University of Toronto


Last year's seminar. 
Back to the Mathematics Department seminars page. 
Sept. 22
Title: On Eisenstein series and the cohomology of arithmetic groups
Speaker: Joachim Schwermer (University of Vienna)
Abstract:
The automorphic cohomology of a reductive kgroup G, defined in terms of
the automorphic spectrum of G, captures essential analytic aspects of the
arithmetic subgroups of G and their cohomology. We discuss the actual
construction of cohomology classes represented by residues or principal
values of derivatives of Eisenstein series. We show that nontrivial
Eisenstein cohomology classes can only arise if the point of evaluation
features a halfintegral property. This rises questions concerning the
analytic behavior of certain automorphic Lfunctions at halfintegral
arguments. The case of the group G obtained from the general linear group
GL_{2} over a generalized quaternion algebra defined over an
algebraic number field will serve as a specific example.
Sept. 29
Title: Quantum affine algebras, canonical bases and qdeformation
of arithmetical functions
Speaker: Henry Kim (University of Toronto)
Abstract:
We interpret GindikinKarpelevich formula and CasselmanShalika formula as
sums over Lusztig's canonical bases, generalizing the results of
BumpNakasuji to arbitrary split reductive groups. We also obtain their
affine analogues. Suggested by these formulas, we define natural
qdeformation of arithmetical functions such as (multi)partition function
and Ramanujan taufunction, and prove various identities among them. In
some examples, we recover classical identities by taking limits. We also
consider qdeformation of Kostant's function. This is a joint work with
KyuHwan Lee.
Oct. 6
Title: Arithmetic cycles on Picard modular varieties
Speaker: Stephen Kudla (University of Toronto)
Abstract:
In this talk I will describe the connections between algebraic and
arithmetic cycles in Picard modular varieties and the Fourier coefficients
of certain hermitian modular forms, especially Eisenstein series. This is
joint work with M. Rapoport.
Oct. 20
Title: Frobenius rigidity
Speaker: Ying Zong (University of Toronto)
Abstract:
We show that the sequence of iterated Frobenius endomorphisms of schemes
of positive characteristics have a strong rigidity. An application to
reduction of shimura varieties will be discussed.
Oct. 27
Title: An ErdosKac theorem for multiplicative order
Speaker: Leo Goldmakher (University of Toronto)
Abstract:
A famous theorem of Erdos and Kac asserts that the number of distinct
prime factors of n behaves like a normally distributed random variable
with mean and variance log log n. In this talk I will describe recent work
on an analogue of this theorem, on the distribution of the multiplicative
order of a fixed integer a (mod p) as p varies over primes. This is joint
work with Greg Martin (UBC).
Nov. 3
Title: Ngo's proof of the fundamental Lemma
Speaker: Joel Kamnitzer (University of Toronto)
Abstract:
I will give an expository talk on Ngo's proof of the fundamental lemma,
focusing on the geometric aspects of the proof. I will discuss affine
Springer fibres, the Hitchin fibration, and Ngo's support theorem.
Nov. 17
Title: The SiegelWeil formula for classical groups
Speaker: Shunsuke Yamana (Osaka City University)
Abstract:
The SiegelWeil formula is an identity between a value of a certain
Eisenstein series and an integral of a theta function. Such an identity
was first proven by Siegel and then extended to classical dual reductive
pairs by Weil under the assumption that the Eisenstein series is
absolutely convergent. Later, Kudla and Rallis extended this formula for
symplecticorthogonal dual pairs in certain cases beyond the range of
absolute convergence. In this talk I survey the work of Kudla and Rallis
as well as my recent work in which the SiegelWeil formula is extended to
all quaternion dual pairs.
Nov. 24
Title: The L^2 restriction of a GL(3) Maass form to GL(2)
Speaker: Xiaoqing Li (SUNYBuffalo)
Abstract:
In this talk, we sill study the L^2 restriction problem of a GL(3)Maass
form to GL(2). By Parseval's formula, the problem becomes bounding
averages of different families of GL(3)xGL(2) Lfunctions. Assuming the
Lindelof hypothesis for these GL(3)xGL(2) Lfunctions as we usually do,
one can prove the L^2 restriction is bounded by q^\vepsilon where q is the
analytic conductor of the varying GL(3) Maass form. However, we will give
an unconditional proof of this bound for selfdual GL(3) Maass forms. For
nonselfdual GL(3) Maass forms, our bounds depend on the bounds for the
first Fourier coefficients of the GL(3) Maass forms. This is a joint work
with Matt Young.
Dec. 1
Title: Pairs of polynomials over Q taking infinitely many
common values
Speaker: Ben Weiss (University of Michigan)
Abstract:
For two polynomials G(X), H(Y) with rational coefficients, when does G(X)
= H(Y) have infinitely many solutions over the rationals? Such G and H
have been classified in various special cases by previous mathematicians.
I will discuss both the known results, the new results in my thesis, and
related problems which have yet to be solved.
Dec. 8
Title: The combinatorics of local models for Shimura varieties
Speaker: Brian Smithling (University of Toronto)
Abstract:
I will discuss some combinatorial problems on the enumeration of certain
Schubert varieties in affine flag varieties. These problems arise in the
theory of local models for Shimura varieties, but they admit a purely
grouptheoretic formulation. Therefore essentially no background on
Shimura varieties or their local models will be necessary.
Jan. 19
Title: The Congruence Subgroup Kernel and the Fundamental Group of
the Reductive BorelSerre Compactification
Speaker: John Scherk (University of Toronto)
Abstract:
Let G be simple, simply connected algebraic group defined over a number
field k, and let S be a finite set of places which includes all the
infinite ones. The congruence subgroup problem asks whether all
Sarithmetic subgroups of G are congruence subgroups, or equivalently,
whether the congruence subgroup kernel is trivial. The cohomology of
compactifications of locally symmetric spaces of G has been studied
extensively, largely because of its relation to the cohomology of
(S)arithmetic subgroups and to automorphic forms. It turns out that the
fundamental group of the reductive BorelSerre compactification is related
to the congruence subgroup kernel. This involves describing the
fundamental group in terms of "elementary matrices".
Jan. 26
Title:
Nekovar duality over padic Lie extensions of global fields
Speaker: Meng Fai Lim (University of Toronto)
Abstract:
In his monograph, Nekovar gave formulations of analogues of Tate local
duality and PoitouTate duality for finitely generated modules over a
complete commutative local Noetherian ring R with finite residue field of
characteristic a fixed prime p. In the usual formulation of Tate local
duality and PoitouTate duality, one takes the Pontryagin dual which does
not in general preserve the property of finite generation. Nekovar takes
the dual with respect to a dualizing complex of Grothendieck so as to have
a duality between bounded complexes of Rmodules with finitely generated
cohomology groups. In this talk, we give a generalization of this result
to the setting of nonabelian padic Lie extensions. This is a joint work
with Romyar Sharifi.
Feb. 2
Title:
Values of Green functions at CM points
Speaker: Stephen Kudla (University of Toronto)
Abstract:
I will survey of various generalizations of the classic GrossZagier
formula giving an explicit formula for the factorization of singular
moduli. The Green functions on Shimura varieties of orthogonal type are
constructed via the Borcherd's regularized theta lifts. Their special
values at various types of CM points can be expressed in terms of
derivatives of Fourier coefficients of incoherent Eisenstein series. In
the case of `big' CM points, these results are joint work with Jan
Bruinier and Tonghai Yang.
Feb. 9
Title:
Classification of Representations
Speaker: James Arthur (University of Toronto)
Abstract:
Suppose that G is a quasisplit orthogonal or symplectic group over a field
F of characteristic 0. We shall describe a classification of the
irreducible representations of G(F) if F is local, and the automorphic
representations in the discrete spectrum of G if F is global.
Feb. 16
Title:
Monodromy Exercise
Speaker: Ying Zong (University of Toronto)
Abstract:
With Kumar Murty, we give a proof of a theorem of Serre on big
monodromy of abelian varieties over number fields. We also prove a new
theorem of big monodromy.
Mar. 2
Title:
Postcritically finite polynomials
Speaker: Patrick Ingram (University of Waterloo)
Abstract:
In classical holomorphic dynamics, rational selfmaps of the Riemann
sphere whose critical points all have finite forward
orbit under iteration are known as postcritically finite (PCF) maps. A
deep result of Thurston shows that if one excludes
examples arising from endomorphisms of elliptic curves, then PCF maps are
in some sense sparse, living in a countable
union of zerodimensional subvarieties of the appropriate moduli space (a
result offering dubious comfort to number
theorists, who tend to work over countable fields). We show that if one
restricts attention to polynomials, then the set of
PCF points in moduli space is actually a set of algebraic points of
bounded height. This allows us to give an elementary
proof of the appropriate part of Thurston's result, but it also provides
an effective means of listing all PCF polynomials
of a given degree, with coefficients of bounded algebraic degree (up to
the appropriate sense of equivalence).
Mar. 23
Title:
Nilpotence join theorem for Lie algebras
Speaker:
Hamid Usefi (University of Toronto)
Abstract:
I will prove that if L is a Lie algebra generated by subideals H and K and
L/L' is finitedimensional, then given positive integers a and b there
exists a positive integer c such that \gamma_c(L) is contained in the sum
of \gamma_a(H) and \gamma_b(K). One of the main ingredients is a similar
result for the augmentation ideals. (Based on a joint work with Salvatore
Siciliano.)
Mar. 30
Title:
Some results about Arthur's packets in the real case, and a topological
application
Speaker:
Laurent Clozel (Universite ParisSud 11)
Abstract:
This is joint work with N. Bergeron. We prove (conditional on some
expected results pertaining to the twisted trace formula) that certain
hyperbolic 7manifolds have trivial 1st Betti number. This depends on the
understanding of the local "packets" defined by Arthur in his
classification of automorphic forms on classical groups. In the lecture I
will explain our partial results in this direction.
Apr. 6 (1st
session)
Title:
A geometric variant of Titchmarsh divisor problem
Speaker:
Amir Akbary (University of Lethbridge)
Abstract:
We formulate a geometric analogue of the Titchmarsh Divisor Problem in the
context of elliptic curves and more generally for abelian varieties. For
any abelian variety A defined over rationals, we study the asymptotic
distribution of the primes which split completely in the division fields
of A. This is a joint work with Dragos Ghioca (UBC).
Apr. 6 (2nd
session)
Title:
Zeros of a Family of Approximations of the Riemann ZetaFunction
Speaker:
Steve Gonek (University of Rochester)
Abstract:
Let U(s) = ∑_{ n ≤ N} n^{s} be a
partial sum of
the Riemann zetafunction ζ(s), where s = σ + it is a complex
variable. We set ζ_{N}(s) = U(s) + χ(s) U(1s).
If
N = √t /
2π, the approximate functional equation for
ζ(s) tells us that ζ_{N}(s) is an excellent
approximation of ζ(s). Here we describe the distribution
of zeros of ζ_{N}(s) when N is independent of t. For
instance, we discuss "critical" strips in which most of the zeros lie
(on and off the Riemann Hypothesis), the number of zeros of
ζ_{N}(s) in rectangles, and lower bounds for the number of
zeros of ζ_{N}(s) on the critical line Re s = ½.
In particular, we show that if N ≥ 1 is fixed and T is sufficiently
large, then 100% of the zeros with ordinates in (T,2T] lie on the critical
line and are simple. This is joint work with Hugh Montgomery.