Working seminar on:
Cohomology of Vector Bundles &
Syzgies
by Jerzy Weyman
Publisher: Cambridge University Press
ISBN: 0521621976
Organized by J.M. Landsberg
jml@math.tamu.edu
Time: Mondays, 3-4:30pm Milner 317 (sometimes every other week)
We will study the defining
equations and more subtle information about varieties
determined by group actions.
For a motivating example, consider the variety X_r of rank at most r
matrices in the
space of n by n matrices. It is the zero set of the (r+1) x (r+1)
minors. We may rephrase
this by writing the space of n by n matrices as C^n \tensor C^n and X_r
is the set of
rank at most r tensors. Now consider the space Z_r of rank at most r
tensors in
C^n\tensor C^n\tensor C^n. It is an important open question in several
areas (computational
complexity, algebraic statistics, phylogenetics) to determine defining
equations for Z_r. We
could take this question as a motivating question for the seminar,
however the techniques
will be applicable to numerous problems.
We will begin by covering the necessary backround material from
commutative algebra,
algebraic geometry and representation theory. These topics are worth
learning in their own right.
Then we will study Kempf's technique of desingularization of orbit
closures via "a collapse" of
a vector bundle and Weyman's generalizations. We will then compute many
many examples.
The goal of the seminar is to learn new mathematics so graduate
students are particularly encouraged
to attend.
First meeting: Monday Jan. 24
Speaker: J.M. Landsberg
Title: What is this seminar about?
Monday Jan. 31
Speaker: Hal Schenck
Title: Algebra tools I
Abstract: I'll cover Sections 1.2.1-1.2.3 of Weymans
book. Section 1.2.1 starts in
with Ext, so in order to make sense of that, I'll talkabout projective and injective modules and
resolutions, functorsand
derived functors; and illustrate with examples how to computethese gadgets. Then we'll introduce the
notions of depth, regularsequences,
and the Cohen-Macaulay and Gorenstein properties. Finally, we'll discuss Koszul complexes and
homology, free resolutions, and
the theorems of Auslander-Buchsbaum (relating depth to projective dimension)
and the Buchsbaum-Eisenbud acyclicity criterion. Mostly this
lecture will be commutative algebra (with examples), but I'll also
try to get to the notion of the derived category, which we'll eventually
need.
Monday Feb. 7 - Meeting in Milner 216 for
this week only
Speaker: Hal Schenck
Title: Algebra tools II
derived functors (Tor and Ext), Sheaves and Cech cohomology.
Monday Feb. 14
Speaker: Hal Schenck
Title: Algebra tools III
Tuesday, Feb. 22, 1pm Milnor 313 (NOTE: special day, time and place!!!)
Speaker: J. Weyman
Title: New applications of geometric method of
calculating syzygies.
Abstract: I will talk about the jet schemes of determinantal varieties and their
Cohen-Macaulay property. This is the joint work in progress with Grag
Smith.
We can prove Cohen-Macaulayness for 1-jets (i.e. tangent schemes) but the
higher jets do not have rational singularities. Computer calculations show
however that their free resolutions have a fascinating pattern.
Monday Feb. 28
Speaker: Hal Schenck
Title: Algebra tools IV
Monday March 7
Speaker: Hal Schenck
Title: Algebra tools V
Monday March 21
Speaker: Hal Schenck
Title: The basic theorem
Monday March 28
Speaker: Frank Sottile
Title: Combinatorics I: Young tableaux etc...
NOTE: this lecture will be mostly independent
from previous lectures so new arrivals are welcome!
Monday April 4
Speaker: Frank Sottile
Title: Combinatorics II
Note: Monday April 11 there will be a special geometry
talk by S. Basu, so the Weyman seminar will meet on
Friday instead:
Friday April 15
Speaker: Frank Sottile
Title: Combinatorics III
Monday April 18
Speaker: Frank Sottile
Title: Combinatorics IV
Monday April 25,
Speaker: F. Sottile
Title: Combinatorics V: Schur complexes
Monday May 2,
Speaker: J.M. Landsberg
Title: Grassmannians I
Monday May 16,
Speaker: J.M. Landsberg
Title: Grassmannians II
There will be a second meeting the week of the 16th,
and then we will resume again in the fall