NTS/Abstracts Spring 2011: Difference between revisions

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== Anton Gershaschenko  ==
== Anton Gershaschenko  ==


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== Shuichiro Takeda, Purdue  ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: On the regularized Siegel-Weil formula for the second terms and
non-vanishing of theta lifts from orthogonal groups
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Abstract: In this talk, we will discuss (a certain form of) the
Siegel-Weil formula for the second terms (the weak second term
identity). If time permits, we will give an application of the
Siegel-Weil formula to non-vanishing problems of theta lifts. (This is
a joint with W. Gan.)
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== Xinyi Yuan  ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Volumes of arithmetic line bundles and equidistribution
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| bgcolor="#DDDDDD"| 
In this talk, I will introduce equidistribution of small points in
algebraic dynamical systems. The result is a corollary of the
differentiability of volumes of arithmetic line bundles in Arakelov
geometry. For example, the equidistribution theorem on abelian varieties
by Szpiro-Ullmo-Zhang is a consequence of the arithmetic Hilbert-Samuel
formula by Gillet-Soule.
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== Jared Weinstein, IAS  ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: Resolution of singularities on the tower of modular curves
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Abstract:
The family of modular curves X(p^n) provides the geometric link between two types of objects:  On the one hand, 2-dimensional representations of the absolute Galois group of Q_p, and on the other, admissible representations of the group GL_2(Q_p).    This relationship, known as the local Langlands correspondence, is realized in the cohomology of the modular curves.  Unfortunately, the Galois-module structure of the cohomology of X(p^n) is obscured by the fact that integral models have very bad reduction.  In this talk we present a new combinatorial picture of the resolution of
singularities of the tower of modular curves, and demonstrate how this
picture encodes some features of the local Langlands correspondence.
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== David Zywina, U Penn  ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: Bounds for Serre's open image theorem
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Abstract.
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== Soroosh Yazdani, UBC and SFU ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: Local Szpiro Conjecture
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For any elliptic curve E over Q, let N(E) and Delta(E)    denote it's conductor and minimal discriminant. Szpiro conjecture    states that for any epsilon>0, there exists a constant C    such that    Abs(Delta(E)) < C (N(E))^{6+\epsilon}    for any elliptic curve E. This conjecture, if true, will have    applications to many Diophantine equations.    Assuming Szpiro conjecture, one expects that there are    only finitely many semistable elliptic curves    E such that    min_{p|N(E)} v_p(\Delta(E)) >6.    We conjecture that, in fact, there are none. In this talk    we study this conjecture in some special cases, and provide    some evidence towards this conjecture.
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== Zhiwei Yun, MIT ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: From automorphic forms to Kloosterman sheaves (joint work with J.Heinloth and B-C.Ngo)
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Abstract: Classical Kloosterman sheaves are rank n local systems on
the punctured line (over a finite field) which incarnate Kloosterman
sums in a geometric way. The arithmetic properties of the Kloosterman
sums (such as estimate of absolute values and distribution of angles)
can be deduced from geometric properties of these sheaves. In this
talk, we will construct generalized Kloosterman local systems with an
arbitrary reductive structure group using the geometric Langlands
correspondence. They provide new examples of exponential sums with
nice arithmetic properties. In particular, we will see exponential
sums whose equidistribution laws are controlled by exceptional groups
E_7,E_8,F_4 and G_2.
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== David Brown, UW Madison ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title
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Abstract.
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== Bryden Cais, UW Madison ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title:  On the restriction of crystalline Galois representations
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| bgcolor="#DDDDDD"| Abstract: We formulate a generalization of a conjecture of Breuil (now a theorem of Kisin) on the restriction of crystalline p-adic Galois
representations to a general class of infinite index subgroups of the Galois group.  Following arguments of Breuil, we will explain the proof of our generalization in the Barsotti-Tate case.
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== Tom Hales, University of Pittsburg ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title
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| bgcolor="#DDDDDD"| 
At the International Congress of Mathematicians in India in
August, Ngo Bao Chau was awarded a Fields medal for his proof of the
"Fundamental Lemma."  This talk is particularly intended for students
and mathematicians who are not specialists in the theory of
Automorphic Representions.  I will describe the significance and some
of the applications of the "Fundamental Lemma."  I will explain why
this problem turned out to be so difficult to solve and will give some
of the key ideas that go into the proof.
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== Jay Pottharst, Boston University ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: Iwasawa theory at nonordinary primes
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| bgcolor="#DDDDDD"| 
Abstract.
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== Melanie Matchett Wood, Stanford and AIM ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: Geometric parametrizations of ideal classes
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| bgcolor="#DDDDDD"| 
In a ring of algebraic integers, the ideal class group measures the
failure of unique factorization.  A classical correspondence due to
Dirichlet and Dedekind allows us to work with ideal classes of
quadratic rings concretely in terms of binary quadratic forms with
integer coefficients.  A recent result of Bhargava gives an analogous
correspondence between ideal classes of cubic rings and certain
trilinear forms.  From another point of view, the ideal class group is
the group of invertible modules of a ring, whose geometric analog is
the Picard group of line bundles on a space.  We discuss how we can
view these correspondences between ideal classes and forms
geometrically, and give new results on parametrizations of ideal
classes of certain rank n rings (e.g. orders in degree n number
fields) by trilinear forms.
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== Samit Dasgupta, UC Santa Cruz ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| On Greenberg's conjecture on derivatives of p-adic L-functions with
trivial zeroes
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| bgcolor="#DDDDDD"| 
In 1991, Ralph Greenberg stated a conjecture about p-adic L-functions
that have a trivial zero at s=1.  Here "trivial" means that the zero
arises from the vanishing of an Euler factor that must be removed in
order to state the interpolation property of the p-adic L-function.
Greenberg's conjecture concerns the value of the derivative of the
p-adic L-function at s=1.  An example of this conjecture is the case
of the p-adic L-function of an elliptic curve E/Q with split
multiplicative reduction at p.  In this case, Greenberg's conjecture
reduces to an earlier conjecture by Mazur, Tate, and Teitelbaum, and
was proven by Greenberg himself in joint work with Glenn Stevens.  In
this talk, we will describe a strategy to prove new cases of
Greenberg's conjecture.  We will concentrate on the case of the
symmetric square of an elliptic curve with good reduction at p.  The
strategy is a generalization of my previous work with Darmon and
Pollack proving certain cases of the Gross--Stark conjecture (which
can also be viewed as a special case of Greenberg's conjecture).  The
method involves studying explicit p-adic families of modular forms on
GSp_4 and their associated Galois representations.
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== David Geraghty, Princeton and IAS ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: Potential automorphy for compatible systems
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| bgcolor="#DDDDDD"| 
Abstract: I will describe a joint work with Barnet-Lamb, Gee and Taylor where we establish a potential automorphy result for compatible systems of Galois representations over totally real and CM fields. This is deduced from a potential automorphy result for single l-adic Galois representations satisfying a `diagonalizability' condition at the places dividing l.
|}                                                                       
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== Toby Gee, Northwestern ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#DDDDDD" align="center"| Title: Potential automorphy for compatible systems
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| bgcolor="#DDDDDD"| 
Abstract: I will continue David Geraghty's talk, and discuss a number of applications.
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== Organizer contact information ==
== Organizer contact information ==

Revision as of 19:02, 16 January 2011

Anton Gershaschenko

Title: Moduli of Representations of Unipotent Groups

Abstract: Representations of reductive groups are discretely parameterized, but unipotent groups can have non-trivial families of representations, so it makes sense try to construct and understand a moduli stack (or space) of representations of a given unipotent group. If you restrict to certain kinds of representations, it is possible to actually get your hands on the moduli stack and to construct a moduli space. I'll summarize the few things I know about the general case and then give you a tour of some interesting features that appear in small examples.



Organizer contact information

David Brown:

Bryden Cais:


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