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A math lecture

Prof. Kra gives a math lecture

Courses: Grad Courses

Undergraduate prerequisite courses

Introduction to Real Analysis.
Rigorous treatment of the real number system. Functions, basic topology of Euclidean space, limits and continuity, derivatives, infinite series, sequences of functions, the space of continuous functions, fixed point theorems, the Riemann integral. Inverse and implicit function theorems, Lebesgue integration.

Introduction to Topology.
Metric spaces, topological spaces, product spaces, compactness, connectedness, separation axioms, and other topics in point set topology.

Introduction to Modern Algebra.
Rigorous treatment of elementary theory of groups, rings, and fields; applications to the ring of integers and polynomial rings. Abstract linear algebra; vector spaces. Bilinear forms, canonical forms, modules.

Beginning graduate-level courses

Foundations of Modern Analysis. (Math 412-1,2,3)
General topology, metric spaces, measure, and integration. Elementary techniques of functional analysis, Banach and Hilbert spaces. Applications. Basic properties of holomorphic unctions including theorems of Cauchy, Morea, and Rouche; residue and open mapping theorems; harmonic functions; entire functions; analytic continuation; conformal mapping.

Algebraic Topology. (Math 442-1,2,3)
First-year introduction to algebraic methods in topology. (If you are contemplating possible research in topology, we urge you not to delay this course until your second year.) Homology and cohomology in complexes, singular homology and cohomology, cup products, homotopy. Graduate algebra, though not a prerequisite, is a useful corequisite.

Introduction to the Geometry of Manifolds. (Math 443-1,2,3)
Manifolds, connections, Riemannian manifolds, curvature, second variation of arc-length, selected other topics.

Dynamical Systems. (Math 447-1,2,3)
Qualitative theory of differentiable dynamical systems, emphasizing global properties such as structural stability theorems.

Probability. (Math 450-1,2)
Probability spaces, random variables, distribution functions, conditional probability, laws of large numbers, central limit theorem. Random walk, Markov chains, martingales, stochastic processes.

Algebra. (Math 470-1,2,3)
Groups: free, permutation, solvable, simple, and linear actions of groups on sets; Sylow theorems. Rings and modules: polynomials and power series, Euclidean domains, pids, ufds. Free and projective modules, structure of modules over pids, Field and Galois theory, algebraic and transcendental extensions, splitting fields and normal extensions, Galois groups, finite and cyclotomic fields. Structure of rings: simple and semisimple rings and modules, Wedderburn theory. Commutative algebra: prime ideals, localization, integral extensions.

Advanced graduate-level courses

Functions of a Complex Variable. (Math 413-1,2)
More advanced topics from complex variables.

Introduction to Riemann Surfaces. (Math 415-1)
Abstract Riemann Surfaces, differential forms, Poincare-Hopf formula, algebraic curves Riemann-Hurwitz formula, Riemann-Roch formula and applications, Jacobi variety and Abel theorem, and Uniformization theorem.

Basic Differential Equations. (Math 426-1)
Introduction to basic differential equations with emphasis on the theory of partial differential equations.

Partial Differential Equations. (Math 427-1,2)
Elliptic equations: the Dirichlet problem, differentiability and analyticity of solutions. Parabolic equations: initial-boundary value problems and asymptotic behavior of solutions. Hyperbolic equations and the Cauchy problem. Fundamental solutions.

Fourier Analysis. (Math 429-1,2)
A short overview of classical Fourier analysis on the circle. Selected topics about Fourier analysis on the line and in Euclidean space.

Introduction to Functional Analysis. (Math 444-1,2)
Topological groups and topological vector spaces; Banach spaces, linear functionals, and operators; applications to functional equations.

Algebraic Topology II. (Math 446-1,2,3)
Cohomology theories and operations, homotopy and obstruction theory, CW complexes, Vector Bundles, K-theory, cobordism.

Stochastic Analysis. (Math 462-1,2)
Principal types of processes (such as stationary, Markov, Gaussian) and their general theory; detailed study of some processes such as the Poisson and Brownian motion processes; selected topics such as sums of independent random variables, waiting lines, branching processes, information theory.

Commutative Algebra. (Math 477-1)
Topics from contemporary research in commutative algebra, such as the theory of depth (regular sequences, Koszul complexes), dimension theory, completions, Hilbert functions, Cohen-Macaulay modules, excellent rings, Hensel rings, and minimal resolutions. Prerequisites: Algebra (last listing under beginning graduate-level courses) or equivalent preparation.

Representation Theory. (Math 478-1)
Topics in the representation theory and cohomology of finite and infinite groups including compact and non-compact Lie groups.

Homological Algebra. (Math 480-1)
Exact sequences, Ext and Tor, homological dimensions.

Algebraic Number Theory. (Math 482-1)
First and second quarters: theory of global and local fields. Third quarter: special topics chosen by the instructor (often includes class field theory).

Algebraic Geometry. (Math 483-1,2,3)
An introduction to classical and scheme-theoretic methods of algebraic geometry. Topics include algebraic vector bundles, sheaf cohomology, the Riemann-Roch theorem for curves, and intersection theory.

Lie Theory. (Math 484-1)
Topics in the theory of Lie algebras and Lie groups including classification.

Algebraic K-theory. (Math 486-1,2,3)
Classical algebraic K-theory (the functors K0 and K1, origins in and relations with topology); the congruence subgroup problem; techniques of computation (exact sequences, localization, resolution and devissage); polynomial and related extensions; higher K-theories (Karoubi-Villamayor, Quillen).

Advanced seminars

Seminars on topics of current research interest include the following:

Independent Study Courses

In addition, advanced graduate students regularly take independent study courses with faculty on specialized topics.

Fall 1998

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Spring 1999

Fall 1999

Winter 2000

Spring 2000

500-Level Courses

Spring 1998

Fall 1998

Winter 1999

Spring 1999

Winter 2000