Abstracts

Le Thai Hoang (UCLA)
Arithmetic Combinatorics in Function Fields
Analogies between the integers and the ring F_q[t] have been long known. However, from an arithmetic combinatorics perspective, these analogies have been little and only recently explored. As it turns out, in many cases existing methods can be transfered directly to F_q[t], while at times extra difficulties will arise. In this talk, I will discuss about analogs of some wellknown results in this setting, including:
GreenTao theorem for F_q[t]: The irreducible polynomials in F_q[t] contain affine spaces of arbitrarily high dimension.
Sarkozy's theorem for F_q[t]: In any subset of positive density in F_q[t], we can find polynomials f, g such that fg = h^2 for some nonzero polynomial h in F_q[t].

Frank Calegari (Northwestern)
The Galois Groups of Graphs
Let Gamma be a finite graph. When is the field generated by the largest eigenvalue of the adjacency matrix of G abelian?
We study this (and related questions) by investigating some surprising properties of small cyclotomic integers. This is joint
work with Noah Snyder and Scott Morrison.

KaiWen Lan (Princeton/IAS)
Vanishing theorems for torsion automorphic sheaves
Given a compact PELtype Shimura variety, a sufficiently
regular weight (defined by mild and effective conditions), and a prime
number p unramified in the linear data and larger than an effective
bound given by the weight, we show that the etale cohomology with
Z_pcoefficients of the given weight vanishes away from the middle
degree, and hence has no ptorsion. (This is joint work with Junecue Suh.)

Frank Calegari (Northwestern)
Galois representations and the FontaineMazur conjecture
We discuss some general conjectures regarding padic Galois representations.

David Geraghty (Harvard)
Congruences between weight 2 Hilbert modular forms
Let F be a totally real field and rhobar an irreducible modular
mod l representation of G_F. We prove an existence theorem for potentially
BarsottiTate modular lifts of rhobar. The key ingredient in the proof is
a result guaranteeing the existence of ordinary lifts, after replacing F
by a solvable extension. We give applications to modularity lifting
theorems for BarsottiTate Galois representations. This is joint work with
Thomas BarnetLamb and Toby Gee.
