From: Brenda Sue Cook (brenda@math.missouri.edu)
Date: Wed Mar 12 2008 - 16:31:37 CDT
COLLOQUIUM
Department of Mathematics
Lars Winther Christensen
Texas Tech University
Finite Gorenstein representation type
implies simple singularity
Abstract: In our first course on linear algebra, we learn to solve
systems of linear equations. For example, how to find all points in
space that satisfy both of the equations
(A) x+y+z=0 and
(B) x-2y+z=0.
The geometry of this problem is simple. The points that solve (A)
form plane, the solution set for (B) is a different plane, and the
set of common solutions is their intersection, which is a line. The
difficulty of solving such systems does not significantly depend on
the number of variables or the number of equations. However,
equations that involve higher powers of the variables, say x^2 or the
product xyz, are surprisingly complicated, and it is a fundamental
goal to understand the geometry of their solution sets, which are
called algebraic varieties.
It is a general principle in mathematics to study an object via
functions defined on it. Families of functions defined on varieties
form commutative rings, and it is a firmly established maxim that
understanding a ring is tantamount to understanding its module category.
Over small rings, notably finite dimensional algebras over a field,
it is sometimes possible to give an exhaustive description of the
module category by classifying its indecomposable objects. When this
is not possible, one hopes to find characteristics of the ring—and
the corresponding variety—reflected in a manageable subcategory of
its modules category. Striking examples of such connections emerged
in the 1980s; in the talk I will survey these results and recent
improvements.
Tuesday, March 18
3:30 p.m.
217 Strickland (GCB)
Refreshments will be served at 3:00 p.m.
in Room 326 Mathematical Sciences (Math Lounge).
This archive was generated by hypermail 2.1.4 : Fri May 02 2008 - 11:40:00 CDT