Graduate Study in Algebraic Geometry

Introduction

Algebraic Geometry is a new emphasis area in the department, whose purpose is the geometric study of solutions of systems of polynomial equations in several variables. It plays a central role in much of modern mathematics, and an understanding of its basic concepts is increasingly important. It interacts with many other subjects, including Mathematical Physics, Commutative Algebra, Algebraic Number Theory, Complex and Differential Geometry, and Computer Vision.

Course Descriptions

Math 510, Riemann Surfaces and Algebraic Curves

This course is designed to be an entry level course for algebraic geometry. It will lead naturally into Math 511, Algebraic Geometry, its planned sequel Algebraic Geometry II, and various topics courses in algebraic geometry. The course consists of an introduction to algebraic geometry in dimension 1 over the field of complex numbers. It covers Riemann surfaces, projective algebraic curves, differential forms, integration, divisors of poles and zeroes, linear systems, the Riemann-Roch theorem, Serre duality, and applications. Prerequisites are Math 500 and Math 542. This course is usually taught in the fall.

Math 511, Algebraic Geometry

This course covers properties of affine and projective varieties defined over algebraically closed fields; rational mappings, birational geometry and divisors, especially on curves and surfaces; introduction to the language of schemes; and Riemann-Roch theorem for curves. This course is usually taught in the spring.

Algebraic Geometry II (under development)

The proposed curriculum for this course is in transition. It is likely that the curriculum for Math 511 will change slightly when the new course comes into existence, but will almost certainly include sheaf cohomology and Serre duality. This course is usually taught in the fall.

Math 524, Linear Analysis on Manifolds

A transcendental algebraic geometry course, following Griffith's and Harris', Principles of Algebraic Geometry, this course focuses on the study of compact complex manifolds, and in the process lays the foundation for further study in algebraic geometry. The foundational results frequently have counterparts in a purely algebraic formulation of algebraic geometry as is typically taught in Math 511, while the techniques are frequently very different. Geometric methods in algebraic geometry frequently come to the forefront in a transcendental approach. Topics covered include complex manifolds, holomorphic vector bundles and sheaves, Hermitian differential geometry, Hodge theory for Kahler manifolds, Dolbeault cohomology, Chern classes, vanishing theorems, algebraic varieties, the Kodaira embedding theorem, low dimensional complex geometry, and complex tori.

Spring 2005 Algebraic Geometry Course Offerings

Math 511, Algebraic Geometry I (Nevins)

Math 525 CAG, Topics in Complex Differential Geometry (Bradlow)

Math 595, Real Algebraic Geometry with Applications (Fossum)

Math 595, Local Cohomology (Dutta)

Math 595 AG, Linear Algebraic Groups (Haboush)

Math 595 TV, Toric Varieties (Haboush)

Fall 2005 Algebraic Geometry Course Offerings