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
Abstract:
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.

See the link http://math.huji.ac.il/~shnidman/HBWA.html

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: The generalized doublingmethod and its applications
Abstract: The doubling method,first introduced by Piatetski-Shapiro and Rallis in the 80s, has had numerousapplications, e.g. to the theta correspondence and to arithmetic problems.In a series of recent works this method was generalized in severalaspects, with an application to functoriality from classical groups to GL(N).The most recent result is a multiplicityone theorem (joint work with Gourevitch and Aizenbud).
I will brieflysurvey the method and talk about some of its applications.

See the link http://math.huji.ac.il/~shnidman/HBWA.html

This semester will be devoted to resolution of singularities -- a process that modifies varieties at the singular locus so that the resulting variety becomes smooth. For many years this topic had the reputation of very technical and complicated, though rather elementary.
In fact, the same resolution algorithm can be described in various settings, including schemes, algebraic varieties or complex analytic spaces.

Title: Mordell-Lang and Integral Points on Abelian Varieties in Characteristic p

This semester will be devoted to resolution of singularities -- a process that modifies varieties at the singular locus so that the resulting variety becomes smooth. For many years this topic had the reputation of very technical and complicated, though rather elementary.
In fact, the same resolution algorithm can be described in various settings, including schemes, algebraic varieties or complex analytic spaces.

Title: The tame site

In the representation theory of reductive p-adic groups, a lot of general structure was developed by Bernstein: supercuspidal representations, cuspidal support, Bernstein components and so on.

Let G be the group SL(2) over a finite extension F of Q_p, p odd. For a fixed r ≥ 0, we identify the elements of the Bernstein center of G supported in the Moy-Prasad G-domain G_{r^+}, by characterizing them spectrally.
We study the behavior of convolution with such elements on orbital integrals of functions in C^∞_c(G(F)), proving results in the spirit of semisimple descent.
These are ‘depth r versions’ of results proved for general reductive groups by J.-F. Dat, R. Bezrukavnikov, A. Braverman and D. Kazhdan.