## 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.

Instructors:
Miller, J. (PI)

## 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.

Instructors:
Yang, T. (PI)

## MATH 217C: Complex Differential Geometry

Complex structures, almost complex manifolds and integrability, Hermitian and Kahler metrics, connections on complex vector bundles, Chern classes and Chern-Weil theory, Hodge and Dolbeault theory, vanishing theorems, Calabi-Yau manifolds, deformation theory.

| Repeatable for credit

Instructors:
Ionel, E. (PI)

## MATH 220: Partial Differential Equations of Applied Mathematics (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.

Terms: Aut
| Units: 3

Instructors:
Ryzhik, L. (PI)

## 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, Navier-Stokes equations for incompressible flow. Prerequisites:
MATH 220A or
CME 302.

Instructors:
Ying, L. (PI)

## MATH 228: Stochastic Methods in Engineering (CME 308)

Review of basic probability; Monte Carlo simulation; state space models and time series; parameter estimation, prediction, and filtering; Markov chains and processes; stochastic control; and stochastic differential equations. Examples from various engineering disciplines. Prerequisites: exposure to probability; background in real variables and analysis.

Instructors:
Papanicolaou, G. (PI)

## MATH 230A: Theory of Probability (STATS 310A)

Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: 116,
MATH 171.

Instructors:
Diaconis, P. (PI)

## MATH 230B: Theory of Probability (STATS 310B)

Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,nn(v) ergodic theory. Prerequisite: 310A or
MATH 230A.

Instructors:
Dembo, A. (PI)

## MATH 230C: Theory of Probability (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.

Instructors:
Chatterjee, S. (PI)

## MATH 231: Orthogonal Polynomials and the Moment Problem

Orthogonal polynomials in one variable (three term recurrence, Favard's theorem, distribution of zeros Verblunsky coefficients). Classical examples (Hermite, Chebychev, Jacobi, Meixner, Askey-Wilson). Applications in probability (Markov chains), Statistics (multivariate distributions with given margins), Numerical analysis (Gaussian Quadriture), Combinatorics (combinatorial interpretation of the classical orthogonal polynomials). The moment problem on R (when is a measure determined by its moments, what happens if not?). Multivariate orthogonal polynomials (with an introduction to symmetric function theory). Connections to group representations.

Instructors:
Diaconis, P. (PI)

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