Differential Geometry I

Core topics in differential and Riemannian geometry including Lie groups, curvature, relations with topology.

Algebraic Topology I

The fundamental group, covering spaces, core topics in homology and cohomology theory including CW complexes, universal coefficients, and Poincare duality.

Real Analysis II

Topics include L^p, Banach and Hilbert spaces, basic functional analysis.

Real Analysis I

Measure and integration theory

Complex Analysis

Complex integration, including Goursat's theorem; classification of singularities, the argument principle, the maximum principle; Riemann Mapping theorem; analytic continuation and Riemann surfaces; range of an analytic function, including Picard's theorem.

Multivariate Statistical Analysis

Multivariate normal distribution theory, correlation and dependence analysis, regression and prediction, dimension-reduction methods, sampling distributions and related inference problems, selected applications in classification theory, multivariate process control, and pattern recognition.

Statistical Estimation

Basic theories of statistical estimation, including optimal estimation in finite samples and asymptotically optimal estimation. A careful mathematical treatment of the primary techniques of estimation utilized by statisticians.

Probability II

Develops the probability basis requisite in modern statistical theories and stochastic processes. (2nd of two courses)

Algebra II

Graduate level linear and abstract algebra including rings, fields, modules, some algebraic number theory and Galois theory. (2nd of two courses)


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