MATH 18:
Foundations for Calculus
This course develops and enriches fundamental skills in foundational math to prepare students for success in calculus and other courses at Stanford that rely on quantitative methods (such as in biology, chemistry, computer science, economics, engineering, and physics). The focus is on proficiency in topics prior to calculus, with an emphasis on both conceptual understanding and problem-solving experience. A primary goal is to help students to hone the mathematical skills necessary to transition to quantitative coursework at Stanford, including how to use understanding of ideas to improve study habits and to avoid excessive rote memorization. Prerequisites: You must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut
| Units: 2
MATH 19:
Calculus
Introduction to differential calculus of functions of one variable. Review of elementary functions (including exponentials and logarithms), limits, rates of change, the derivative and its properties, applications of the derivative. Prerequisites: periodic trigonometric functions, advanced algebra, and analysis of elementary functions (including exponentials and logarithms). You must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 19ACE:
Calculus, ACE
Additional problem solving session for Math 19 guided by a course assistant. Concurrent enrollment in Math 19 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win
| Units: 1
MATH 20:
Calculus
The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initial-value problems, exponential and logistic models, direction fields, and parametric curves. Prerequisite: Math 19 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 20ACE:
Calculus, ACE
Additional problem solving session for Math 20 guided by a course assistant. Concurrent enrollment in Math 20 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win, Spr
| Units: 1
MATH 21:
Calculus
This course addresses a variety of topics centered around the theme of "calculus with infinite processes", largely the content of BC-level AP Calculus that isn't in the AB-level syllabus. It is needed throughout probability and statistics at all levels, as well as to understand approximation procedures that arise in all quantitative fields (including economics and computer graphics). After an initial review of limit rules, the course goes on to discuss sequences of numbers and of functions, as well as limits "at infinity" for each (needed for any sensible discussion of long-term behavior of a numerical process, such as: iterative procedures and complexity in computer science, dynamic models throughout economics, and repeated trials with data in any field). Integration is discussed for rational functions (a loose end from Math 20) and especially (improper) integrals for unbounded functions and "to infinity": this shows up in contexts as diverse as escape velocity for a rocket, the present value of a perpetual yield asset, and important calculations in probability (including the famous "bell curve" and to understand why many statistical tests work as they do). The course then turns to infinite series (how to "sum" an infinite collection of numbers), some useful convergence and divergence rests for these, and the associated killer app: power series and their properties, as well as Taylor approximations, all of which provide the framework that underlies virtually all mathematical models used in any quantitative field. Prerequisite: Math 20 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr, Sum
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 21ACE:
Calculus, ACE
Additional problem solving session for Math 21 guided by a course assistant. Concurrent enrollment in Math 21 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win, Spr
| Units: 1
MATH 51:
Linear Algebra, Multivariable Calculus, and Modern Applications
This course provides unified coverage of linear algebra and multivariable differential calculus, and the free course e-text connects the material to many fields. Linear algebra in large dimensions underlies the scientific, data-driven, and computational tasks of the 21st century. The linear algebra portion includes orthogonality, linear independence, matrix algebra, and eigenvalues with applications such as least squares, linear regression, and Markov chains (relevant to population dynamics, molecular chemistry, and PageRank); the singular value decomposition (essential in image compression, topic modeling, and data-intensive work in many fields) is introduced in the final chapter of the e-text. The multivariable calculus portion includes unconstrained optimization via gradients and Hessians (used for energy minimization), constrained optimization (via Lagrange multipliers, crucial in economics), gradient descent and the multivariable Chain Rule (which underlie many machine learning algorithms, such as backpropagation), and Newton's method (an ingredient in GPS and robotics). The course emphasizes computations alongside an intuitive understanding of key ideas. The widespread use of computers makes it important for users of math to understand concepts: novel users of quantitative tools in the future will be those who understand ideas and how they fit with examples and applications. This is the only course at Stanford whose syllabus includes nearly all the math background for CS 229, which is why CS 229 and CS 230 specifically recommend it (or other courses resting on it). For frequently asked questions about the differences between Math 51 and CME 100, see the FAQ on the placement page on the Math Department website. Prerequisite: Math 21 or equivalent (e.g. 5 on the AP Calculus BC test or suitable score on certain international exams: https://studentservices.stanford.edu/my-academics/earn-my-degree/undergraduate-degree-progress/test-transfer-credit/external-test-2). If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr, Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 51A:
Linear Algebra, Multivariable Calculus, and Modern Applications, ACE
Students attend one of the regular MATH 51 lectures with a longer discussion section of four hours per week instead of two. Active mode: students in small groups discuss and work on problems from a worksheet distributed 2 or 3 days in advance, with a TA providing guidance and answering questions. Application required: https://forms.gle/ruykWBk6zJMgXRB49
Last offered: Spring 2022
| Units: 6
| UG Reqs: GER:DB-Math, WAY-FR
MATH 51ACE:
Linear Algebra, Multivariable Calculus, and Modern Applications, ACE
Additional problem solving session for Math 51 guided by a course assistant. Concurrent enrollment in Math 51 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win, Spr
| Units: 1
MATH 52:
Integral Calculus of Several Variables
Iterated integrals, line and surface integrals, vector analysis with applications to vector potentials and conservative vector fields, physical interpretations. Divergence theorem and the theorems of Green, Gauss, and Stokes. Prerequisite: Math 21 and Math 51 or equivalents.
Terms: Win, Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 52ACE:
Integral Calculus of Several Variables, ACE
Additional problem solving session for Math 52 guided by a course assistant. Concurrent enrollment in Math 52 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Last offered: Winter 2023
| Units: 1
MATH 53:
Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications
Ordinary differential equations and initial value problems, linear systems of such equations with an emphasis on second-order constant-coefficient equations, stability analysis for non-linear systems (including phase portraits and the role of eigenvalues), and numerical methods. Partial differential equations and boundary-value problems, Fourier series and initial conditions, and Fourier transform for non-periodic phenomena. Throughout the development we harness insights from linear algebra, and software widgets are used to explore course topics on a computer (no coding background is needed). The free e-text provides motivation from applications across a wide array of fields (biology, chemistry, computer science, economics, engineering, and physics) described in a manner not requiring any area-specific expertise, and it has an appendix on Laplace transforms with many worked examples as a complement to the Fourier transform in the main text. Prerequisite: Math 21 and Math 51, or equivalents.
Terms: Aut, Win, Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 53ACE:
Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications, ACE
Additional problem solving session for Math 53 guided by a course assistant. Concurrent enrollment in Math 53 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Last offered: Winter 2023
| Units: 1
MATH 56:
Proofs and Modern Mathematics
How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but at a pace and level of abstraction not as intense as Math 61CM/DM, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the course develops some of the main results in that material systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. This course may be paired with Math 51; though that course is not a pre- or co-requisite.
Terms: Aut, Win
| Units: 4
| UG Reqs: WAY-FR
MATH 61CM:
Modern Mathematics: Continuous Methods
This is the first part of a theoretical (i.e., proof-based) sequence in multivariable calculus and linear algebra, providing a unified treatment of these topics. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space as local graphs, integration on Euclidean space, and many examples. The linear algebra content is covered jointly with Math 61DM. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor. This series provides the necessary mathematical background for majors in all Computer Science, Data Science, Economics, Mathematics, Natural Sciences, and Engineering.
Terms: Aut
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 61DM:
Modern Mathematics: Discrete Methods
This is the first part of a theoretical (i.e., proof-based) sequence in discrete mathematics and linear algebra. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. The linear algebra content is covered jointly with Math 61CM. Students should have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.This series provides the necessary mathematical background for majors in Computer Science, Data Science, Economics, Mathematics, and most Natural Sciences and some Engineering majors. Those who plan to major in Physics or in Engineering, majors requiring Math 50's beyond Math 51, are recommended to take Math 60CM.
Terms: Aut
| Units: 5
| UG Reqs: WAY-FR
MATH 62CM:
Modern Mathematics: Continuous Methods
A proof-based introduction to manifolds and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisites: Math 61CM.
Terms: Win
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 62DM:
Modern Mathematics: Discrete Methods
This is the second part of a theoretical (proof-based) sequence with a focus on discrete mathematics. The central objects discussed in this course are finite fields. These are beautiful structures in themselves, and very useful in large areas of modern mathematics, and beyond. Our goal will be to construct these, understand their structure, and along the way discuss unexpected applications in combinatorics and number theory. Highlights of the course include a complete proof of a polynomial time algorithm for primality testing, Sidon sets and finite projective planes, and an understanding of a lovely magic trick due to Persi Diaconis. Prerequisite: Math 61DM or 61CM.
Terms: Win
| Units: 5
| UG Reqs: WAY-FR
MATH 63CM:
Modern Mathematics: Continuous Methods
A proof-based course on ordinary differential equations. Topics include the inverse and implicit function theorems, implicitly-defined submanifolds of Euclidean space, linear systems of differential equations and necessary tools from linear algebra, stability and asymptotic properties of solutions to linear systems, existence and uniqueness theorems for nonlinear differential equations, behavior of solutions near an equilibrium point, and Sturm-Liouville theory. Prerequisite: Math 61CM or Math 61DM.
Terms: Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 63DM:
Modern Mathematics: Discrete Methods
Third part of a proof-based sequence in discrete mathematics, though independent of the second part (62DM). The first half of the quarter gives a brisk-paced coverage of probability and random processes with an intensive use of generating functions and a rich variety of applications. The second half treats entropy, Bayesian inference, Markov chains, game theory, probabilistic methods in solving non-probabilistic problems. We use continuous calculus, e.g. in handling the Gaussian, but anything needed will be reviewed in a self-contained manner. Prerequisite: Math 61DM or 61CM
Terms: Spr
| Units: 5
| UG Reqs: WAY-FR
MATH 75SI:
Learn to Give a Math Talk
This class focuses on preparing and presenting math talks. The first few weeks introduce the main skills, learning from panels, guest lectures, and discussions. In the remaining weeks, participants practice presenting math topics to classmates that are selected through class votes, and feedback is provided for improving presentations. The material in the topics is new to the students, but not too advanced, so participants learn some math also. The course is aimed at students who have taken at least one proof-based math class, and preference is given to students who have less experience presenting math to others. Application to enroll is necessary: https://forms.gle/7GSA2Vh7xZhm5AUaA
Terms: Win
| Units: 2
MATH 77Q:
Probability and gambling
One of the earliest probabilistic discussions was in 1654 between two French mathematicians, Pascal and Fermat, on the following question: 'If a pair of six-sided dice is thrown 24 times, should you bet even money on the occurrence of at least one `double six'?' Shortly after the discussion, Huygens, a Dutch scientist, published De Ratiociniis in Ludo Aleae (The Value of all Chances in Games of Fortune) in 1657; this is considered to be the first treatise on probability. Due to the inherent appeal of games of chance, probability theory soon became popular, and the subject underwent rapid development in the 18th century with contributions from mathematical giants, such as Bernoulli, de Moivre, and Laplace. There are two fairly different lines of thought associated with applications of probability: the solution of betting/gambling and the analysis of statistical data related to quantitative subjects such as mortality tables and insurance rates. In this Introsem, we will discuss poker and other games of chance, such as daily fantasy sports, from the perspective of risk analysis. This Introsem does not require any programming knowledge, but some experience with Excel, MATLAB, R, and/or Python will enhance your experience in our discussion of daily fantasy sports. Students should be familiar with all material from Math 51. No prior knowledge of sports and games of chance is required. Students must apply through the IntroSem application process.
Terms: Win, Spr
| Units: 3
| UG Reqs: WAY-FR
MATH 87Q:
Mathematics of Knots, Braids, Links, and Tangles
Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics.
Terms: Spr
| Units: 3
| UG Reqs: WAY-FR
MATH 101:
Math Discovery Lab
MDL is a discovery-based project course in mathematics. Students work independently in small groups to explore open-ended mathematical problems and discover original mathematics. Students formulate conjectures and hypotheses; test predictions by computation, simulation, or pure thought; and present their results to classmates. WIM. Admission is by application. Motivated students with a mathematical background of at least Math 51 or 61CM or 61DM (or equivalent) are encouraged to apply. Please visit https://mathematics.stanford.edu/math-101 for more information about the course and application.
Terms: Win
| Units: 4
| UG Reqs: WAY-FR
MATH 104:
Applied Matrix Theory
Linear algebra for applications in science and engineering. The course introduces the key mathematical ideas in matrix theory, which are used in modern methods of data analysis, scientific computing, optimization, and nearly all quantitative fields of science and engineering. While the choice of topics is motivated by their use in various disciplines, the course will emphasize the theoretical and conceptual underpinnings of this subject. Topics include orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares methods, and algorithms for solving systems of linear equations; applications include clustering, principal component analysis and dimensionality reduction, regression. MATH 113 offers a more theoretical treatment of linear algebra. MATH 104 and ENGR 108 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of ENGR 108 is on a few linear algebra concepts, and many applications. Prerequisites: MATH 51 and programming experience on par with CS 106A.
Terms: Aut, Win, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 106:
Functions of a Complex Variable
Complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. (Math 116 offers a more theoretical treatment.) Prerequisite: 52.
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 107:
Graph Theory
An introductory course in graph theory establishing fundamental concepts and results in variety of topics. Topics include: basic notions, connectivity, cycles, matchings, planar graphs, graph coloring, matrix-tree theorem, conditions for hamiltonicity, Kuratowski's theorem, Ramsey and Turan-type theorem. Prerequisites: 51 or equivalent and some familiarity with proofs is required.
Terms: Win
| Units: 4
| UG Reqs: WAY-FR
MATH 108:
Introduction to Combinatorics and Its Applications
Topics: graphs, trees (Cayley's Theorem, application to phylogony), eigenvalues, basic enumeration (permutations, Stirling and Bell numbers), recurrences, generating functions, basic asymptotics. Prerequisites: 51 or equivalent.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 109:
Groups and Symmetry
Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. Honors math majors and students who intend to do graduate work in mathematics should take 120. WIM. Prerequisite: Math 51.
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 110:
Number Theory for Cryptography
Number theory and its applications to modern cryptography. Topics include: congruences, primality testing and factorization, public key cryptography, and elliptic curves, emphasizing algorithms. Includes an introduction to proof-writing. This course develops math background useful in CS 255. WIM. Prerequisite: Math 51
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 113:
Linear Algebra and Matrix Theory
Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; dual space and dual basis; eigenvectors and eigenvalues; diagonalization. Includes an introduction to proof-writing. (Math 104 offers a more application-oriented treatment.) Prerequisites: Math 51
Terms: Aut, Win, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 115:
Functions of a Real Variable
The development of 1-dimensional real analysis (the logical framework for why calculus works): sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Includes introduction to proof-writing. Prerequisite: Math 51 or Math 56.
Terms: Aut, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 116:
Complex Analysis
Holomorphic and analytic functions, power series, Cauchy integral and Cauchy integral formula, meromorphic functions and differential forms, calculus of residues and applications, analytic continuation, conformal mappings, Riemann mapping theorem, Laurent series and conformal classification of annuli, harmonic functions and Dirichlet problem, introduction to Riemann surfaces, theory of elliptic functions and integrals. ( Math 106 offers a less theoretical treatment). Prerequisites: 51,52 and 171, or 61cm and 62cm.
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 117:
Advanced Complex Analysis
Review of holomorphic and meromorphic 1-forms, product development, Gamma-function and Riemann zeta-function, Fourier series and integrals, Fourier and Laplace transforms, differential geometric and analytic approaches to Riemann surfaces and conformal mappings, introduction to hyperbolic geometry, Laplace and d-bar equations and their solvability, the Uniformization Theorem, divisors and line bundles, Riemann-Roch theorem, Abel Jacobi theory. Prerequisite: Math 116.
Terms: Win
| Units: 4
| UG Reqs: WAY-FR
MATH 118:
Mathematics of Computation
Notions of analysis and algorithms central to modern scientific computing: continuous and discrete Fourier expansions, the fast Fourier transform, orthogonal polynomials, interpolation, quadrature, numerical differentiation, analysis and discretization of initial-value and boundary-value ODE, finite and spectral elements. Prerequisites: MATH 51 and 53.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 120:
Groups and Rings
Recommended for Mathematics majors and required of honors Mathematics majors. A more advanced treatment of group theory than in Math 109, also including ring theory. Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains. WIM course. Prerequisite: Math 51 and some prior proof-writing experience.
Terms: Aut, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 121:
Galois Theory
Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Prerequisite: Math 120 and (also recommended) 113.
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 122:
Modules and Group Representations
Modules over PID. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite: Math 113 and 120.
Terms: Spr
| Units: 4
| UG Reqs: WAY-FR
MATH 131P:
Partial Differential Equations
An introduction to techniques for solving PDE's. Topics include physical examples (such as the heat equation, wave equation, and Laplace's equation in 2 and 3 dimensions) and separation of variables with various coordinate systems to relate them to Sturm-Liouville problems using Fourier, Bessel, and Legendre series. Prerequisite: Math 53.
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 136:
Stochastic Processes (STATS 219)
Introduction to measure theory, Lp spaces and Hilbert spaces. Random variables, expectation, conditional expectation, conditional distribution. Uniform integrability, almost sure and Lp convergence. Stochastic processes: definition, stationarity, sample path continuity. Examples: random walk, Markov chains, Gaussian processes, Poisson processes, Martingales. Construction and basic properties of Brownian motion. Prerequisite: STATS 116 or MATH 151 or equivalent. Recommended: MATH 115 or equivalent. http://statweb.stanford.edu/~adembo/math-136/
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 137:
Mathematical Methods of Classical Mechanics
Newtonian mechanics. Lagrangian formalism. E. Noether's theorem. Oscillations. Rigid bodies. Introduction to symplectic geometry. Hamiltonian formalism. Legendre transform. Variational principles. Geometric optics. Introduction to the theory of integrable systems. Prerequisites: Math 53 and 147 or Math 62CM and 63CM.
Last offered: Spring 2019
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 138:
Celestial Mechanics
Mathematically rigorous introduction to the classical N-body problem: the motion of N particles evolving according to Newton's law. Topics include: the Kepler problem and its symmetries; other central force problems; conservation theorems; variational methods; Hamilton-Jacobi theory; the role of equilibrium points and stability; and symplectic methods. Prerequisites: 53, and 115 or 171.
Last offered: Autumn 2014
| Units: 4
| UG Reqs: GER:DB-Math
MATH 142:
Hyperbolic Geometry
An introductory course in hyperbolic geometry. Topics may include: different models of hyperbolic geometry, hyperbolic area and geodesics, Isometries and Mobius transformations, conformal maps, Fuchsian groups, Farey tessellation, hyperbolic structures on surfaces and three manifolds, limit sets. Prerequisites: some familiarity with the basic concepts of differential geometryand the topology of surfaces and manifolds is strongly recommended
Last offered: Autumn 2021
| Units: 4
| UG Reqs: WAY-FR
MATH 143:
Differential Geometry
Geometry of curves and surfaces in three-space. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces. Prerequisite: Math 52.
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 144:
Introduction to Topology and Geometry
Point set topology, including connectedness, compactness, countability and separation axioms. The inverse and implicit function theorems. Smooth manifolds, immersions and submersions, embedding theorems. Prerequisites: Math 61CM or both Math 113 and Math 171.
Terms: Win
| Units: 4
| UG Reqs: WAY-FR
MATH 145:
Algebraic Geometry
An introduction to the methods and concepts of algebraic geometry. The point of view and content will vary over time, but include: affine varieties, Hilbert basis theorem and Nullstellensatz, projective varieties, algebraic curves. Required: 120. Strongly recommended: additional mathematical maturity via further basic background with fields, point-set topology, or manifolds.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 147:
Differential Topology
Introduction to smooth methods in topology including tranvsersality, intersection number, fixed point theorems, as well as differential forms and integration. Prerequisites: Math 144 or equivalent.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 148:
Algebraic Topology
Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 151:
Introduction to Probability Theory
A proof-oriented development of basic probability theory. Counting; axioms of probability; conditioning and independence; expectation and variance; discrete and continuous random variables and distributions; joint distributions and dependence; Central Limit Theorem and laws of large numbers. CS majors can petition to use Math 151 in place of CS 109, provided they expect to take either CS 228 or CS 229 as well. Prerequisite: Math 61CM, or Math 52 and either Math 56 or Math 115 (or equivalent).
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 152:
Elementary Theory of Numbers
Euclid's algorithm, fundamental theorems on divisibility; prime numbers; congruence of numbers; theorems of Fermat, Euler, Wilson; congruences of first and higher degrees; quadratic residues; introduction to the theory of binary quadratic forms; quadratic reciprocity; partitions. Prerequisite: Math 51 and proof-writing experiences (e.g., Math 56).
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 154:
Algebraic Number Theory
Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations. Prerequisites: 120 and 121, especially modules over principal ideal domains and Galois theory of finite fields.
Last offered: Spring 2023
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 155:
Analytic Number Theory
Introduction to Dirichlet series and Dirichlet characters, Poisson summation, Gauss sums, analytic continuation for Dirichlet L-functions, applications to prime numbers (e.g., prime number theorem, Dirichlet's theorem). Prerequisites: Complex analysis (Math 106 or 116), Math 152 (or comparable familiarity with the Euclidean algorithm, multiplicative group modulo n, and quadratic reciprocity), and experience with basic analysis arguments.
Terms: Aut
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 158:
Probability and Stochastic Differential Equations for Applications (CME 298)
Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with some applications in science and/or engineering. Prerequisites: Math 53 and introductory probability (such as Stats 116 or Math 151).
Terms: Spr
| Units: 4
| UG Reqs: WAY-FR
MATH 159:
Discrete Probabilistic Methods
Modern discrete probabilistic methods suitable for analyzing discrete structures of the type arising in number theory, graph theory, combinatorics, computer science, information theory and molecular sequence analysis. Prerequisite: STATS 116/MATH 151 or equivalent. Typically in alternating years.
Terms: Aut
| Units: 4
| UG Reqs: WAY-FR
MATH 161:
Set Theory
Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations. The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents. Well-orderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. Prerequisite: Math 56 or comfort with proof-writing.
Last offered: Autumn 2022
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 171:
Fundamental Concepts of Analysis
Recommended for Mathematics majors and required of honors Mathematics majors. A more advanced and general version of Math 115, introducing and using metric spaces. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: Math 61CM, or 61DM, or Math 51 and Math 115. WIM
Terms: Aut, Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 172:
Lebesgue Integration and Fourier Analysis
Similar to 205A, but for undergraduate Math majors and graduate students in other disciplines. Topics include Lebesgue measure on Euclidean space, Lebesgue integration, L^p spaces, the Fourier transform, the Hardy-Littlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 173:
Theory of Partial Differential Equations
A rigorous introduction to PDE accessible to advanced undergraduates. Elliptic, parabolic, and hyperbolic equations in many space dimensions including basic properties of solutions such as maximum principles, causality, and conservation laws. Methods include the Fourier transform as well as more classical methods. The Lebesgue integral will be used throughout, but a summary of its properties will be provided to make the course accessible to students who have not had 172 or 205A. In years when Math 173 is not offered, Math 220 is a recommended alternative (with similar content but a different emphasis). Prerequisite: 171 or equivalent.
Terms: Spr
| Units: 4
| UG Reqs: WAY-FR
MATH 175:
Elementary Functional Analysis
Linear operators on Hilbert space. Spectral theory of compact operators; applications to integral equations. Elements of Banach space theory. Prerequisite: 115 or 171.
Terms: Win
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 177:
Geometric Methods in the Theory of Ordinary Differential Equations
Hamiltonian systems and their geometry. First order PDE and Hamilton-Jacobi equation. Structural stability and hyperbolic dynamical systems. Completely integrable systems. Perturbation theory.
Last offered: Spring 2018
| Units: 4
| UG Reqs: WAY-FR
MATH 193:
Polya Problem Solving Seminar
Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Students present solutions to the class. Open to anyone with an interest in mathematics.
Terms: Aut
| Units: 1
| Repeatable
4 times
(up to 4 units total)
MATH 193X:
Polya Problem Solving Seminar
Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Open to anyone with an interest in mathematics. Taught concurrently with Math 193, but students who sign up for Math 193X are given homework and exams. Students cannot enroll in both 193 and 193X during the same quarter. Math majors may use Math 193X towards their elective requirements.
Terms: Aut
| Units: 3
MATH 197:
Senior Honors Thesis
Honors math major working on senior honors thesis under an approved advisor carries out research and reading. Satisfactory written account of progress achieved during term must be submitted to advisor before term ends. May be repeated 3 times for a max of 9 units. Contact department student services specialist to enroll.
Terms: Aut, Win, Spr, Sum
| Units: 1-6
| Repeatable
3 times
(up to 9 units total)
Instructors: ;
Abouzaid, M. (PI);
Barrett, C. (PI);
Borga, J. (PI);
Charikar, M. (PI);
Chodosh, O. (PI);
Conrad, B. (PI);
Eliashberg, Y. (PI);
Icard, T. (PI);
Linderman, S. (PI);
Manolescu, C. (PI);
Montanari, A. (PI);
Reingold, O. (PI);
Rotskoff, G. (PI);
Ryzhik, L. (PI);
Schramm, T. (PI);
Schwarz, K. (PI);
Soundararajan, K. (PI);
Vondrak, J. (PI)
MATH 198:
Practical Training
Only for undergraduate students majoring in mathematics. Students obtain employment in a relevant industrial or research activity to enhance their professional experience. Students submit a concise report detailing work activities, problems worked on, and key results. May be repeated for credit up to 3 units. Prerequisite: qualified offer of employment and consent of department. Prior approval by Math Department is required; you must contact the Math Department's Student Services staff for instructions before being granted permission to enroll.
Terms: Sum
| Units: 1
| Repeatable
3 times
(up to 3 units total)
MATH 199:
Reading Topics
For Math majors only. Undergraduates pursue a reading program under the direction of a Math faculty member; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for Math majors. Departmental approval required; please contact the Student Services Specialist for the enrollment proposal form at least 2 weeks before the final study list deadline. May be repeated for credit. Enrollment beyond a third section requires additional approval.
Terms: Aut, Win, Spr, Sum
| Units: 3
| Repeatable
for credit
MATH 205A:
Real Analysis
Basic measure theory and the theory of Lebesgue integration. Prerequisite: Math 171. Math 172 is also recommended.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
MATH 205B:
Real Analysis
Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: 171 and 205A or equivalent. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
MATH 205C:
Topics in Harmonic Analysis
Topics of contemporary interest in Fourier, harmonic, and/or microlocal analysis. Prerequisite: Math 205B.NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 210A:
Modern Algebra I
Basic commutative ring and module theory, tensor algebra, homological constructions, linear and multilinear algebra, canonical forms and Jordan decomposition. Prerequisite: Math 121. It is also recommended to have taken at least one of Math 122, Math 145, or Math 154.nNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
MATH 210B:
Modern Algebra II
Continuation of 210A. Topics in field theory, commutative algebra, algebraic geometry, and finite group representations. Prerequisites: 210A, and 121 or equivalent.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
MATH 210C:
Lie Theory
Topics in Lie groups, Lie algebras, and/or representation theory. Prerequisite: Math 210A and familiarity with the basics of finite group representations. When the course is on Lie groups, familiarity with tangent spaces and integration on manifolds is assumed. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 215A:
Algebraic Topology
Topics: fundamental group and covering spaces, basics of homotopy theory, homology and cohomology (simplicial, singular, cellular), products, introduction to topological manifolds, orientations, Poincare duality. Prerequisites: 120 and 144.nNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
MATH 215B:
Differential Topology
Topics: Basics of differentiable manifolds (tangent spaces, vector fields, tensor fields, differential forms), embeddings, tubular neighborhoods, integration and Stokes' Theorem, deRham cohomology, intersection theory via Poincare duality, Morse theory. Prerequisite: 215ANOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
MATH 215C:
Differential Geometry
This course will be an introduction to Riemannian Geometry. Topics will include the Levi-Civita connection, Riemann curvature tensor, Ricci and scalar curvature, geodesics, parallel transport, completeness, geodesics and Jacobi fields, and comparison techniques. Prerequisites 147 or 215B. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
MATH 216A:
Introduction to Algebraic Geometry
Algebraic varieties, and introduction to schemes, morphisms, sheaves, and the functorial viewpoint. May be repeated for credit. Prerequisites: 210AB or equivalent. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 216B:
Introduction to Algebraic Geometry
Continuation of 216A. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
| Repeatable
for credit
MATH 216C:
Introduction to Algebraic Geometry
Continuation of 216B. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 217C:
Complex Differential Geometry
Complex structures, almost complex manifolds and integrability, Hermitian and Kahler metrics, connections on complex vector bundles, Chern classes and Chern-Weil theory, Hodge and Dolbeault theory, vanishing theorems, Calabi-Yau manifolds, deformation theory.
Last offered: Winter 2015
| Units: 3
| Repeatable
for credit
MATH 220A:
Partial Differential Equations of Applied Mathematics (CME 303)
Introduction to partial differential equations: basic properties of elliptic, parabolic, and hyperbolic equations; Hamilton-Jacobi equations and applications to optimal control; stochastic modeling, forward and backward Kolmogorov equations; Fourier transform and Fourier series. Prerequisite: multivariable calculus, rigorous courses on basic real analysis and ordinary differential equations. Note: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
MATH 220B:
Computational Methods of Applied Mathematics (CME 306)
Numerical methods for solving elliptic, parabolic, and hyperbolic partial differential equations. Algorithms for gradient and Hamiltonian systems. Algorithms for stochastic differential equations and Monte Carlo methods. Algorithms for computational harmonic analysis. Prerequisites: advanced undergraduate level PDE and advanced undergraduate level numerical analysis. Note: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
MATH 221B:
Mathematical Methods of Imaging
This is a project based course where the first half is an introduction to imaging: coherent and incoherent, passive and active, migration, time-reversal and optimization-based imaging, with applications to ultrasound, radar, sonar and seismic imaging. The projects come from a close study of recent papers that use neural networks for imaging when large data sets are available and some special applications such as satellite imaging.
Terms: Spr
| Units: 3
MATH 228:
Stochastic Methods in Engineering (CME 308, MS&E 324)
The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering. Diffusion approximations, Brownian motion and an introduction to stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis.
Terms: Spr
| Units: 3
MATH 230A:
Theory of Probability I (STATS 310A)
Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: STATS 116, MATH 171.
Terms: Aut
| Units: 3
MATH 230B:
Theory of Probability II (STATS 310B)
Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,n(v) ergodic theory. http://statweb.stanford.edu/~adembo/stat-310b. Prerequisite: 310A or MATH 230A.
Terms: Win
| Units: 3
MATH 230C:
Theory of Probability III (STATS 310C)
Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Invariance principle, random walks, LIL and functional CLT. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Prerequisite: 310B or MATH 230B. http://statweb.stanford.edu/~adembo/stat-310c/
Terms: Spr
| Units: 3
MATH 231:
Mathematics and Statistics of Gambling (STATS 334)
Probability and statistics are founded on the study of games of chance. Nowadays, gambling (in casinos, sports and the Internet) is a huge business. This course addresses practical and theoretical aspects. Topics covered: mathematics of basic random phenomena (physics of coin tossing and roulette, analysis of various methods of shuffling cards), odds in popular games, card counting, optimal tournament play, practical problems of random number generation. Prerequisites: Statistics 116 and 200.
Last offered: Autumn 2020
| Units: 3
MATH 232:
Topics in Probability
A topics course in probability and related areas. The topic will be announced by the instructor. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 233A:
Topics in Combinatorics
A topics course in combinatorics and related areas. The topic will be announced by the instructor.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 233B:
Topics in Combinatorics
A topics course in combinatorics and related areas. The topic will be announced by the instructor.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2021
| Units: 3
| Repeatable
for credit
MATH 233C:
Topics in Combinatorics
A topics course in combinatorics and related areas. The topic will be announced by the instructor. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 234:
Large Deviations Theory (STATS 374)
Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Prerequisite: MATH 230A or STATS 310. Offered every 2-3 years. http://statweb.stanford.edu/~adembo/large-deviations/
Last offered: Spring 2019
| Units: 3
MATH 235:
Modern Markov Chains (STATS 318)
Tools for understanding Markov chains as they arise in applications. Random walk on graphs, reversible Markov chains, Metropolis algorithm, Gibbs sampler, hybrid Monte Carlo, auxiliary variables, hit and run, Swedson-Wong algorithms, geometric theory, Poincare-Nash-Cheeger-Log-Sobolov inequalities. Comparison techniques, coupling, stationary times, Harris recurrence, central limit theorems, and large deviations.
Terms: Win
| Units: 3
MATH 235A:
Topics in combinatorics
This advanced course in extremal combinatorics covers several major themes in the area. These include extremal combinatorics and Ramsey theory, the graph regularity method, and algebraic methods.
Last offered: Spring 2019
| Units: 3
| Repeatable
for credit
(up to 99 units total)
MATH 235B:
Modern Markov Chain Theory
This is a graduate-level course on the use and analysis of Markov chains. Emphasis is placed on explicit rates of convergence for chains used in applications to physics, biology, and statistics. Topics covered: basic constructions (metropolis, Gibbs sampler, data augmentation, hybrid Monte Carlo); spectral techniques (explicit diagonalization, Poincaré, and Cheeger bounds); functional inequalities (Nash, Sobolev, Log Sobolev); probabilistic techniques (coupling, stationary times, Harris recurrence). A variety of card shuffling processes will be studies. Central Limit and concentration.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2016
| Units: 3
| Repeatable
for credit
(up to 99 units total)
MATH 235C:
Topics in Markov Chains
Classical functional inequalities (Nash, Faber-Krahn, log-Sobolev inequalities), comparison of Dirichlet forms. Random walks and isoperimetry of amenable groups (with a focus on solvable groups). Entropy, harmonic functions, and Poisson boundary (following Kaimanovich-Vershik theory).
Last offered: Spring 2016
| Units: 3
| Repeatable
for credit
(up to 99 units total)
MATH 236:
Introduction to Stochastic Differential Equations
Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Introduction to stochastic control and Bayesian filtering. Prerequisite: Math 136 or equivalent and basic familiarity with parabolic partial differential equations. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such other courses taken.
Terms: Win
| Units: 3
MATH 237A:
Topics in Financial Math: Market microstructure and trading algorithms
Introduction to market microstructure theory, including optimal limit order and market trading models. Random matrix theory covariance models and their application to portfolio theory. Statistical arbitrage algorithms.
Last offered: Spring 2023
| Units: 3
| Repeatable
10 times
(up to 30 units total)
MATH 238:
Mathematical Finance (STATS 250)
Stochastic models of financial markets. Risk neutral pricing for derivatives, hedging strategies and management of risk. Multidimensional portfolio theory and introduction to statistical arbitrage. Prerequisite: Math 136 or equivalent. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as other courses taken.
Terms: Win
| Units: 3
MATH 243:
Functions of Several Complex Variables
Holomorphic functions in several variables, Hartogs phenomenon, d-bar complex, Cousin problem. Domains of holomorphy. Plurisubharmonic functions and pseudo-convexity. Stein manifolds. Coherent sheaves, Cartan Theorems A&B. Levi problem and its solution. Grauert's Oka principle. Prerequisites: MATH 215A and experience with manifolds. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2021
| Units: 3
| Repeatable
for credit
MATH 244:
Riemann Surfaces
Riemann surfaces and holomorphic maps, algebraic curves, maps to projective spaces. Calculus on Riemann surfaces. Elliptic functions and integrals. Riemann-Hurwitz formula. Riemann-Roch theorem, Abel-Jacobi map. Uniformization theorem. Hyperbolic surfaces. (Suitable for advanced undergraduates.) Prerequisites: MATH 106 or MATH 116, and familiarity with surfaces equivalent to MATH 143, MATH 146, or MATH 147.
Last offered: Autumn 2021
| Units: 3
| Repeatable
for credit
MATH 245A:
Topics in Algebraic Geometry
Topics of contemporary interest in algebraic geometry. May be repeated for credit.NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 245B:
Topics in Algebraic Geometry
Topics of contemporary interest in algebraic geometry. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2023
| Units: 3
| Repeatable
for credit
MATH 245C:
Topics in Algebraic Geometry
Topics of contemporary interest in algebraic geometry. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Spring 2023
| Units: 3
| Repeatable
for credit
MATH 249A:
Topics in number theory
Topics of contemporary interest in number theory. May be repeated for credit.NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Autumn 2022
| Units: 3
| Repeatable
for credit
MATH 249B:
Topics in Number Theory
Topics of contemporary interest in number theory. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
| Repeatable
for credit
MATH 249C:
Topics in Number Theory
Topics of contemporary interest in number theory. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 256A:
Partial Differential Equations
The theory of linear and nonlinear partial differential equations, beginning with linear theory involving use of Fourier transform and Sobolev spaces. Topics: Schauder and L2 estimates for elliptic and parabolic equations; De Giorgi-Nash-Moser theory for elliptic equations; nonlinear equations such as the minimal surface equation, geometric flow problems, and nonlinear hyperbolic equations.NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Spring 2023
| Units: 3
MATH 256B:
Partial Differential Equations
Continuation of 256A.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 257A:
Symplectic Geometry and Topology
Linear symplectic geometry and linear Hamiltonian systems. Symplectic manifolds and their Lagrangian submanifolds, local properties. Symplectic geometry and mechanics. Contact geometry and contact manifolds. Relations between symplectic and contact manifolds. Hamiltonian systems with symmetries. Momentum map and its properties. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2023
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 257B:
Symplectic Geometry and Topology
Continuation of 257A. May be repeated for credit.NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Spring 2022
| Units: 3
| Repeatable
for credit
MATH 257C:
Symplectic Geometry and Topology
Continuation of 257B. May be repeated for credit.
Last offered: Spring 2021
| Units: 3
MATH 258:
Topics in Geometric Analysis
Topics in Geometric Analysis. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 262:
Applied Fourier Analysis and Elements of Modern Signal Processing (CME 372)
Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Computational topics include fast Fourier transforms (FFT) and nonuniform FFTs. Applications include Fourier imaging (the theory of diffraction, computed tomography, and magnetic resonance imaging) and the theory of compressive sensing. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2021
| Units: 3
MATH 263A:
Topics in Representation Theory
Topics in Contemporary Representation Theory. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 263B:
Topics in Representation Theory
Topics in Contemporary Representation Theory. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
| Units: 3
| Repeatable
for credit
MATH 263C:
Topics in Representation Theory
Topics in Contemporary Representation Theory. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 269:
Topics in symplectic geometry
May be repeated for credit.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 270:
Geometry and Topology of Complex Manifolds
Complex manifolds, Kahler manifolds, curvature, Hodge theory, Lefschetz theorem, Kahler-Einstein equation, Hermitian-Einstein equations, deformation of complex structures. May be repeated for credit.
Last offered: Winter 2017
| Units: 3
| Repeatable
for credit
MATH 271:
The H-Principle
The language of jets. Thom transversality theorem. Holonomic approximation theorem. Applications: immersion theory and its generaliazations. Differential relations and Gromov's h-principle for open manifolds. Applications to symplectic geometry. Microflexibility. Mappings with simple singularities and their applications. Method of convex integration. Nash-Kuiper C^1-isometric embedding theorem.
Terms: Win
| Units: 3
MATH 272:
Topics in Partial Differential Equations
NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
| Repeatable
for credit
MATH 273:
Topics in Mathematical Physics (STATS 359)
Covers a list of topics in mathematical physics. The specific topics may vary from year to year, depending on the instructor's discretion. Background in graduate level probability theory and analysis is desirable.
Last offered: Autumn 2018
| Units: 3
| Repeatable
for credit
MATH 275A:
Topics in Applied Math I
Topics in Applied Mathematics I. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 275B:
Topics in Applied Math II
Topics in Applied Mathematics II. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win
| Units: 3
| Repeatable
for credit
MATH 275C:
Topics in Applied Mathematics III: The Mathematics of AI
This course introduces the mathematics knowledge involved in machine learning and artificial intelligence on two levels. In the first half of the quarter, we introduce math needed to understand machine learning practices, i.e. data, models, and algorithms. Topics include advanced notions in linear algebra, probability, statistics, and optimization theories. In the second half of the quarter, we focus on math used to study and analyze machine learning in scientific research. Topics include approximation theory, concentration inequalities, functional analysis, and optimization. This course focuses on the mathematical tools for studying machine learning, rather than implementations of machine learning methods. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Spring 2023
| Units: 3
| Repeatable
for credit
MATH 276:
Mathematical Problems in Machine Learning (STATS 375)
Mathematical tools to understand modern machine learning systems. Generalization in machine learning, the classical view: uniform convergence, Radamacher complexity. Generalization from stability. Implicit (algorithmic) regularization. Infinite-dimensional models: reproducing kernel Hilbert spaces. Random features approximations to kernel methods. Connections to neural networks, and neural tangent kernel. Nonparametric regression. Asymptotic behavior of wide neural networks. Properties of convolutionalnetworks. Prerequisites: EE364A or equivalent; Stat310A or equivalent.
Terms: Spr
| Units: 3
MATH 282A:
Low Dimensional Topology
The theory of surfaces and 3-manifolds. Curves on surfaces, the classification of diffeomorphisms of surfaces, and Teichmuller space. The mapping class group and the braid group. Knot theory, including knot invariants. Decomposition of 3-manifolds: triangulations, Heegaard splittings, Dehn surgery. Loop theorem, sphere theorem, incompressible surfaces. Geometric structures, particularly hyperbolic structures on surfaces and 3-manifolds. May be repeated for credit up to 6 total units. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Spring 2023
| Units: 3
| Repeatable
for credit
MATH 282B:
Homotopy Theory
Homotopy groups, fibrations, spectral sequences, simplicial methods, Dold-Thom theorem, models for loop spaces, homotopy limits and colimits, stable homotopy theory. May be repeated for credit up to 6 total units. Prerequisite: Math 215A. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2023
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 282C:
Fiber Bundles and Cobordism
Possible topics: principal bundles, vector bundles, classifying spaces. Connections on bundles, curvature. Topology of gauge groups and gauge equivalence classes of connections. Characteristic classes and K-theory, including Bott periodicity, algebraic K-theory, and indices of elliptic operators. Spectral sequences of Atiyah-Hirzebruch, Serre, and Adams. Cobordism theory, Pontryagin-Thom theorem, calculation of unoriented and complex cobordism. May be repeated for credit up to 6 total units.
Last offered: Spring 2018
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 283A:
Topics in Topology
Topics of contemporary interest in topology. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Autumn 2022
| Units: 3
| Repeatable
for credit
MATH 286:
Topics in Differential Geometry
Topics of contemporary interest in differential geometry. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Winter 2023
| Units: 3
| Repeatable
for credit
MATH 298:
Graduate Practical Training
Only for mathematics graduate students. Students obtain employment in a relevant industrial or research activity to enhance their professional experience. Students submit a concise report detailing work activities, problems worked on, and key results. May be repeated for credit up to 3 units. Prerequisite: qualified offer of employment and consent of department. Prior approval by Math Department is required; you must contact the Math Department's Student Services staff for instructions before being granted permission to enroll.
Terms: Aut, Win, Spr, Sum
| Units: 1
| Repeatable
for credit
MATH 355:
Graduate Teaching Seminar
Required of and limited to second year Mathematics graduate students.
Terms: Aut
| Units: 1
MATH 360:
Advanced Reading and Research
Terms: Aut, Win, Spr, Sum
| Units: 1-10
| Repeatable
for credit
Instructors: ;
Abouzaid, M. (PI);
Borga, J. (PI);
Bump, D. (PI);
Candes, E. (PI);
Carlsson, G. (PI);
Chatterjee, S. (PI);
Chodosh, O. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Eliashberg, Y. (PI);
Fox, J. (PI);
Ionel, E. (PI);
Kerckhoff, S. (PI);
Luk, J. (PI);
Malinnikova, E. (PI);
Manolescu, C. (PI);
Mazzeo, R. (PI);
Montanari, A. (PI);
Papanicolaou, G. (PI);
Ryzhik, L. (PI);
Soundararajan, K. (PI);
Taylor, R. (PI);
Tokieda, T. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Vondrak, J. (PI);
White, B. (PI);
Ying, L. (PI);
Zhu, X. (PI)
MATH 382:
Qualifying Examination Seminar
Terms: Win, Sum
| Units: 1-3
| Repeatable
for credit
MATH 802:
TGR Dissertation
Terms: Aut, Win, Spr, Sum
| Units: 0
| Repeatable
for credit
Instructors: ;
Abouzaid, M. (PI);
Bump, D. (PI);
Candes, E. (PI);
Carlsson, G. (PI);
Chatterjee, S. (PI);
Chodosh, O. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Eliashberg, Y. (PI);
Fox, J. (PI);
Ionel, E. (PI);
Kerckhoff, S. (PI);
Luk, J. (PI);
Malinnikova, E. (PI);
Manolescu, C. (PI);
Mazzeo, R. (PI);
Papanicolaou, G. (PI);
Ryzhik, L. (PI);
Soundararajan, K. (PI);
Taylor, R. (PI);
Tokieda, T. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Vondrak, J. (PI);
White, B. (PI);
Ying, L. (PI);
Zhu, X. (PI)
