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: trigonometry, 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) in order to register for this course.
Terms: Aut, Win, Sum

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors: ;
Cant, D. (PI);
Dore, D. (PI);
Kimport, S. (PI);
Ohrt, C. (PI);
Wang, Y. (PI);
Chaturvedi, S. (TA);
Chen, S. (TA);
Datta, I. (TA);
Helfer, J. (TA);
Kimport, S. (GP);
Lim, B. (TA);
Lolas, P. (TA);
Mackey, W. (TA);
Zou, J. (TA)
MATH 20:
Calculus
The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus, and the Mean Value Theorem for integrals. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initialvalue 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) in order to register for this course.
Terms: Aut, Win, Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors: ;
Kemeny, M. (PI);
Kimport, S. (PI);
Lucianovic, M. (PI);
Schaeffer, G. (PI);
Solis, P. (PI);
Chen, D. (TA);
Hui, Y. (TA);
Kimport, S. (GP);
Libkind, S. (TA);
Lucianovic, M. (GP);
McConnell, S. (TA);
Zhou, Z. (TA)
MATH 21:
Calculus
Review of limit rules. Sequences, functions, limits at infinity, and comparison of growth of functions. Review of integration rules, integrating rational functions, and improper integrals. Infinite series, special examples, convergence and divergence tests (limit comparison and alternating series tests). Power series and interval of convergence, Taylor polynomials, Taylor series and applications. 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) in order to register for this course.
Terms: Aut, Win, Spr

Units: 4

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 21A:
Calculus, ACE
Students attend MATH 21 lectures with different recitation sessions: two hours per week instead of one, emphasizing engineering applications. Prerequisite: application; see https://web.stanford.edu/dept/soe/osa/ace.fb
Terms: Aut, Win, Spr

Units: 5

Grading: Letter or Credit/No Credit
MATH 51:
Linear Algebra, Multivariable Calculus, and Modern Applications
This course provides unified coverage of linear algebra and multivariable differential calculus. It discusses applications connecting the material to many quantitative fields. Linear algebra in large dimensions underlies the scientific, datadriven, and computational tasks of the 21st century. The linear algebra portion of the course includes orthogonality, linear independence, matrix algebra, and eigenvalues as well as ubiquitious applications: least squares, linear regression, Markov chains (relevant to population dynamics, molecular chemistry, and PageRank), singular value decomposition (essential in image compression, topic modeling, and dataintensive work in the natural sciences), and more. The multivariable calculus material includes unconstrained optimization via gradients and Hessians (used for energy minimization in physics and chemistry), 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 (a crucial part of how GPS works). The course emphasizes computations alongside an intuitive understanding of key ideas, making students wellprepared for further study of mathematics and its applications to other fields. The widespread use of computers makes it more important, not less, for users of math to understand concepts: in all scientific fields, novel users of quantitative tools in the future will be those who understand ideas and how they fit with applications and examples. 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: 21, 42, or the math placement diagnostic (offered through the Math Department website) in order to register for this course.
Terms: Aut, Win, Spr, Sum

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors: ;
Church, T. (PI);
Conrad, B. (PI);
Lucianovic, M. (PI);
Mazzeo, R. (PI);
McConnell, S. (PI);
Ohrt, C. (PI);
Taylor, C. (PI);
Wieczorek, W. (PI);
Arana Herrera, F. (TA);
Conrad, B. (GP);
Lucianovic, M. (GP);
Nguyen, D. (TA);
Stanton, C. (TA);
Ungemach, W. (TA);
Wieczorek, W. (GP);
Zhang, S. (TA);
Zhou, Z. (TA);
Zou, J. (TA)
MATH 51A:
Linear Algebra, Multivariable Calculus, and Modern Applications, ACE
Students attend MATH 51 lectures with different recitation sessions: four hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see https://engineering.stanford.edu/studentsacademics/engineeringdiversityprograms/additionalcalculusengineersace
Terms: Aut, Win, Spr

Units: 6

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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: 51 or equivalents.
Terms: Aut, Win, Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 53:
Ordinary Differential Equations with Linear Algebra
Ordinary differential equations and initial value problems, systems of linear differential equations with constant coefficients, applications of secondorder equations to oscillations, matrix exponentials, Laplace transforms, stability of nonlinear systems and phase plane analysis, numerical methods. Prerequisite: 51 or equivalents.
Terms: Aut, Win, Spr, Sum

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 61CM:
Modern Mathematics: Continuous Methods
This is the first part of a theoretical (i.e., proofbased) 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, inverse and implicit function theorems, and many examples. The linear algebra content is covered jointly with Math 61DM. Students should know 1variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BClevel Advanced Placement calculus exam, or consent of the instructor.
Terms: Aut

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 61DM:
Modern Mathematics: Discrete Methods
This is the first part of a theoretical (i.e., proofbased) 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 BClevel Advanced Placement calculus exam, or consent of the instructor.nnThis sequence is not appropriate for students planning to major in natural sciences, economics, or engineering, but is suitable for majors in any other field (such as MCS ("data science"), computer science, and mathematics).
Terms: Aut

Units: 5

UG Reqs: WAYFR

Grading: Letter or Credit/No Credit
MATH 62CM:
Modern Mathematics: Continuous Methods
A continuation of themes from Math 61CM, centered around: manifolds, multivariable integration, and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space and an introduction to general manifolds (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisite: Math 61CM.
Terms: Win

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter (ABCD/NP)
MATH 62DM:
Modern Mathematics: Discrete Methods
This is the second part of a proofbased sequence in discrete mathematics. This course covers topics in elementary number theory, group theory, and discrete Fourier analysis. For example, we'll discuss the basic examples of abelian groups arising from congruences in elementary number theory, as well as the nonabelian symmetric group of permutations. Prerequisites: 61DM or 61CM.
Terms: Win

Units: 5

UG Reqs: WAYFR

Grading: Letter (ABCD/NP)
MATH 63CM:
Modern Mathematics: Continuous Methods
A proofbased course on ordinary differential equations, continuing themes from Math 61CM and Math 62CM. Topics include 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 with some applications to manifolds, behavior of solutions near an equilibrium point, and SturmLiouville theory. Prerequisites: Math 61CM and Math 62CM.
Terms: Spr

Units: 5

UG Reqs: GER:DBMath, WAYFR

Grading: Letter (ABCD/NP)
MATH 63DM:
Modern Mathematics: Discrete Methods
Third part of a proofbased sequence in discrete mathematics. This course covers several topics in probability (random variables, independence and correlation, concentration bounds, the central limit theorem) and topology (metric spaces, pointset topology, continuous maps, compactness, Brouwer's fixed point and the BorsukUlam theorem), with some applications in combinatorics. Prerequisites: 61DM or 61CM
Terms: Spr

Units: 5

UG Reqs: WAYFR

Grading: Letter (ABCD/NP)
MATH 83N:
Proofs and Modern Mathematics
How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higherlevel 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 onevariable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the seminar develops 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 algebraic notions such as groups, rings, and fields. This seminar may be paired with Math 51; though that course is not a pre or corequisite.
Terms: Aut

Units: 3

UG Reqs: WAYFR

Grading: Letter or Credit/No Credit
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: Win

Units: 3

UG Reqs: WAYFR

Grading: Letter (ABCD/NP)
MATH 104:
Applied Matrix Theory
Linear algebra for applications in science and engineering: orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, leastsquares, the condition number of a matrix, algorithms for solving linear systems. MATH 113 offers a more theoretical treatment of linear algebra. MATH 104 and EE 103/CME 103 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of EE 103 is on a few linear algebra concepts, and many applications. Prerequisites: MATH 51 and programming experience on par with CS 106.
Terms: Aut, Win, Spr, Sum

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
Instructors: ;
Kazeev, V. (PI);
Taylor, C. (PI);
Velcheva, K. (PI);
Ying, L. (PI);
Guijarro Ordonez, J. (TA);
Khoo, Y. (GP);
Liu, Y. (TA);
Luo, S. (TA);
Sloman, L. (TA);
Truong Vu, N. (TA);
Wang, G. (TA);
Wolf, A. (TA)
MATH 106:
Functions of a Complex Variable
Complex numbers, analytic functions, CauchyRiemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. (Math 116 offers a more theoretical treatment.) Prerequisite: 52.
Terms: Spr, Sum

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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, matrixtree theorem, conditions for hamiltonicity, Kuratowski's theorem, Ramsey and Turantype theorem. Prerequisites: 51 or equivalent and some familiarity with proofs is required.
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
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: Win

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 109:
Applied Group Theory
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.
Terms: Aut

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 110:
Applied Number Theory and Field Theory
Number theory and its applications to modern cryptography. Topics: congruences, finite fields, primality testing and factorization, public key cryptography, error correcting codes, and elliptic curves, emphasizing algorithms. WIM.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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; eigenvectors and eigenvalues; diagonalization. (Math 104 offers a more applicationoriented treatment.)
Terms: Aut, Win, Spr, Sum

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 114:
Introduction to Scientific Computing (CME 108)
Introduction to Scientific Computing Numerical computation for mathematical, computational, physical sciences and engineering: error analysis, floatingpoint arithmetic, nonlinear equations, numerical solution of systems of algebraic equations, banded matrices, least squares, unconstrained optimization, polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, truncation error, numerical stability for time dependent problems and stiffness. Implementation of numerical methods in MATLAB programming assignments. Prerequisites: MATH 51, 52, 53; prior programming experience (MATLAB or other language at level of CS 106A or higher).
Terms: Win, Sum

Units: 3

UG Reqs: GER:DBEngrAppSci, WAYAQR, WAYFR

Grading: Letter or Credit/No Credit
MATH 115:
Functions of a Real Variable
The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Honors math majors and students who intend to do graduate work in mathematics should take 171. Prerequisite: 21.
Terms: Aut, Spr, Sum

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 116:
Complex Analysis
Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. (Math 106 offers a less theoretical treatment.) Prerequisites: 52, and 115 or 171.
Terms: Aut

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
MATH 120:
Groups and Rings
Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 109 but altered content and more theoretical orientation. 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 nonPID. Unique factorization domains. WIM.
Terms: Aut, Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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 120. Also recommended: 113.
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
MATH 131P:
Partial Differential Equations
An introduction to PDE; particularly suitable for nonMath majors. Topics include physical examples of PDE's, method of characteristics, D'Alembert's formula, maximum principles, heat kernel, Duhamel's principle, separation of variables, Fourier series, Harmonic functions, Bessel functions, spherical harmonics. Students who have taken MATH 171 should consider taking MATH 173 rather than 131P. Prerequisite: 53.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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/math136/
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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: 51, 52, 53, or 61CM, 62CM, 63CM.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 143:
Differential Geometry
Geometry of curves and surfaces in threespace and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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, pointset topology, or manifolds.
Terms: Aut, alternate years, not given next year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 146:
Analysis on Manifolds
Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 62CM or 52 and familiarity with linear algebra and analysis arguments at the level of 113 and 115 respectively.
Terms: Aut

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 148:
Algebraic Topology
Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Terms: Win, alternate years, not given next year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 151:
Introduction to 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. Prerequisite: 52 or consent of instructor.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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.
Terms: Spr, alternate years, not given next year

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 158:
Basic Probability and Stochastic Processes with Engineering 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 emphasis on applications from economics, physics and engineering, such as filtering and control. Prerequisites: exposure to basic probability.
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
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: Spr

Units: 3

UG Reqs: WAYFR

Grading: Letter or Credit/No Credit
MATH 161:
Set Theory
Informal and axiomatic set theory: sets, relations, functions, and settheoretical operations. The ZermeloFraenkel axiom system and the special role of the axiom of choice and its various equivalents. Wellorderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. Prerequisite: students should be comfortable doing proofs.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 171:
Fundamental Concepts of Analysis
Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 115 but altered content and more theoretical orientation. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: 61CM or 61DM or 115 or consent of the instructor. WIM
Terms: Aut, Spr

Units: 3

UG Reqs: GER:DBMath, WAYFR

Grading: Letter or Credit/No Credit
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 HardyLittlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor.
Terms: Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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 for credit

Grading: Satisfactory/No Credit
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: 16

Repeatable for credit

Grading: Letter (ABCD/NP)
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 for credit

Grading: Satisfactory/No Credit
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 topics not in regular department course offerings. Credit can fulfill the elective requirement for math majors. May be repeated for credit. Undergraduates may take this course at most 3 times, only enroll in one section per quarter, and complete up to 9 units total. Please contact the student services specialist for the enrollment proposal form at least 2 weeks before enrollment for the quarter closes.
Terms: Aut, Win, Spr, Sum

Units: 13

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 205A:
Real Analysis
Basic measure theory and the theory of Lebesgue integration. Prerequisite: 171 or equivalent.
Terms: Aut

Units: 3

Grading: Letter or Credit/No Credit
MATH 205B:
Real Analysis
Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: 171 and 205A or equivalent.
Terms: Win

Units: 3

Grading: Letter or Credit/No 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: 122 or equivalent.
Terms: Aut

Units: 3

Grading: Letter or Credit/No Credit
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.
Terms: Win

Units: 3

Grading: Letter or Credit/No Credit
MATH 210C:
Lie Theory
Topics in Lie groups, Lie algebras, and/or representation theory. Prerequisite: math 210B. May be repeated for credit.
Terms: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No 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: 113, 120, and 171.
Terms: Aut

Units: 3

Grading: Letter or Credit/No Credit
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: 215A
Terms: Win

Units: 3

Grading: Letter or Credit/No Credit
MATH 215C:
Differential Geometry
This course will be an introduction to Riemannian Geometry. Topics will include the LeviCivita connection, Riemann curvature tensor, Ricci and scalar curvature, geodesics, parallel transport, completeness, geodesics and Jacobi fields, and comparison techniques. Prerequisites 146 or 215B
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
MATH 220:
Partial Differential Equations of Applied Mathematics (CME 303)
Firstorder partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: Basic coursework in multivariable calculus and ordinary differential equations, and some prior experience with a proofbased treatment of the material as in Math 171 or Math 61CM (formerly Math 51H).
Terms: Aut

Units: 3

Grading: Letter or Credit/No Credit
MATH 226:
Numerical Solution of Partial Differential Equations (CME 306)
Hyperbolic partial differential equations: stability, convergence and qualitative properties; nonlinear hyperbolic equations and systems; combined solution methods from elliptic, parabolic, and hyperbolic problems. Examples include: Burger's equation, Euler equations for compressible flow, NavierStokes equations for incompressible flow. Prerequisites: MATH 220 or CME 302.
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
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

Grading: Letter or Credit/No Credit
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, BorelCantelli 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: 24

Grading: Letter or Credit/No Credit
MATH 230B:
Theory of Probability II (STATS 310B)
Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 01 laws, RadonNikodym 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. Prerequisite: 310A or MATH 230A.
Terms: Win

Units: 23

Grading: Letter or Credit/No Credit
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/stat310c/
Terms: Spr

Units: 24

Grading: Letter or Credit/No Credit
MATH 233A:
Topics in Combinatorics
A topics course in combinatorics and related areas. The topic will be announced by the instructor.
Terms: Aut

Units: 3

Repeatable for credit

Grading: Letter or Credit/No 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 23 years. http://statweb.stanford.edu/~adembo/largedeviations/
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
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.
Terms: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
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. Prerequisite: 136 or equivalent and differential equations.
Terms: Win

Units: 3

Grading: Letter or Credit/No Credit
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.
Terms: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 238:
Mathematical Finance (STATS 250)
Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Corequisites: MATH 236 and 227 or equivalent.
Terms: Win

Units: 3

Grading: Letter or Credit/No Credit
MATH 245A:
Topics in Algebraic Geometry
Topics of contemporary interest in algebraic geometry. May be repeated for credit.
Terms: Aut

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 245B:
Topics in Algebraic Geometry
May be repeated for credit.
Terms: Win

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 249A:
Topics in number theory
Topics of contemporary interest in number theory. May be repeated for credit.
Terms: Aut

Units: 3

Repeatable for credit

Grading: Letter or Credit/No 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 GiorgiNashMoser theory for elliptic equations; nonlinear equations such as the minimal surface equation, geometric flow problems, and nonlinear hyperbolic equations.
Terms: Aut

Units: 3

Grading: Letter or Credit/No Credit
MATH 256B:
Partial Differential Equations
Continuation of 256A.
Terms: Win

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 257C:
Symplectic Geometry and Topology
Continuation of 257B. May be repeated for credit.
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
MATH 263A:
Algebraic Combinatorics and Symmetric Functions
Symmetric function theory unifies large parts of combinatorics. Theorems about permutations, partitions, and graphs now follow in a unified way. Topics: The usual bases (monomial, elementary, complete, and power sums). Schur functions. Representation theory of the symmetric group. LittlewoodRichardson rule, quasisymmetric functions, combinatorial Hopf algebras, introduction to Macdonald polynomials. Throughout, emphasis is placed on applications (e.g. to card shuffling and random matrix theory). Prerequisite: 210A and 210B, or equivalent.
Terms: Aut

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 263C:
Topics in Representation Theory
May be repeated for credit.
Terms: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 269:
Topics in symplectic geometry
May be repeated for credit.
Terms: Aut

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 272:
Topics in Partial Differential Equations
Terms: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No 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.
Terms: Aut

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 275:
Topics in Applied Mathematics: A World of Flows
The purpose of this course is to show beautiful surprises and instructive paradoxes in a maximal diversity of fluid phenomena, and to understand them with minimal models. Some deep currents will develop across multiple lectures. The prerequisites are fluency in the socalled `mathematical methods¿¿vector calculus, complex analysis, Fourier transform/series, ODEs, PDEs¿plus a willingness to wade into physics (classical more than quantum) at the advanced undergraduate level.
Terms: Win

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 282B:
Homotopy Theory
Homotopy groups, fibrations, spectral sequences, simplicial methods, DoldThom theorem, models for loop spaces, homotopy limits and colimits, stable homotopy theory. May be repeated for credit up to 6 total units.
Terms: Win

Units: 3

Repeatable for credit

Grading: Letter or Credit/No 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

Grading: Satisfactory/No Credit
MATH 305:
Applied mathematics through toys and magic
This course is a series of casestudies in doing applied mathematics on surprising phenomena we notice in daily life. Almost every class will show demos of these phenomena (toys and magic) and suggest open projects. The topics range over a great variety and cut across areas traditionally pigeonholed as physics, biology, engineering, computer science, mathematics ¿ but, instead of developing sophisticated mathematics on simple material, our aim is to extract simple mathematical understanding from sophisticated material which, at first, we may not yet know how to pigeonhole. In each class I will try to make the discussion selfcontained and to give everybody something to take home, regardless of the background.
Terms: Win

Units: 3

Grading: Letter or Credit/No Credit
MATH 355:
Graduate Teaching Seminar
Required of and limited to firstyear Mathematics graduate students.
Terms: Spr

Units: 1

Grading: Satisfactory/No Credit
MATH 360:
Advanced Reading and Research
Terms: Aut, Win, Spr, Sum

Units: 110

Repeatable for credit

Grading: Letter or Credit/No Credit
Instructors: ;
Bump, D. (PI);
Candes, E. (PI);
Carlsson, G. (PI);
Chatterjee, S. (PI);
Church, T. (PI);
Cohen, R. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Eliashberg, Y. (PI);
Fox, J. (PI);
Galatius, S. (PI);
Ionel, E. (PI);
Kerckhoff, S. (PI);
Li, J. (PI);
Luk, J. (PI);
Mazzeo, R. (PI);
Ohrt, C. (PI);
Papanicolaou, G. (PI);
Poulson, J. (PI);
Ryzhik, L. (PI);
Schoen, R. (PI);
Soundararajan, K. (PI);
Taylor, R. (PI);
Vakil, R. (PI);
Varolgunes, U. (PI);
Vasy, A. (PI);
Venkatesh, A. (PI);
Vondrak, J. (PI);
White, B. (PI);
Wright, A. (PI);
Ying, L. (PI)
MATH 382:
Qualifying Examination Seminar
Terms: Win, Sum

Units: 13

Repeatable for credit

Grading: Satisfactory/No Credit
MATH 802:
TGR Dissertation
Terms: Aut, Win, Spr, Sum

Units: 0

Repeatable for credit

Grading: TGR
Instructors: ;
Bump, D. (PI);
Candes, E. (PI);
Carlsson, G. (PI);
Chatterjee, S. (PI);
Church, T. (PI);
Cohen, R. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Eliashberg, Y. (PI);
Fox, J. (PI);
Galatius, S. (PI);
Ionel, E. (PI);
Kerckhoff, S. (PI);
Li, J. (PI);
Luk, J. (PI);
Mazzeo, R. (PI);
Papanicolaou, G. (PI);
Rajaratnam, B. (PI);
Ryzhik, L. (PI);
Schoen, R. (PI);
Soundararajan, K. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Venkatesh, A. (PI);
Vondrak, J. (PI);
White, B. (PI);
Ying, L. (PI)