This seminar is organized by Henry Kim. If you would like to speak at the seminar, please email Henry (henrykim at math dot toronto dot edu).
For inquiries regarding this web page, please email Jonathan (jkorman at math dot toronto dot edu).

Fall 2004 Term Schedule

Winter 2005 Term Schedule

Summary

January 26
Title: Residues and Grothendieck duality-II
Speaker: Pramath Sastry (U of T)
Abstract: I plan to talk on recent advances in duality---especially the series of (very) long papers/books, beginning with Brian Conrad's, and ending in work done independently and sometimes in collaboration with Joe Lipman and S. Nayak.

February 2
Title: INDPRO methods in representation theory (an introductory overview)
Speaker: Elliot Lawes (U of T)
Abstract: I'll begin by explaining the terms in the title, mostly by example. (E.g. An IndPro-finite group, like GL_2(F_q((t))) is the direct limit of the inverse limit of a collection of appropriately-behaved finite sets/groups.) I'll then tell the story of the Satake Isomorphism, from its inception in the IndPro-finite (i.e. locally compact) context, to its recent incarnation in the IndPro-algebraic context. If time permits, I'll finish with a beginning: IndProIndPro methods! This isn't just art for art's sake, but provides a concrete motivation for the IndPro perspective. Throughout, I'll try to provide a ``map of the subject'' rather than present technical information. Graduate students are encouraged to attend.

February 9
Title: Multiple L Values and Periods
Speaker: Ramesh Sreekantan (Tata Institute of Fundamental Research)
Abstract: There are generalizations of the Riemann Zeta function to functions of several variables called the Multiple Zeta functions. Like the usual zeta function, their values at positive integer points, called multiple Zeta values, are interesting. There are also several relations between different multiple Zeta Values. While originally defined by Euler, recently these numbers have been studied from a different point of view. They are periods in the sense of Kontsevich and Zagier. Further they appear as periods of a mixed Hodge structure on the fundamental group of the complex projective plane with the points 0,1 and the point at infinity removed. In this talk we define a generalization of such numbers called Multiple L Values of modular forms. We show that they have similar properties to the Multiple Zeta Values and further, some of the values are periods. Further, in some cases we can show that these numbers are periods of a mixed Hodge structure on the fundamental group of a modular curve.

March 2
Title: Irreducible specialization in characteristic 2
Speaker: Brian Conrad (University of Michigan)
Abstract:A few years ago, in joint work with K. Conrad and R. Gross, it was shown that the function-field analogue of classical heuristics on prime specialization of irreducible polynomials over Z are false, due to a new phenomenon ("Mobius bias") unlike anything known in characteristic 0; it was also shown that this Mobius bias permits a plausible correction for the conjecture in the function field case. Though those results (and the associated conjecture) have been extended to higher genus at the expense of using much heavier amounts of algebraic and rigid geometry, in the case of characteristic 2 there remain some basic vexing questions that can be illustrated by very concrete examples. In this talk, we present such concrete examples and discuss some related theorems (proved jointly with K. Conrad and R. Gross), thereby motivating the formulation of some open problems that the speaker has no clue how to solve.

March 9
Title: Carmichael Numbers
Speaker: Vicentiu Tipu (U of T)
Abstract:I am planning to give an overview of the ideas used by Alford, Granville and Pomerance in their 1994 paper to prove that there are infinitely many Carmichael numbers, and to discuss a more special type of Carmichael numbers, and some results (one of them my own) related to them. The arguments are mostly elementary, so hopefully it will appeal to a wide audience..

March 16
Title: Motivic Tamagawa Numbers of Algebraic Groups
Speaker: Ajneet Dhillon (Western Ontario)
Abstract: We begin by recalling the theory of Tamagawa numbers in the geometric case for split groups. Motivated by this these results a conjectural formula for the motive of the moduli stack of $G$-torsors over a curve is presented. The formula is then related to some formulas of Atiyah and Bott for the Poincare polynomials of gauge groups. If time permits we will discuss the cases in which the conjecture is known. I will give a brief account of algebraic stacks at the start of the talk, enough to understand what is going on.

March 23
Title: Large Sieve Inequalities
Speaker: Liangyi Zhao (U of T)
Abstract: In this talk, I will talk about the classical large sieve and my recent work (theorems and conjectures) extending the classical theory.

March 30
Title: Elliptic Curves and Random Matrix Theory
Speaker: Michael Rubinstein (University of Waterloo)
Abstract: I'll discuss, in the context of random matrix theory, the value distribution of L-functions associated to the quadratic twists of an elliptic curve.

April 1 and 4
Title: Non-Abelian L-Functions for Global Fields
Speaker: Lin Weng (Kyushu University)
Abstract: In this two talks, we are going to introduce and examine non-abelian zeta and $L$-functions for number fields. In the first talk, we start our discussion on non-abelian zeta functions for number fields. This is a natural generalization of Dedekind zeta functions and is based on the notion of stable lattices and a new type of cohomology. Basic properties will be given. Moreover, we will show that such zeta functions can be rewritten in terms of certain integrations of Epstein type of zeta functions. This then leads to a more general type of non-abelian $L$-functions. Simply put, non-abelian $L$ functions are defined as integrations of Eisenstein series associated with $L^2$-automorphic forms with integration domain compact subsets obtained by truncating the corresponding fundamental domain using stability. As such, we then will use Langlands' theory of Eisenstein series to show that these non-abelian $L$-functions admit a unique meromorphic continuation, satisfy functional equations. In the second talk, we will give a detailed study of a relation between Arthur's analytic truncation and our geometric truncation. This is motivated by a similar discussion of Lafforgue on vector bundles over curves. Consequently, we introduce the so-called abelian and essential non-abelian parts of these non-abelian $L$-functions. Motivated by Arthur's work on inner products of Eisenstein series and Jacquet-Lapid-Rogawski's work on automorphic preriods, we will indicate how abelian part of non-abelian $L$-functions can be evaluated via a generalized version of Rankin-Selberg (and Zagier) method.