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.