

DECEMBER 2008 


Automorphic Forms and Galois Representations Seminar 
Topic: 
Compatabilities at p of Hilbert modular forms and their padic Galois representations 
Presenter: 
Christopher Skinner, Princeton University 
Date: 
Wednesday, December 10, 2008, Time: 1:30 p.m., Location: Fine Hall 314 


Department Colloquium 
Topic: 
The geometry underlying DonaldsonThomas theory 
Presenter: 
Kai Behrend, University of British Columbia 
Date: 
Wednesday, December 10, 2008, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
DonaldsonThomas invariants are algebraic analogues of Casson invariants. They are virtual counts of stable coherent sheaves on CalabiYau threefolds. Ideally, the moduli spaces giving rise to these invariants should be critical sets of "holomorphic ChernSimons functions". Currently, such holomorphic ChernSimons functions exist at best locally (see my seminar talk on Monday), and it is unlikely that they exist globally. I will describe geometric structures on the moduli spaces (some conjectural) that exist globally and reflect the fact that the moduli spaces look as if they were the zero loci of holomorphic maps. These are: symmetric obstruction theories, which prove that DonaldsonThomas invariants are weighted Euler characteristics of moduli spaces, and derived scheme structures, exhibiting the moduli spaces as the classical schemes underlying schemes of Gerstenhaber algebras. 


Discrete Mathematics Seminar 
Topic: 
Packing cycles with modularity 
Presenter: 
Paul Wollan, University of Hamburg 
Date: 
Thursday, December 11, 2008, Time: 2:15 p.m., Location: Fine Hall 224 
Abstract: 
Erdős and Posa proved that a there exists a function $f$ such that any graph either has $k$ disjoint cycles or there exists a set of $f(k)$ vertices that intersects every cycle. The analogous statement is not true for odd cycles  there exist numerous examples of graphs that do not have two disjoint odd cycles, and yet no bounded number of vertices intersects every odd cycle. However, Reed has given a partial characterization of when there does not exist a bounded size set of vertices intersecting every odd cycle.
We will discuss recent work on when a graph has many disjoint cycles of nonzero length modulo $m$ for arbitrary $m$. When $m$ is odd, we see that again there exists a function $f$ such that any graph either has $k$ disjoint cycles of nonzero length modulo $m$ or there exists a set of at most $f(k)$ vertices intersecting every such cycle of nonzero length. When $m$ is even, no such function $f(k)$ exists. However, the partial characterization of Reed can be extended to describe when a graph has neither many disjoint cycles of nonzero length modulo $m$ nor a small set of vertices intersecting every such cycle. 


Number Theory Seminar 
Topic: 
Langlands functoriality and the inverse problem in Galois theory 
Presenter: 
Gordon Savin, University of Utah 
Date: 
Thursday, December 11, 2008, Time: 4:30 p.m., Location: IAS SH101 
Abstract: 
In a couple of recent works with C. Khare and M. Larsen we contruct finte groups of Lie type B_n, C_n and G_2 as Galois groups over rational numbers. The method combines some established, special cases of the functoriality principle with ladic representations attached to selfdual automrophic representations of GL(n). 


Topology Seminar 
Topic: 
Primitivestable representations of the free group 
Presenter: 
Yair Minsky, Yale University 
Date: 
Thursday, December 11, 2008, Time: 4:30 p.m., Location: Fine Hall 314 
Abstract: 
Automorphisms of the free group F_n act on its representations into a given group G. When G is a simple compact Lie group and n>2, Gelander showed that this action is ergodic. We consider the case G=PSL(2,C), where the variety of (conjugacy classes of) representations has a natural invariant decomposition, up to sets of measure 0, into discrete and dense representations. This turns out NOT to be the relevant decomposition for the dynamics of the outer automorphism group. Instead we describe a set called the "primitivestable" representations containing discrete as well as dense representations, onwhich the action is properly discontinuous. 

