ABSTRACTS:

Henry Cohn
Title: Horosphere Packings

Abstract: We study packings with horospheres in hyperbolic space, and highlight the role of number theory. We construct packings using certain semisimple group schemes over Z, analyze the packings in terms of number-theoretic properties of these groups, and show how to use trace formula arguments to prove density bounds. This is joint work with Thomas Hales, Jacob Lurie, and Peter Sarnak.


CheeWhye Chin
Title: Lafforgue's Work on the Langlands Correspondence over Function Fields

Abstract: This will be an expository account of Lafforgue's theorem on the Langlands Correspondence for GL(r) over function fields. I will give precise statements of the fundamental results of Lafforgue, and indicate the key ingredients used in his proof. Time permitting, I will also discuss the applications of this theorem to arithmetic geometry and representation theory.


Michael Bennett
Title: Perfect Powers from Progressions

Abstract: In this talk, we will discuss how techniques from the theory of Galois representations and modular forms can be brought to bear on a classical question of Erdos on whether the product of consecutive terms in an arithmetic progression can be a perfect power.


Karl Rubin
Title: Kolyvagin Systems

Abstract: In its simplest setting, Kolyvagin's construction attaches to each cyclotomic unit 1- zetapn a "derivative" element in Q(zetap)x/ (Q(zetap)x)pk for appropriate k. Kolyvagin then uses these elements to bound the ideal class group of Q(zetap).

It turns out that this collection of derivative elements has more structure than had previously been recognized. In this talk I will discuss joint work with Barry Mazur, in which we attempt to understand more fully what these derivative elements are, and what they are good for.


Joe Buhler
Title: The probability that a p-adic polynomial splits

Abstract: This talk will begin by calculating the probability that a uniformly chosen polynomial, with degree n and coefficients in the p-adic integers, factors completely into a product of linear polynomials. (Warm-up question for you: find the answer for the case n = 2, i.e., quadratic polynomials.) This was motivated by ideas on generalizing Serre's ``mass formula'' to not necessarily irreducible polynomials; further results in this direction will be discussed. This is joint work with Asher Auel.


Stephen Choi
Title: Small Prime Solutions for Quadratic Equations with Five Variables

Abstract: Initiated by a diophantine problem considered by A. Baker, the solubility and small solutions over primes of the following equation were studied by M.C. Liu and K. M. Tsang:

b=a1p1 + a2p2 + a3p3

When a1 = a2 = a3, their results recover the celebrated Vinogradov's three prime theorem. In this talk, we will discuss their results for other diophantine equations especially the quadratic equations

b=a1(p1)2 + a2(p2)2 + a3(p3)2 + a4(p4)2 . + a5(p5)2



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