Speaker: Spencer Leslie (Duke)
Title: The endoscopic fundamental lemma for unitary symmetric spaces
Abstract: Motivated by the study of certain cycles in locally symmetric
spaces and periods of automorphic forms on unitary groups, I propose a
theory of endoscopy for certain symmetric spaces. The main result is the
fundamental lemma for the unit function. After explaining where the
fundamental lemma fits into this broader picture, I will describe its
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Abstract. We investigate the approximation rate of a typical element of the Cantor set by dyadic rationals. This is a manifestation of the times two times three phenomenon, and is joint work with Demi Allen and Han Yu.
Title: Cup products oncurves over finite fields
Abstract: This is joint work with Ted Chinburg.
Let C be a smooth projective curve over a finite field k, and
let l be a prime number different from the characteristic of k.
In this talk I will discuss triple cup products on the first etale
cohomology group of C with coefficients in the constant
sheaf of l-th roots of unity. These cup products are important
for finding explicit descriptions of the l-adic completion of the
etale fundamental group of C and also for cryptographic
Title: Ramanujan Conjectures, Density Hypotheses and Applications for Arithmetic Groups.
Abstract: The Generalized Ramanujan Conjecture (GRC) for GL(n) is a central open problem in modern number theory. Its resolution is known to yield applications in many fields, such as: Diophantine approximation and arithmetic groups. For instance, Deligne's proof of the Ramanujan-Petersson conjecture for GL(2) was a key ingredient in the work of Lubotzky, Phillips and Sarnak on Ramanujan graphs.
Title: The Grothendieck--Serre conjecture for classical groups in low dimensions
A famous conjecture of Grothendieck and Serre predicts that if G is a reductive group scheme over a semilocal regular domain R and X is a G-torsor, then X has a point over the fraction field of R if and only if it has an R-point. Many instances of the conjecture have been established over the years. Most notably, Panin and Fedorov--Panin proved the conjecture when R contains a field.