Seminar Schedule
October 22 - October 26, 2007
All Seminars are held in Boyd Graduate Studies Bldg. unless otherwise noted.
MONDAY, October 22, 2007
Algebra
2:30pm, Room 410
Speaker: Cornelius Pillen, University of South Alabama
Title of talk: Cohomology of finite groups of Lie type
Abstract: Let $G$ be a reductive algebraic group over a field $k$ of prime characteristic $p$ which is split over the prime field $\Fp$. Let $\Fr : G \to G$ denote the Frobenius map. Then the fixed points of the $r$th iterate of the Frobenius map, denoted $\gfpr$, is a finite Chevalley group. The question of interest in this talk is to determine the least $i > 0$ such that the cohomology group $\opH^i(\gfpr,k) \neq 0$.
Quillen showed that $\opH^i(GL_n(\mathbb{F}_{p^r}),k) = 0$ for all $0 < i < r(p - 1)$ for all $n$. In that work, he noted that for all finite Chevalley groups there exist constants $C$ depending on the root system such that $\opH^i(\gfpr,k) = 0$ for $0 < i < Cr$. However no explict value of $C$ is given except for $G = SL_2$ (and $p$ odd) in which case one can take $C = (p-1)/2$. Further, it was not shown whether these vanishing ranges were sharp. Indeed, in the case of $SL_2$, one can see that these bounds are not sharp in general. Later work by Friedlander and by Hiller extended Quillen's results and found vanishing ranges for groups of all types. Since then, few if any results have been obtained in this direction.
The goal here is to exploit techniques developed by Bendel, Nakano and the presenter which relate $\opH^i(\gfpr,k)$ to extensions over $G$. In particular, for a group $G$ of classical type and $r=1$, under the assumption that $p > h$ (the Coxeter number), we improve on the vanishing ranges of Hiller and in many cases find sharp bounds.
Topology/Geometry Joint Seminar
2:30pm, Room 303
Speaker: Vladimir Oliker, Emory
Title of the talk: Embedding $S^{n}$ into $R^{n+1}$ with prescribed
integral Gauss curvature and optimal mass transport on $S^n$.
Faculty and Graduate Social
3:00pm, Room 409
Coffee, Tea, Cookies
TUESDAY, October 23, 2007
VIGRE - Graduate Student Seminar
2:00pm, Room 304
Speaker: Jesse Ratzkin, University of Georgia
Title of talk: The Maximum Principle
Abstract: The maximum principle is one of the most important tools in differential equations. Unfortunately, most people never see it until taking a graduate course in partial differential equations. I will begin by explaining why, in a way, you already know the maximum principle. Then I will discuss some of its applications in PDE and geometry.
FRG Analysis and Additive Combinatorics Working Group
3:30pm, Room 410
WEDNESDAY, October 24, 2007
Algebraic Geometry
2:30pm, Room 410
No meeting this week
Faculty and Graduate Student Social
3:00pm, Room 409
Coffee, Cookies, Tea
Arithmetic Geometry/Number Theory
3:30pm, Room 304
No meeting this week
Mathematical Physics
3:45pm, Room 302
Speaker: Brad Bassler, University of Georgia, Philosophy Department
Title of talk: Caratheodory's formulation of thermodynamics, cont.
THURSDAY, October 25, 2007
Fall Break
FRIDAY, October 26, 2007
Fall Break