Prerequisites: Basic graduate algebra, commutative algebra at the level of Atiyah and Macdonald, and some basic algebraic geometry, i.e. ideals/varieties/Nullstellensatz/regular and rational maps. 

Commutative algebra and algebraic geometry are fields with a large number of interesting examples. This course will cover the main computational methods in commutative algebra and algebraic geometry. There will be many examples, and we will use Macaulay2 for examples, and to create new algorithms. I expect that you will know some commutative algebra and algebraic geometry, but we will summarize in each class what we will assume or need from these fields.

Class will be most Monday's from 10 am - 1 pm.  Generally, this means a one hour lecture, followed by some time doing problems or examples, followed by another lecture related to the topic that day. We will not use a specific textbook, but will draw on a number of sources.

Depending on participant interest and background, topics might include: 

1. Groebner bases: basics and basic applications, such as ideal quotients, Hilbert functions, syzygies, elimination theory.

2. Monomial ideals, Stanley-Reisner theory, simplicial complexes.

3. Free resolutions, Hom, tensor product, Ext and Tor, depth, Auslander-Buchsbaum, Serre, regularity, Cohen-Macaulay rings, regular sequences, flatness (really, computations involving these notions).

4. Components of algebraic varieties: primary decomposition, minimal  primes.

5. Integral closure, Noether normalization (including desingularization of plane curves).

6. Divisors, line bundles, maps to projective space.  Fibers of maps to projective space. Deciding if a line bundle is nef, ample, very ample.

7. Sheaf cohomology, local cohomology, Bernstein-Gelfand-Gelfand, D-modules.  Note: we do not assume knowledge of sheaves or schemes, we will develop what we need.

8. Blowups, tangent cones, normal cones, Rees algebras, multiplicity, Mora's algorithm, Localization, Hilbert-Samuel functions.  

9. Intersection theory of smooth varieties.  Chern classes, Hirzebruch-Riemann-Roch.

*Note: This course is intended as audit-only, but, arrangements are possible for receiving credit on a case-by-case basis.