Graduate Courses

Fundamentals of complex analysis, and further topics such as conformal mapping, hyperbolic geometry, canonical products, elliptic functions and modular forms.

Prerequisites::

Basic complex analysis, topology of covering spaces, differential forms.

:

Fundamentals of algebraic curves as complex manifolds of dimension one. Topics may include branched coverings, sheaves and cohomology, potential theory, uniformization and moduli.

Recommended Prep:

Knowledge of the material in Mathematics 213a.

:

Commutative Algebra lies at the foundations of Number Theory and Algebraic Geometry. It plays an important role in Algebraic Topology, Geometry and other fields. We will cover the main topics of Commutative Algebra and give a taste of its applications. Starting from generalities on rings, modules and ideals, localization and primary decomposition in Noetherian rings and modules, we then move to integral extensions, going-up and going-down, Noether normalization and Hilbert’s Nullstellensatz, dimension theory. The final part of the class will cover graded rings, Hilbert polynomials and homological methods (if time permits, including regular local rings).

Recommended Prep:

Mathematics 122, 123.

:

Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.

Recommended Prep:

Knowledge of the material in Mathematics 114, 123 and 132.

:

A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.

Recommended Prep:

Knowledge of the material in Mathematics 129.

:

Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate’s thesis or Euler systems.

Recommended Prep:

Knowledge of the material in Mathematics 223a.

:

Basic properties and examples of smooth manifolds, Lie groups, and vector bundles; Riemannian geometry (metrics, geodesics, Levi-Civita connections, and Riemann curvature tensors); principal bundles and associated vector bundles with their connections and characteristic classes.

Recommended Prep:

Knowledge of the material in Mathematics 132 and 136.

:

Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.

Recommended Prep:

Knowledge of the material in Mathematics 131 and 132.

:

Continuation of Mathematics 231a. Topics will be chosen from: Cohomology products, homotopy theory, bundles, obstruction theory, characteristic classes, spectral sequences, Postnikov towers, and topological applications.

Recommended Prep:

Knowledge of the material in Mathematics 231a.

:

Introduction to complex algebraic curves, surfaces, and varieties.

Recommended Prep:

Knowledge of the material in Mathematics 123, 132, and 137.

:

This is a continuation of the material covered in the first semester, with a focus on coherent sheaves, cohomology, and their applications to the theory of curves and surfaces. Occasionally it may cover Hodge structures, Lefschetz theorems, or deformations.  See the course Canvas website for more about the semester’s course focus.

Recommended Prep:

Knowledge of the material in Mathematics 232ar.

:

This course introduces basic concepts of mathematical biology and evolutionary dynamics: reproduction, selection, mutation,  genetic drift, quasi-species, finite and infinite population dynamics, game dynamics, evolution of cooperation, language, spatial models, evolutionary graph theory, infection dynamics, virus dynamics, somatic evolution of cancer.

Recommended Prep:

Mathematics 19a,b or 21a,b or 22a,b or 23a,b or 25a,b or 55a,b; or an equivalent background in mathematics.

:

Research seminar on evolutionary dynamics, spanning mathematical and computational models of evolution in biological and social systems. Students attend a weekly lecture and conduct an original research project.

Recommended Prep:

Experience with mathematical biology at the level of Mathematics 153.

:

Become an effective instructor. This course focuses on observation, practice, feedback, and reflection providing insight into teaching and learning. Involves iterated videotaped micro-teaching sessions, accompanied by individual consultations. Required of all mathematics graduate students.

: