Abstracts

Frank Calegari (Northwestern)
What is an arithmetic group?
This will be an introductory talk on some questions and concepts related to the title.

Alex Gamburd (UCSC/Northwestern)
Sumproduct, expanders, and sieving

Ramin TaklooBighash (UIC)
GrossPrasad for GSp(4)
The conjectures of Gross and Prasad predict precise branching laws for representations of
orthogonal groups when restricted to orthogonal subgroups locally and globally. In this talk I will
explain a recent work verifying certain special cases with interesting arithmetic applications.
This is joint work with Dipendra Prasad.

Matthew Emerton (Northwestern)
Elliptic curves and modular forms
This is an introductory lecture on the connections between the arithmetic of elliptic curves and the theory of modular forms.

John Flynn (Northwestern)
Galois groups and the image size of rational maps

Mark Reeder (Boston College)
Simple supercuspidal representations and simple wild parameters
This talk is about the interaction between Lie groups and local Galois theory.
The local Langlands conjecture predicts that irreducible squareintegrable representations of a reductive
group G over a padic field k should be parametrized by certain homomorphisms from the absolute Galois
group of k into a complex Lie group LG which is in some sense dual to G. A recent conjecture of
HiragaIchinoIkeda enhances this conjecture to predict the formal degree of the representation in terms of
gamma factors of the parameter. A refinement of the HII prediction leads to the simplest examples of the
local Langlands conjecture, which (surprisingly) seem to have gone unnoticed until now. This is joint work
with Benedict Gross.

Tong Liu (Purdue)
Lattices in semistable representations
Let p be an odd prime. We construct and study integral (torsion) padic Hodge data, such as WeilDeligne representation, associated to integral (torsion) semistable representations. As an application, we prove the compatibility between local Langlands correspondence and Fontaine's construction at p for Galois representations attached to Hilbert modular forms.

Stephen Kudla (Toronto)
Arithmetic cycles for unitary groups
In this talk, I will discuss some recent work with M. Rapoport on arithmetic
cycles for Shimura varieties associated to U(n1,1).
In particular, we establish a relation between the arithmetic degrees of certain 0cycles and
the nonsingular, nondegenerate Fourier coefficients of the derivatives
of certain incoherent Eisenstein series on U(n,n).

Fucheng Tan (MIT)
Galois eigenvarieties
I generalize Kisin's construction of Galois theoretic eigencurve
on GL_2(Q) to GL_N and GSp_{2g} over arbitrary number fields. A criterion for smoothness of de Rham points in a Galois
eigenvariety and a lower bound of dimension of the eigenvariety will be given. Time permitting, I will prove (in many cases)
the de Rhamness of Galois representations attached to Siegel modular forms of low weight and modular forms on GL_2 over
imaginary quadratic fields.

David Helm (UT Austin)
On padic families of admissible representations of GL_2(Q_l)
The local Langlands correspondence for GL_2 associates an admissible
representation of GL_2(Q_l) to every Frobeniussemisimple twodimensional
representation of the Weil group W of Q_l. It is an interesting question
to try to extend this correspondence to padic families of representations
of W that is, given a padic family of representations of W, construct a
corresponding family of admissible representations of GL_2(Q_l). Recent work of Emerton gives a set of
properties that uniquely characterise such
a family. We show how to construct such families using deformationtheoretic arguments.
