofs (PCPs), and prove some fundamental consequences of it as well as more recent advances. More specifically, the first half of the course will be devoted to the (algebraic) proof of the basic PCP Theorem and basic relation to approximation problems. We will then move on to more advanced topics, such as hardness amplification, the long-code framework, the Unique-Games Conjecture and its implications, and the 2-to-2 Gam…
18.409 · Graduate · Spring 2002
This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.
18.409 · Graduate · Fall 2009
This course covers a collection of geometric techniques that apply broadly in modern algorithm design.
18.409 · Graduate · Spring 2015
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.
18.413 · Undergraduate · Spring 2004
This course introduces students to iterative decoding algorithms and the codes to which they are applied, including Turbo Codes, Low-Density Parity-Check Codes, and Serially-Concatenated Codes. The course will begin with an introduction to the fundamental problems of Coding Theory and their mathematical formulations. This will be followed by a study of Belief Propagation–the probabilistic heuristic which underlies iterative decoding algorithms. Belief Propagation will then be applied to the dec…
18.417 · Graduate · Fall 2004
This course introduces the basic computational methods used to understand the cell on a molecular level. It covers subjects such as the sequence alignment algorithms: dynamic programming, hashing, suffix trees, and Gibbs sampling. Furthermore, it focuses on computational approaches to: genetic and physical mapping; genome sequencing, assembly, and annotation; RNA expression and secondary structure; protein structure and folding; and molecular interactions and dynamics.
18.433 · Undergraduate · Fall 2003
Combinatorial Optimization provides a thorough treatment of linear programming and combinatorial optimization. Topics include network flow, matching theory, matroid optimization, and approximation algorithms for NP-hard problems.
18.440 · Undergraduate · Spring 2014
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.
18.445 · Graduate · Spring 2015
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.
18.465 · Graduate · Spring 2007
The main goal of this course is to study the generalization ability of a number of popular machine learning algorithms such as boosting, support vector machines and neural networks. Topics include Vapnik-Chervonenkis theory, concentration inequalities in product spaces, and other elements of empirical process theory.
18.465 · Graduate · Spring 2005
<p>This graduate-level course focuses on one-dimensional nonparametric statistics developed mainly from around 1945 and deals with order statistics and ranks, allowing very general distributions.</p> <p>For multidimensional nonparametric statistics, an early approach was to choose a fixed coordinate system and work with order statistics and ranks in each coordinate. A more modern method, to be followed in this course, is to look for rotationally or affine invariant procedures. These can be base…
18.600 · Undergraduate · Fall 2019
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.
18.642 · Undergraduate · Fall 2024
<p>This is an updated version of <em>18.S096 Topics in Mathematics with Applications in Finance</em> from Fall 2013. Please visit the <em>18.S096</em> site for more materials and lecture recordings. An investment game is also available as an additional learning resource.</p> <p>The purpose of the class is to expose undergraduate and graduate students to the mathematical concepts and techniques used in the financial industry. The course will consist of a set of mathematics lectures on topic…
18.650 · Undergraduate · Fall 2016
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.
18.650 (formerly 18.443) · Undergraduate · Spring 2015
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.
18.650 (formerly 18.443) · Undergraduate · Fall 2006
<p>This course offers a broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics include: hypothesis testing and estimation, confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, and correlation.</p> <p>OCW offers an earlier version of this course, from Fall 2003. This newer version focuses less on estimation theory and more on multiple linear regression models. In addition, a number of Mat…
18.655 · Graduate · Spring 2016
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.
18.657 · Graduate · Fall 2015
<p>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.</p> <p>You can read more about Prof. Rigollet’s work and courses on his website.</p>
18.700 · Undergraduate · Fall 2013
This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with <em>18.06 Linear Algebra</em>, more emphasis is placed on theory and proofs.
18.701 · Undergraduate · Fall 2010
This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.
18.702 · Undergraduate · Spring 2011
This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.
18.703 · Undergraduate · Spring 2013
This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics.
18.704 · Undergraduate · Fall 2004
This is a seminar for mathematics majors, where the students present the lectures. No prior experience giving lectures is necessary.
18.704 · Undergraduate · Fall 2008
In this undergraduate level seminar series, topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.
18.705 · Graduate · Fall 2008
In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.
18.706 · Graduate · Spring 2023
Noncommutative algebra studies algebraic phenomena involving multiplication for which commutativity law fails, such as product of matrices in linear algebra; such phenomena arise in various disciplines ranging from quantum physics to number theory.
18.712 · Undergraduate · Fall 2010
The goal of this course is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces.
18.725 · Graduate · Fall 2015
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 m…
18.725 · Graduate · Fall 2003
This course covers the fundamental notions and results about algebraic varieties over an algebraically closed field. It also analyzes the relations between complex algebraic varieties and complex analytic varieties.
18.726 · Graduate · Spring 2009
This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.
18.727 · Graduate · Spring 2006
The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.
18.727 · Graduate · Spring 2008
The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.
18.735 · Graduate · Fall 2009
Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems.
18.745 · Graduate · Fall 2020
This course is the first half of the year-long introductory graduate sequence 18.745/18.755 on Lie groups and Lie algebras. Topics include foundations of the theory of Lie groups and Lie algebras; theorems of Engel and Lie; the universal enveloping algebra, the Poincare-Birkhoff-Witt theorem; free Lie algebras; the Campbell-Hausdorff formula; classification and structure of finite dimensional complex simple Lie algebras; their finite dimensional representations; and the Weyl character formula.
18.755 · Graduate · Fall 2004
<p>This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry. The text for this class is <em>Differential Geometry, Lie Groups and Symmetric Spaces</em> by Sigurdur Helgason (American Mathematical Society, 2001).</p> <p>Much of the course material is based on Chapter I (first half) and Chapter II of the text. The text however develops basic Riemannian Geometry, Complex Manifolds, as well as a detailed theory of Semisimple Lie G…
18.755 · Graduate · Spring 2024
This course is the second half of the year-long introductory graduate sequence <em>Lie Groups and Lie Algebras I</em> <em>& Lie Groups and Lie Algebras II</em>. Topics include classical groups, Haar measure on locally compact groups, the representation-theoretic understanding of the hydrogen atom, representations of compact (in particular, finite) groups, the Peter-Weyl theorem with proof, maximal tori, Cartan and Iwasawa decompositions, classification of real reductive Lie groups, topology…
18.757 · Graduate · Fall 2023
The goal of this course is to give an introduction to the representation theory of compact and non-compact Lie groups. It will rely on some material from <em>18.745 Lie Groups and Lie Algebras I</em> and <em>18.755 Lie Groups and Lie Algebras II</em>, but full familiarity with this material is not required. Topics include continuous representations; algebras of measures on a Lie group; K-finite, smooth, and analytic vectors; admissible representations; unitary representations; the Harish-Chandr…
18.769 · Graduate · Spring 2009
This course will give a detailed introduction to the theory of tensor categories and review some of its connections to other subjects (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.
18.781 · Undergraduate · Spring 2012
This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions.
18.782 · Undergraduate · Fall 2013
This course is an introduction to arithmetic geometry, a subject that lies at the intersection of algebraic geometry and number theory. Its primary motivation is the study of classical Diophantine problems from the modern perspective of algebraic geometry.
18.783 · Undergraduate · Spring 2021
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.
18.783 · Undergraduate · Fall 2025
This course offers a computationally focused introduction to elliptic curves, emphasizing their deep connections to number theory and cryptography. Core topics include point-counting, isogenies, pairings, and the theory of complex multiplication. Throughout the course, these concepts are applied to practical problems such as integer factorization, primality proving, and elliptic curve cryptography.
18.785 · Graduate · Fall 2021
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.
18.786 · Graduate · Spring 2006
This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet’s units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets.
18.786 · Graduate · Spring 2010
This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet’s units theorem, local fields, ramification, discriminants.
18.786 · Graduate · Spring 2016
This course is the continuation of <em>18.785 Number Theory I</em>. 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.
18.821 · Undergraduate · Spring 2013
<p><em>Project Laboratory in Mathematics</em> is a course designed to give students a sense of what it’s like to do mathematical research. In teams, students explore puzzling and complex mathematical situations, search for regularities, and attempt to explain them mathematically. Students share their results through professional-style papers and presentations.</p> <p>This course site was created specifically for educators interested in offering students a taste of mathematical research. This si…
18.900 · Undergraduate · Spring 2023
This course introduces students to selected aspects of geometry and topology, using concepts that can be visualized easily. We mix geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). The course is suitable for students with no prior exposure to differential geometry or topology. Think of it as a moderate hike, overlooking various parts of the geometry and topology landscape. Bits are flat, bits are uphill, there are occasional rock…
18.901 · Undergraduate · Fall 2004
This course introduces topology, covering topics fundamental to modern analysis and geometry. It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group.
18.904 · Undergraduate · Spring 2011
This course is a seminar in topology. The main mathematical goal is to learn about the fundamental group, homology and cohomology. The main non-mathematical goal is to obtain experience giving math talks.
18.905 · Graduate · Fall 2016
This is a course on the singular homology of topological spaces. Topics include: Singular homology, CW complexes, Homological algebra, Cohomology, and Poincare duality.
18.906 · Graduate · Spring 2020
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.
18.915 · Graduate · Fall 2014
This is a literature seminar with a focus on classic papers in Algebraic Topology. It is named after the late MIT professor Daniel Kan. Each student gives one or two talks on each of three papers, chosen in consultation with the instructor, reads all the papers presented by other students, and writes reactions to the papers. This course is useful not only to students pursuing algebraic topology as a field of study, but also to those interested in symplectic geometry, representation theory, and …
18.917 · Graduate · Fall 2007
The goal of this course is to describe some of the tools which enter into the proof of Sullivan’s conjecture.
18.950 · Undergraduate · Fall 2008
This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.
18.965 · Graduate · Fall 2004
Geometry of Manifolds analyzes topics such as the differentiable manifolds and vector fields and forms. It also makes an introduction to Lie groups, the de Rham theorem, and Riemannian manifolds.
18.966 · Graduate · Spring 2007
This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.
18.969 · Graduate · Fall 2006
This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is i…
18.969 · Graduate · Spring 2009
This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor.
18.994 · Undergraduate · Fall 2004
In this course, students take turns in giving lectures. For the most part, the lectures are based on Robert Osserman’s classic book <em>A Survey of Minimal Surfaces</em>, Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.
18.996 · Graduate · Spring 2002
<p>We will discuss numerous research problems that are related to the internet. Sample topics include: routing algorithms such as BGP, communication protocols such as TCP, algorithms for intelligently selecting a resource in the face of uncertainty, bandwidth sensing tools, load balancing algorithms, streaming protocols, determining the structure of the internet, cost optimization, DNS-related problems, visualization, and large-scale data processing. The seminar is intended for students who are…
18.996A · Graduate · Spring 2004
This is an advanced topics course in model theory whose main theme is simple theories. We treat simple theories in the framework of compact abstract theories, which is more general than that of first order theories. We cover the basic properties of independence (i.e., non-dividing) in simple theories, the characterization of simple theories by the existence of a notion of independence, and hyperimaginary canonical bases.
18.997 · Graduate · Spring 2004
In this graduate-level course, we will be covering advanced topics in combinatorial optimization. We will start with non-bipartite matchings and cover many results extending the fundamental results of matchings, flows and matroids. The emphasis is on the derivation of purely combinatorial results, including min-max relations, and not so much on the corresponding algorithmic questions of how to find such objects. The intended audience consists of Ph.D. students interested in optimization, combin…
18.A34 · Undergraduate · Fall 2018
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.
18.S66 · Undergraduate · Spring 2003
The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is <em>2<sup>n-1</sup></em>. We will be concerned primarily with <em>bijective proofs</em>, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical bac…
18.S096 · Undergraduate · Fall 2013
<p>This is the first version of <em>18.642 Topics in Mathematics with Applications in Finance</em> from Fall 2024. Please visit the <em>18.642</em> site for more materials and lecture recordings. An investment game is also available as an additional learning resource.</p> <p>The purpose of the class is to expose undergraduate and graduate students to the mathematical concepts and techniques used in the financial industry. Mathematics lectures are mixed with lectures illustrating the corres…
18.S096 · Undergraduate · Fall 2015
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.
18.S096 · Undergraduate · January IAP 2023
<p>We all know that calculus courses such as <em>18.01 Single Variable Calculus</em> and <em>18.02 Multivariable Calculus</em> cover univariate and vector calculus, respectively. Modern applications such as machine learning and large-scale optimization require the next big step, “matrix calculus” and calculus on arbitrary vector spaces.</p> <p>This class covers a coherent approach to matrix calculus showing techniques that allow you to think of a matrix holistically (not just as an array of sca…
18.S097 · Undergraduate · January IAP 2019
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.
18.S190 · Undergraduate · January IAP 2023
This course provides a basic introduction to metric spaces. It covers metrics, open and closed sets, continuous functions (in the topological sense), function spaces, completeness, and compactness.
18.S996 · Graduate · Spring 2013
The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common scientific models. The power of the language will be tested by its ability to penetrate into taken-for-granted ideas, either by exposing existing weaknesses or flaws in our understanding, or by highlighting hidden commonalities across scientific fields.
18.S997 · Undergraduate · Fall 2011
This course is intended to assist undergraduates with learning the basics of programming in general and programming MATLAB® in particular.
18.S997 · Graduate · Spring 2015
<p>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.</p> <p>You can read more about Prof. Rigollet’s work and courses on his website</p>
18.S997 · Graduate · Fall 2012
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.
RES.18-001 · Undergraduate · Fall 2023
First published in 1991 by Wellesley-Cambridge Press, this updated 3rd edition of the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor’s Manual and a student Study Guide.
RES.18-002 · Undergraduate · Spring 2008
<p>This course was offered as a non-credit program during the Independent Activities Period (IAP), January 2008. A more recent version is available as course <em>18.S997 Introduction To MATLAB Programming</em>, including video lectures.</p> <p>The course, intended for students with no programming experience, provides the foundations of programming in MATLAB®. Variables, arrays, conditional statements, loops, functions, and plots are explained. At the end of the course, students should be able t…
RES.18-003 · Undergraduate · Spring 2005
<p>This online textbook provides an overview of Calculus in clear, easy to understand language designed for the non-mathematician.</p> <p>Online Publication</p>
RES.18-004 · Undergraduate · Spring 2009
<p>“<em>Getting an education at MIT is like trying to drink from a firehose.</em>”</p> <p>— folk saying</p> <p><em>The Torch or The Firehose: A Guide to Section Teaching</em>, by MIT Mathematics Professor Arthur Mattuck, is a guide to recitation teaching at MIT. During a typical recitation section, a teaching assistant (TA) meets with a small group of students to review the most recent lecture, expand on the concepts, work through practice problems, and conduct a discussion with the students. W…
RES.18-005 · Undergraduate · Spring 2010
<p>Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject. The series is divided into three sections:</p> <p>Introduction</p> <ul> <li>Why Professor Strang created these videos</li> <li>How to use the materials</li> </ul> <p>Highlights of Calculus</p> <ul> <li>Five videos reviewing the key topics and ide…
RES.18-006 · Undergraduate · Fall 2010
<p>Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshman-level introductory calculus course. The series was first released in 1970 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time.</p> <p>About the Instructor</p> <p>Herb Gross has taught math as senior lecturer at MIT and was the founding math department chair at Bunker Hill Community College.…
RES.18-007 · Undergraduate · Fall 2011
<p>Calculus Revisited is a series of videos and related resources that covers the materials normally found in freshman- and sophomore-level introductory mathematics courses. <em>Multivariable Calculus</em> is the second course in the series, consisting of 26 videos, 4 Study Guides, and a set of Supplementary Notes. The series was first released in 1971 as a way for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for the first time.</p> …
RES.18-008 · Undergraduate · Fall 2011
<p>Calculus Revisited is a series of videos and related resources that covers the materials normally found in freshman- and sophomore-level introductory mathematics courses. <em>Complex Variables, Differential Equations, and Linear Algebra</em> is the third course in the series, consisting of 20 Videos, 3 Study Guides, and a set of Supplementary Notes. Students should have mastered the first two courses in the series (Single Variable Calculus and Multivariable Calculus) before taking …
RES.18-009 · Undergraduate · Fall 2015
<p><em>Learn Differential Equations: Up Close with</em> <em>_Gilbert Strang</em> and_ <em>Cleve Moler</em> is an in-depth series of videos about differential equations and the MATLAB® ODE suite. These videos are suitable for students and life-long learners to enjoy.</p> <p>About the Instructors</p> <p>Gilbert Strang is the MathWorks Professor of Mathematics at MIT. His research focuses on mathematical analysis, linear algebra and PDEs. He has written textbooks on linear algebra, computational s…
RES.18-010 · Undergraduate · Spring 2020
<p>This collection of videos presents Professor Strang’s updated vision of how linear algebra could be taught.</p> <p>It starts with six brief videos, recorded in 2020, containing many ideas and suggestions about the recommended order of topics in teaching and learning linear algebra. Topics include <em>A New Way to Start Linear Algebra</em>, <em>The Column Space of a Matrix,</em> <em>The Big Picture of Linear Algebra,</em> <em>Orthogonal Vectors,</em> <em>Eigenvalues and Eigenvectors,</em> and…
RES.18-011 · Non-Credit · Fall 2021
<p>Algebra I is the first semester of a year-long introduction to modern algebra. Algebra is a fundamental subject, used in many advanced math courses and with applications in computer science, chemistry, etc. The focus of this class is studying groups, linear algebra, and geometry in different forms.</p> <p>These notes, which were created by students in a recent on-campus 18.701 Algebra I class, are offered here to supplement the materials included in OCW’s version of 18.701. They have not bee…
RES.18-012 · Undergraduate · Spring 2022
<p>Algebra II is the second semester of a year-long introduction to modern algebra. The course focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.</p> <p>These notes, which were created by students in a recent on-campus 18.702 Algebra II class, are offered here to supplement the materials included in OCW’s version of 18.702. They have not been checked for accuracy by the…
RES.18-015 · Non-Credit · Spring 2024
The goal of these lectures is to provide an introduction to Fourier analysis. The first topic is Fourier series, in particular, the Gibbs phenomenon and the dependence of their convergence properties on the suitability method used. The second topic is the Fourier transform, first the <em>L</em>¹ theory and then, using Hermite functions, the <em>L</em>² theory. The third topic is L. Schwartz’s theory of tempered distributions. The fourth topic is the theory of weak convergence of probability mea…
RES.18-016 · Undergraduate · Fall 2024
<p>These lecture notes and exercises (with solutions) cover MIT’s multivariable calculus sequence as taught in Fall 2024.</p> <p>The course <em>18.02 Multivariable Calculus</em> is a General Institute Requirement (GIR); every MIT student must pass this class in order to graduate. The first third of the course is dedicated to briefly covering some basic linear algebra. The rest of the course covers the traditional multivariable calculus topics including vectors and matrices, partial derivatives,…