Abstracts
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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 well-known results in this setting, including:
-Green-Tao 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 f-g = h^2 for some nonzero polynomial h in F_q[t].
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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.
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Kai-Wen Lan (Princeton/IAS)
Vanishing theorems for torsion automorphic sheaves
Given a compact PEL-type 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_p-coefficients of the given weight vanishes away from the middle
degree, and hence has no p-torsion. (This is joint work with Junecue Suh.)
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Frank Calegari (Northwestern)
Galois representations and the Fontaine-Mazur conjecture
We discuss some general conjectures regarding p-adic Galois representations.
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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
Barsotti-Tate 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 Barsotti-Tate Galois representations. This is joint work with
Thomas Barnet-Lamb and Toby Gee.
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