### Bulletin Archive

This archived information is dated to the 2008-09 academic year only and may no longer be current.

**For currently applicable policies and information, see the current Stanford Bulletin.**

This archived information is dated to the 2008-09 academic year only and may no longer be current.

**For currently applicable policies and information, see the current Stanford Bulletin.**

Primarily for graduate students; undergraduates may enroll with consent of instructor.

MATH 205A. Real Analysis

Basic measure theory and the theory of Lebesgue integration. Prerequisite: 171 or equivalent.

3 units, Aut (Ryzhik, L)

MATH 205B. Real Analysis

Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: 171 and 205A or equivalent.

3 units, Win (Vasy, A)

MATH 205C. Real Analysis

Continuation of 205B.

3 units, Spr (Katznelson, Y)

MATH 210A. Modern Algebra

Groups, rings, and fields; introduction to Galois theory. Prerequisite: 120 or equivalent.

3 units, Aut (Milgram, R)

MATH 210B. Modern Algebra

Galois theory. Ideal theory, introduction to algebraic geometry and algebraic number theory. Prerequisite: 210A.

3 units, Win (Brumfiel, G)

MATH 210C. Modern Algebra

Continuation of 210B. Representations of groups and noncommutative algebras, multilinear algebra.

3 units, Spr (Bump, D)

MATH 215A. Complex Analysis, Geometry, and Topology

Analytic functions, complex integration, Cauchy's theorem, residue theorem, argument principle, conformal mappings, Riemann mapping theorem, Picard's theorem, elliptic functions, analytic continuation and Riemann surfaces.

3 units, Aut (Li, J)

MATH 215B. Complex Analysis, Geometry, and Topology

Topics: fundamental group and covering spaces, homology, cohomology, products, basic homotopy theory, and applications. Prerequisites: 113, 120, and 171, or equivalent; 215A is not a prerequisite for 215B.

3 units, Win (Galatius, S)

MATH 215C. Complex Analysis, Geometry, and Topology

Differentiable manifolds, transversality, degree of a mapping, vector fields, intersection theory, and Poincare duality. Differential forms and the DeRham theorem. Prerequisite: 215B or equivalent.

3 units, Spr (Cohen, R)

MATH 216A. Introduction to Algebraic Geometry

Algebraic curves, algebraic varieties, sheaves, cohomology, Riemann-Roch theorem. Classification of algebraic surfaces, moduli spaces, deformation theory and obstruction theory, the notion of schemes. May be repeated for credit.

3 units, not given this year

MATH 216B. Introduction to Algebraic Geometry

Continuation of 216A. May be repeated for credit.

3 units, not given this year

MATH 217A. Differential Geometry

Smooth manifolds and submanifolds, tensors and forms, Lie and exterior derivative, DeRham cohomology, distributions and the Frobenius theorem, vector bundles, connection theory, parallel transport and curvature, affine connections, geodesics and the exponential map, connections on the principal frame bundle. Prerequisite: 215C or equivalent.

3 units, Win (Schoen, R)

MATH 217B. Differential Geometry

Riemannian manifolds, Levi-Civita connection, Riemann curvature tensor, Riemannian exponential map and geodesic normal coordinates, Jacobi fields, completeness, spaces of constant curvature, bi-invariant metrics on compact Lie groups, symmetric and locally symmetric spaces, equations for Riemannian submanifolds and Riemannian submersions. Prerequisite: 217A.

3 units, Spr (Brendle, S)

MATH 220. Partial Differential Equations of Applied Mathematics

(Same as CME 303.) First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: foundation in multivariable calculus and ordinary differential equations.

3 units, Aut (Nolen, J)

MATH 221. Mathematical Methods of Imaging

Mathematical methods of imaging: array imaging using Kirchhoff migration and beamforming, resolution theory for broad and narrow band array imaging in homogeneous media, topics in high-frequency, variable background imaging with velocity estimation, interferometric imaging methods, the role of noise and inhomogeneities, and variational problems that arise in optimizing the performance of imaging algorithms and the deblurring of images. Prerequisite: 220.

3 units, not given this year

MATH 222. Computational Methods for Fronts, Interfaces, and Waves

High-order methods for multidimensional systems of conservation laws and Hamilton-Jacobi equations (central schemes, discontinuous Galerkin methods, relaxation methods). Level set methods and fast marching methods. Computation of multi-valued solutions. Multi-scale analysis, including wavelet-based methods. Boundary schemes (perfectly matched layers). Examples from (but not limited to) geometrical optics, transport equations, reaction-diffusion equations, imaging, and signal processing.

3 units, not given this year

MATH 224. Topics in Mathematical Biology

Mathematical models for biological processes based on ordinary and partial differential equations. Topics: population and infectious diseases dynamics, biological oscillators, reaction diffusion models, biological waves, and pattern formation. Prerequisites: 53 and 131, or equivalents.

3 units, not given this year

MATH 227. Partial Differential Equations and Diffusion Processes

Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites: MATH 131 and MATH 136/STATS 219, or equivalents.

3 units, Win (Ryzhik, L)

MATH 228A. Ergodic Theory

Measure preserving transformations and flows, ergodic theorems, mixing properties, spectrum, Kolmogorov automorphisms, entropy theory. Examples. Classical dynamical systems, mostly geodesic and horocycle forms on homogeneous spaces of SL(2,R). May be repeated for credit. Prerequisites: 205A,B.

3 units, not given this year

MATH 230A. Theory of Probability

(Same as STATS 310A.) Mathematical tools: asymptotics, metric spaces; measure and integration; Lp spaces; some Hilbert spaces theory. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Weak convergence and characteristic functions; central limit theorems; local limit theorems; Poisson convergence. Prerequisites: 116, MATH 171.

2-4 units, Aut (Diaconis, P)

MATH 230B. Theory of Probability

(Same as STATS 310B.) Stopping times, 0-1 laws, Kolmogorov consistency theorem. Uniform integrability. Radon-Nikodym theorem, branching processes, conditional expectation, discrete time martingales. Exchangeability. Large deviations. Laws of the iterated logarithm. Birkhoff's and Kingman's ergodic theorems. Recurrence, entropy. Prerequisite: 310A or MATH 230A.

2-4 units, Win (Dembo, A)

MATH 230C. Theory of Probability

(Same as STATS 310C.) Infinitely divisible laws. Continuous time martingales, random walks and Brownian motion. Invariance principle. Markov and strong Markov property. Processes with stationary independent increments. Prerequisite: 310B or MATH 230B.

2-4 units, Spr (Dembo, A)

MATH 231A. An Introduction to Random Matrix Theory

(Same as STATS 351A.) Patterns in the eigenvalue distribution of typical large matrices, which also show up in physics (energy distribution in scattering experiments), combinatorics (length of longest increasing subsequence), first passage percolation and number theory (zeros of the zeta function). Classical compact ensembles (random orthogonal matrices). The tools of determinental point processes.

3 units, Aut (Diaconis, P)

MATH 231B. The Spectrum of Large Random Matrices

Asymptotics of eigenvalues of large random matrices, focusing on Wigner matrices and the Gaussian unitary ensemble: the combinatorics of non-crossing partitions and word graphs, concentration inequalities, Cauchy-Stieltjes transform, Hermite polynomials, Fredholm determinants, Laplace asymptotic method, special functions (Airy, Painleve), and stochastic calculus. Prerequisities: STATS 310A or MATH 205A.

3 units, Win (Dembo, A)

MATH 231C. Free Probability

Background from operator theory, addition and multiplication theorems for operators, spectral properties of infinite-dimensional operators, the free additive and multiplicative convolutions of probability measures and their classical counterparts, asymptotic freeness of large random matrices, and free entropy and free dimension. Prerequisite: STATS 310B or equivalent.

3 units, Spr (Staff)

MATH 232. Topics in Probability: Malliavin Calculus, Fractional Brownian Motion and Applications

Malliavin calculus: derivative and divergence operators, Skorohod integral. Fractional Brownian motion: relavance for financial mathematics, Ito and Tanaka formula, driving force for the heat equation. Ito formula for irregular Gaussian processes and other applications of Malliavin calculus. May be repeated for credit. Prerequisites: MATH 236, STATS 310C or equivalent.

3 units, Win (Staff)

MATH 233. Probabilistic Methods in Analysis

Proofs and constructions in analysis obtained from basic results in Probability Theory and a 'probabilistic way of thinking.' Topics: Rademacher functions, Gaussian processes, entropy.

3 units, Win (Katznelson, Y)

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.

3 units, Win (Papanicolaou, G)

MATH 238. Mathematical Finance

(Same as 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.

3 units, Win (Papanicolaou, G)

MATH 239. Computation and Simulation in Finance

Monte Carlo, finite difference, tree, and transform methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Prerequisite: 238 or equivalent.

3 units, Spr (Toussaint, A)

MATH 240. Topics in Financial Mathematics: Fixed Income Models

Introduction to continuous time models for arbitrage-free pricing of interest rate derivatives. Bonds, yields, and the construction of yield curves. Caps, floors, swaps, swaptions, and bond options. Short rate models. Yield curve models. Forward measures. Forward and futures. LIBOR and swap market models. Prerequisite: MATH 238.

3 units, Spr (Toussaint, A)

MATH 244. Riemann Surfaces

Compact Riemann surfaces and algebraic curves; cohomology of sheaves; Serre duality; Riemann-Roch theorem and application; Jacobians; Abel's theorem. May be repeated for credit.

3 units, Spr (Kerckhoff, S)

MATH 245A. Topics in Algebraic Geometry: Moduli Theory

Intersection theory on the moduli spaces of stable curves, stable maps, and stable vector bundles. May be repeated for credit.

3 units, not given this year

MATH 245B. Topics in Algebraic Geometry: Dessin d'Enfants

Grothendieck's theory of dessin d'enfants, a study of graphs on surfaces and their connection with the absolute Galois group of the rational numbers. Belyi's theorem, representations of the absolute Galois group as automorphisms of profinite groups, Grothendieck-Teichmuller theory, quadratic differentials, and the combinatorics of moduli spaces of surfaces. May be repeated for credit.

3 units, not given this year

MATH 247. Topics in Group Theory

Topics include the Burnside basis theorem, classification of p-groups, regular and powerful groups, Sylow theorems, the Frattini argument, nilpotent groups, solvable groups, theorems of P. Hall, group cohomology, and the Schur-Zassenhaus theorem. The classical groups and introduction to the classification of finite simple groups and its applications. May be repeated for credit.

3 units, Win (Diaconis, P)

MATH 248. Algebraic Number Theory

Introduction to modular forms and L-functions. May be repeated for credit.

1-3 units, not given this year

MATH 248A. Algebraic Number Theory

Structure theory and Galois theory of local and global fields, finiteness theorems for class numbers and units, adelic techniques. Prerequisites: MATH 210A,B.

3 units, Aut (Conrad, B)

MATH 249A. Introduction to Modular Forms

The analytic theory of holomorphic and non-holomorphic modular forms and associated L-functions. Topics include Hecke operators, L-functions, Weil's converse theorem, trace formulas, sub-convexity for L-functions and applications, and Selberg's eigenvalue conjecture. May be repeated for credit. Prerequisites: 205A,B,C, or comparable knowledge of analysis.

3 units, Aut (Soundararajan, K)

MATH 249B. Topics in Number Theory: Class Field Theory

Classification of abelian extensions of local and global fields; classical, adelic, and cohomological formulations; applications to L-functions. May be repeated for credit.

3 units, Win (Conrad, B)

MATH 249C. Topics in Number Theory: Class Field Theory and the Langlands Conjectures

3 units, Spr (Staff)

MATH 254. Geometric Methods in the Theory of Ordinary Differential Equations

Topics may include: structural stability and perturbation theory of dynamical systems; hyperbolic theory; first order PDE; normal forms, bifurcation theory; Hamiltonian systems, their geometry and applications. May be repeated for credit.

3 units, not given this year

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.

3 units, Spr (Vasy, A)

MATH 256B. Partial Differential Equations

Continuation of 256A.

3 units, Win (Liu, T)

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.

3 units, Aut (Ionel, E)

MATH 257B. Symplectic Geometry and Topology

Continuation of 257A. May be repeated for credit.

3 units, Win (Ionel, E)

MATH 258. Topics in Geometric Analysis

May be repeated for credit.

3 units, Win (White, B)

MATH 261A. Functional Analysis

Geometry of linear topological spaces. Linear operators and functionals. Spectral theory. Calculus for vector-valued functions. Operational calculus. Banach algebras. Special topics in functional analysis. May be repeated for credit.

3 units, not given this year

MATH 263A. Lie Groups and Lie Algebras

Definitions, examples, properties. Semi-simple Lie algebras, their structure and classification. Cartan decomposition: real Lie algebras. Representation theory: Cartan-Stiefel diagram, weights. Weyl character formula. Orthogonal and symplectic representations. May be repeated for credit. Prerequisite: 210 or equivalent.

3 units, Win (Bump, D)

MATH 263B. Lie Groups and Lie Algebras

Continuation of 263A. May be repeated for credit.

3 units, Spr (Staff)

MATH 264. Matrix Valued Spherical Functions and Orthogonal Polynomials

Theory of spherical functions on locally compact groups and on Lie groups. Families of orthogonal polynomials with respect to a weight matrix function on the real line, and the corresponding algebra of differential operators. Spherical functions associated to the complex projective space as orthogonal polynomials. Topics may include some applications to quasi birth and death processes. My be repeated for credit. Prerequisities: 114, 205A, and 217A.

3 units, Aut (Staff)

MATH 266. Computational Signal Processing and Wavelets

Theoretical and computational aspects of signal processing. Topics: time-frequency transforms; wavelet bases and wavelet packets; linear and nonlinear multiresolution approximations; estimation and restoration of signals; signal compression. May be repeated for credit.

3 units, not given this year

MATH 269A. Affine Complex Manifolds and Symplectic Geometry

Plurisubharmonic functions and pseudoconvexity: geometric theory. Construction of pseudoconvex shapes. Complex analysis on Stein manifolds. Symplectic geometry of Stein manifolds. Existence theorem for Stein complex manifolds. May be repeated for credit.

3 units, Aut (Eliashberg, Y)

MATH 269B. Affine Complex Manifolds and Symplectic Geometry

Symplectic convexity and Weinstein manifolds. Symplectic topology of subcritical Weinstein manifolds. From Weinstein to Stein structure. Morse-Smale theory for plurisubharmonic functions on Stein manifolds. Deformation theory for Stein complex structures. Symplectic field theory of Weinstein manifolds. May be repeated for credit.

3 units, Win (Eliashberg, Y)

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.

3 units, Win (Li, J)

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.

3 units, Spr (Eliashberg, Y)

MATH 272A. Topics in Partial Differential Equations

3 units, Aut (Tzou, L)

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.

3 units, Aut (Kerckhoff, S)

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.

3 units, Win (Carlsson, G)

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.

3 units, Spr (Milgram, R)

MATH 284A. Geometry and Topology in Dimension 3

The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.

3 units, Win (Staff)

MATH 284B. Geometry and Topology in Dimension 3

The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.

3 units, Spr (Staff)

MATH 286. Topics in Differential Geometry

May be repeated for credit.

3 units, Win (Mazzeo, R), Spr (Schoen, R)

MATH 290B. Finite Model Theory

(Same as PHIL 350B.) Classical model theory deals with the relationship between formal languages and their interpretation in finite or infinite structures; its applications to mathematics using first-order languages. The recent development of the model theory of finite structures in connection with complexity classes as measures of computational difficulty; how these classes are defined within certain languages that go beyond first-order logic in expressiveness, such as fragments of higher order or infinitary languages, rather than in terms of models of computation.

3 units, not given this year

MATH 292A. Set Theory

(Same as PHIL 352A.) The basics of axiomatic set theory; the systems of Zermelo-Fraenkel and Bernays-Gödel. Topics: cardinal and ordinal numbers, the cumulative hierarchy and the role of the axiom of choice. Models of set theory, including the constructible sets and models constructed by the method of forcing. Consistency and independence results for the axiom of choice, the continuum hypothesis, and other unsettled mathematical and set-theoretical problems. Prerequisites: PHIL160A,B, and MATH 161, or equivalents.

3 units, not given this year

MATH 292B. Set Theory

(Same as PHIL 352B.) The basics of axiomatic set theory; the systems of Zermelo-Fraenkel and Bernays-Gödel. Topics: cardinal and ordinal numbers, the cumulative hierarchy and the role of the axiom of choice. Models of set theory, including the constructible sets and models constructed by the method of forcing. Consistency and independence results for the axiom of choice, the continuum hypothesis, and other unsettled mathematical and set-theoretical problems. Prerequisites: PHIL160A,B, and MATH 161, or equivalents.

3 units, not given this year

MATH 293A. Proof Theory

(Same as PHIL 353A.) Gentzen's natural deduction and sequential calculi for first-order propositional and predicate logics. Normalization and cut-elimination procedures. Relationships with computational lambda calculi and automated deduction. Prerequisites: 151, 152, and 161, or equivalents.

3 units, not given this year

MATH 295. Computation and Algorithms in Mathematics

Use of computer and algorithmic techniques in various areas of mathematics. Computational experiments. Topics may include polynomial manipulation, Groebner bases, computational geometry, and randomness. May be repeated for credit.

3 units, not given this year

MATH 355. Graduate Teaching Seminar

Required of and limited to first-year Mathematics graduate students.

1 unit, Spr (Staff)

MATH 360. Advanced Reading and Research

1-9 units, Aut (Staff), Win (Staff), Spr (Staff), Sum (Staff)

MATH 361. Research Seminar Participation

Participation in a faculty-led seminar which has no specific course number.

1-3 units, Aut (Staff), Win (White, B), Spr (Kerckhoff, S), Sum (Staff)

MATH 380. Seminar in Applied Mathematics

Guest speakers on recent advances in applied mathematics.May be repeated for credit.

1 unit, Aut (Staff), Win (Staff), Spr (Staff)

MATH 381. Seminar in Analysis

1-3 units, by arrangement

MATH 384. Seminar in Geometry

1 unit, by arrangement

MATH 385. Seminar in Topology

1-3 units, by arrangement

MATH 386. Mathematics Colloquium

Guest speakers on recent advances in mathematics. May be repeated for credit.

1 unit, Aut (Staff), Win (Bump, D), Spr (Staff)

MATH 387. Seminar in Number Theory

May be repeated for credit.

1 unit, Aut (Staff), Win (Staff), Spr (Staff)

MATH 388. Seminar in Probability and Stochastic Processes

1-3 units, by arrangement

MATH 389. Seminar in Mathematical Biology

1-3 units, by arrangement

MATH 391. Research Seminar in Logic and the Foundations of Mathematics

(Same as PHIL 391.) Contemporary work. May be repeated a total of three times for credit.

1-3 units, Spr (Mints, G; Feferman, S)

MATH 395. Classics in Geometry and Topology

Original papers in geometry and in algebraic and geometric topology. May be repeated for credit.

3 units, Aut (Brumfiel, G), Win (Staff), Spr (Cohen, R)

MATH 396. Graduate Progress

Results and current research of graduate and postdoctoral students. May be repeated for credit.

1 unit, Aut (Staff), Win (Staff), Spr (Staff)

MATH 397. Physics for Mathematicians

Topics from physics essential for students studying geometry and topology. Topics may include quantum mechanics, quantum field theory, path integral approach and renormalization, statistical mechanics, and string theory. May be repeated for credit.

1 unit, Win (Staff)

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