USC Commutative Algebra Seminar - Fall 2010
The seminar meets on Mondays from 12:20pm-1:10pm in LeConte 312.
- Monday, September 13, 12:20pm, LeConte 312.
Speaker: Adela Vraciu
Title: The resolution of Frobenius powers of ideals in a diagonal surface ring.
- Monday, September 20, 12:20pm, LeConte 312.
Speaker: Adela Vraciu
Title: The resolution of Frobenius powers of ideals in a diagonal surface ring, II.
- Monday, September 27, 12:20pm, LeConte 312.
Speaker: Brett Barwick
Title: An introduction to the Quillen-Suslin Theorem.
Abstract: In J.P. Serre's 1955 article "Faisceaux
algebriques coherents" he posed the following problem, which became
known as "Serre's Problem": Do there exist finitely generated projective
modules over k[x_1,...,x_n] which are not free? This problem stimulated
a large amount of research over the next 20 years until it was resolved
independently by D. Quillen and A. Suslin in 1976. This talk aims to
give an introduction to Serre's problem and its reduction to a
corresponding statement about completion of unimodular rows over
polynomial rings to square invertible matrices.
- Monday, October 4, 12:20pm, LeConte 312.
Speaker: Brett Barwick
Title: An introduction to the Quillen-Suslin Theorem, II.
Abstract: We will sketch an algorithmic approach
given in 1992 by Logar-Sturmfels (with a few modifications) that one
can use to complete a unimodular matrix over R = k[x_1,...,x_n] to a
square invertible matrix, which is the essential step for computing a
free generating set for a projective module over R.
- Monday, October 11, 12:20pm, LeConte 312.
Speaker: Andrew Kustin
Title: The Generic Hilbert-Burch matrix.
Abstract: Let X be the set of 3 by 2 matrices
whose entries are homogeneous forms of degree c in the polynomial ring
k[x,y] and let Y be the set of 3 by 1 matrices whose entries are
homogeneous forms of degree 2c. Notice that X may be identified with
an ordinary affine space of dimension 6c+6 and Y may be identified
with an ordinary affine space of dimension 6c+3. The function X -> Y
which is given by taking the three 2 by 2 minors induces a polynomial
function from 6c+6 space to 6c+3 space. Does this function have
a polynomial inverse?
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