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: ;
Kimport, S. (PI);
Madnick, J. (PI);
Mueller, S. (PI);
Falcone, P. (TA);
Izzo, Z. (TA);
Kimport, S. (GP);
Lolas, P. (TA);
Sprunger, C. (TA);
Stanton, C. (TA);
Wei, F. (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: ;
Kimport, S. (PI);
Madnick, J. (PI);
Savvas, M. (PI);
Wilson, J. (PI);
Zhu, X. (PI);
Cant, D. (TA);
Guijarro Ordonez, J. (TA);
Kimport, S. (GP);
Lolas, P. (TA);
Velcheva, K. (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 (ABCD/NP)
MATH 51:
Linear Algebra and Differential Calculus of Several Variables
Geometry and algebra of vectors, matrices and linear transformations, eigenvalues of symmetric matrices, vectorvalued functions and functions of several variables, partial derivatives and gradients, derivative as a matrix, chain rule in several variables, critical points and Hessian, leastsquares, , constrained and unconstrained optimization in several variables, Lagrange multipliers. 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
MATH 51A:
Linear Algebra and Differential Calculus of Several Variables, ACE
Students attend MATH 51 lectures with different recitation sessions: three hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see https://web.stanford.edu/dept/soe/osa/ace.fb
Terms: Aut, Win, Spr

Units: 6

UG Reqs: GER:DBMath, WAYFR

Grading: Letter (ABCD/NP)
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

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

Grading: Letter (ABCD/NP)
MATH 79SI:
Proof Positive: Principles of Mathematics
What is a mathematical proof, and where do proofs come from? Students will become comfortable with fundamental techniques of mathematical proof through practice with interesting and accessible examples from many areas of math. Students will additionally hone their communication skills and develop their ability to formulate and answer precise mathematical questions. Topics include direct proof, proof by contrapositive, proof by contradiction, many applications of mathematical induction, constructing good definitions, and useful writing habits. The course is designed to prepare students who have completed or are concurrently enrolled in MATH 51 to succeed in introductory proofbased math classes at the level of MATH 115 or MATH 120, or to simply appreciate the nature of proof at a deeper level than is seen in high school geometry. To be considered for enrollment, please email masonr@stanford.edu and attend the first class meeting on Tuesday, April 3 at 3PM in 300303.
Terms: Spr

Units: 1

Grading: Satisfactory/No Credit
MATH 80Q:
Capillary Surfaces: Explored and Unexplored Territory
Preference to sophomores. Capillary surfaces: the interfaces between fluids that are adjacent to each other and do not mix. Recently discovered phenomena, predicted mathematically and subsequently confirmed by experiments, some done in space shuttles. Interested students may participate in ongoing investigations with affinity between mathematics and physics.
Terms: Win

Units: 3

UG Reqs: WAYFR, WAYSMA

Grading: Letter (ABCD/NP)
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: WAYFR

Grading: Letter (ABCD/NP)
MATH 101:
Math Discovery Lab
MDL is a discoverybased project course in mathematics. Students work independently in small groups to explore openended 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. No lecture component; inclass meetings reserved for student presentations, attendance mandatory. Admission is by application: http://math101.stanford.edu. Motivated students with any level of mathematical background are encouraged to apply. WIM
Terms: Win

Units: 3

UG Reqs: WAYFR

Grading: Letter or Credit/No Credit
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: Win, Spr

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
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: Win, 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

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: Spr

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

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

Grading: Letter or Credit/No Credit
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 initialvalue and boundaryvalue ODE, finite and spectral elements. Prerequisites: MATH 51 and 53.
Terms: Win

Units: 3

UG Reqs: GER:DBMath

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: Aut

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: Aut

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: Win

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 147:
Differential Topology
Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, BorsukUlam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: 115 or 171.
Terms: Spr

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 155:
Analytic Number Theory
Topics in analytic number theory such as the distribution of prime numbers, the prime number theorem, twin primes and Goldbach's conjecture, the theory of quadratic forms, Dirichlet's class number formula, Dirichlet's theorem on primes in arithmetic progressions, and the fifteen theorem. Prerequisite: 152, or familiarity with the Euclidean algorithm, congruences, residue classes and reduced residue classes, primitive roots, and quadratic reciprocity.
Terms: Spr

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 162:
Philosophy of Mathematics (PHIL 162)
Mathematics is a very peculiar human activity. It delivers a type of knowledge that is particularly stable, often conceived as a priori and necessary. Moreover, this knowledge is about abstract entities, which seem to have no connection to us, spatiotemporal creatures, and yet it plays a crucial role in our scientific endeavors. Many philosophical questions emerge naturally: What is the nature of mathematical objects? How can we learn anything about them? Where does the stability of mathematics comes from? What is the significance of results showing the limits of such knowledge, such as Gödel's incompleteness theorem? The first part of the course will survey traditional approaches to philosophy of mathematics ("the big Isms") and consider the viability of their answers to some of the previous questions: logicism, intuitionism, Hilbert's program, empiricism, fictionalism, and structuralism. The second part will focus on philosophical issues emerging from the actual practice of mathematics. We will tackle questions such as: Why do mathematicians reprove the same theorems? What is the role of visualization in mathematics? How can mathematical knowledge be effective in natural science? To conclude, we will explore the aesthetic dimension of mathematics, focusing on mathematical beauty. Prerequisite: PHIL150 or consent of instructor.
Terms: Win

Units: 4

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 163:
The Greek Invention of Mathematics (CLASSICS 136)
How was mathematics invented? A survey of the main creative ideas of ancient Greek mathematics. Among the issues explored are the axiomatic system of Euclid's Elements, the origins of the calculus in Greek measurements of solids and surfaces, and Archimedes' creation of mathematical physics. We will provide proofs of ancient theorems, and also learn how such theorems are even known today thanks to the recovery of ancient manuscripts.
Terms: Aut

Units: 35

UG Reqs: GER:DBHum, WAYAII

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 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: Win

Units: 3

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: Aut

Units: 3

UG Reqs: GER:DBMath

Grading: Letter or Credit/No Credit
MATH 177:
Geometric Methods in the Theory of Ordinary Differential Equations
Hamiltonian systems and their geometry. First order PDE and HamiltonJacobi equation. Structural stability and hyperbolic dynamical systems. Completely integrable systems. Perturbation theory.
Terms: Spr

Units: 3

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:
Independent Work
For math majors only. Undergraduates pursue a reading program; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for math majors. Approval of Undergraduate Affairs Committee is required to use credit for honors majors area requirement. Contact department student services specialist to enroll.
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 205C:
Real Analysis
Continuation of 205B.
Terms: Spr

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, and algebraic geometry. 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 216A:
Introduction to Algebraic Geometry
Algebraic curves, algebraic varieties, sheaves, cohomology, RiemannRoch theorem. Classification of algebraic surfaces, moduli spaces, deformation theory and obstruction theory, the notion of schemes. May be repeated for credit. Prerequisites: 210ABC or equivalent.
Terms: Aut

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 216B:
Introduction to Algebraic Geometry
Continuation of 216A. May be repeated for credit.
Terms: Win

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 216C:
Introduction to Algebraic Geometry
Continuation of 216B. May be repeated for credit.
Terms: Spr

Units: 3

Repeatable for credit

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 220A 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: 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 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.
Terms: Spr

Units: 3

Grading: Letter or Credit/No Credit
MATH 233A:
Topics in Combinatorics
Geometry of polynomials and nonconstructive proofs in combinatorics: The independence polynomial, the Lovasz Local Lemma and Shearer's Lemma. Realrooted polynomials, stable polynomials, Ramanujan graphs and the KadisonSinger problem. Strongly Rayleigh measures and negative dependence. Applications in algorithms.
Terms: Win

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 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 244:
Riemann Surfaces
Riemann surfaces and holomorphic maps, algebraic curves, maps to projective spaces. Calculus on Riemann surfaces. Elliptic functions and integrals. RiemannHurwitz formula. RiemannRoch theorem, AbelJacobi 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.
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 249B:
Topics in Number Theory
Terms: Win

Units: 3

Repeatable for credit

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 257B:
Symplectic Geometry and Topology
Continuation of 257A. May be repeated for credit.
Terms: Aut

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 263C:
Topics in Representation Theory
May be repeated for credit.
Terms: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 271:
The HPrinciple
The language of jets. Thom transversality theorem. Holonomic approximation theorem. Applications: immersion theory and its generaliazations. Differential relations and Gromov's hprinciple for open manifolds. Applications to symplectic geometry. Microflexibility. Mappings with simple singularities and their applications. Method of convex integration. NashKuiper C^1isometric embedding theorem.
Terms: Win

Units: 3

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

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 282A:
Low Dimensional Topology
The theory of surfaces and 3manifolds. 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 3manifolds: triangulations, Heegaard splittings, Dehn surgery. Loop theorem, sphere theorem, incompressible surfaces. Geometric structures, particularly hyperbolic structures on surfaces and 3manifolds. May be repeated for credit up to 6 total units.
Terms: Aut

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: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
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 Ktheory, including Bott periodicity, algebraic Ktheory, and indices of elliptic operators. Spectral sequences of AtiyahHirzebruch, Serre, and Adams. Cobordism theory, PontryaginThom theorem, calculation of unoriented and complex cobordism. May be repeated for credit up to 6 total units.
Terms: Spr

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 283A:
Topics in Topology
Terms: Win

Units: 3

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 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);
Papanicolaou, G. (PI);
Poulson, J. (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);
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)