Algebraic Geometry investigates the dynamic interplay between algebraic equations and the intricate geometry of their solution sets, known as algebraic varieties. The field has seen tremendous advances in subtle internal questions concerning the classification of algebraic varieties, their topology, and the structure of their singularities, but deep and fundamental questions remain. At the same time, algebraic geometry provides basic examples, tools, and insights for commutative algebra, differential geometry, complex analysis, representation theory, number theory, and mathematical physics. It thus enjoys exciting, vibrant interaction with those fields that is full of surprises.
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.
Math 512, Modern Algebraic Geometry
This course provides an introduction to algebraic geometry via schemes, studying the topics of varieties, sheaves, schemes, sheaves of modules, divisors and invertible sheaves, and Kahler differentials. Prerequisites are Math 500 or equivalent; and either (a) Math 510 or (b) Math 511 or (c) consent of the instructor. The course is usually taught in the fall.
Math 514, Complex Algebraic Geometry
This course introduces the geometry of Kähler manifolds and the associated Hodge structure of cohomology. Note that one of the Clay Institute's Millenium Open Problems is to prove the Hodge conjecture: that every Hodge class of a non-singular projective variety over C is a rational linear combination of cohomology classes of algebraic cycles. Kähler manifolds lie at the intersection of complex geometry, Riemannian geometry, and symplectic geometry. Moreover, every smooth projective variety is a Kähler manifold. All of this structure is reflected in a rich theory of geometric and topological invariants. This course develops techniques from sheaf theory and linear elliptic theory to study the cohomology of Kähler manifolds.
Algebraic geometry does not exist in a vacuum, and students are encouraged to acquire expertise in related areas of mathematics. While the specific areas recommended depend on each student's own research focus, most algebraic geometry students will want to know commutative algebra.
Math 502, Commutative Algebra
This course studies commutative rings and modules, prime ideals, localization, noetherian rings, primary decomposition, integral extensions and Noether normalization, the Nullstellensatz, dimension, flatness, Hensel's lemma, graded rings, Hilbert polynomial, valuations, regular rings, singularities, unique factorization, homological dimension, depth, completion. Possible further topics: smooth and etale extensions, ramification, Cohen-Macaulay modules, complete intersections.