פעילויות השבוע

The usual weighted average of points $(z_0, ..., z_q)$ in the real vector space $R^n$, with weights $(w_0, ..., w_q)$, is translation invariant. Hence it can be seen as an average of points in a torsor Z over the Lie group $G = R^n$ (A $G$-torsor is a $G$-manifold with a simply transitive action.)

In this talk I will explain how this averaging process can be generalized to a torsor Z over a unipotent Lie group $G$. (In differential geometry, a unipotent group is a simply connected nilpotent Lie group. $R^n$ is an abelian unipotent group.)

I will explain how to construct the unipotent weighted average, and discuss its properties (functoriality, symmetry and simpliciality). If time permits, I will talk about torsors over a base manifold, and families of sections parametrized by simplices. I will indicate how I came about this idea, while working on a problem in deformation quantization.

Such an averaging process exists only for unipotent groups. For instance, it does not exist for a torus $G$ (an abelian Lie group that‘s not simply connected). In algebraic geometry the unipotent averaging has arithmetic significance, but this is not visible in differential geometry.

Notes for the talk can be founds here: https://www.math.bgu.ac.il/~amyekut/lectures/average-diff-geom/abstract.html

Given an integral vector, there are several geometric and arithmetic objects one can attach to it. For example, its direction (as a point on the unit sphere), the lattice obtained by projecting the integers to the othonormal hyperplane to the vector, and the vector of residues modulo a prime p to name a few. In this talk I will discuss results pertaining to the statistical properties of these objects as we let the integral vector vary in natural ways.

אנסה להסביר כיצד נקודת מבט הסתברותית עוזרת לקבל באופן אלמנטרי תובנות שונות בתורת המספרים.

If f is a cuspidal modular eigenform of weight k>1, Ribet proved that its associated p-adic Galois representation is irreducible for all primes. More generally, it is conjectured that the p-adic Galois representations associated to cuspidal automorphic representations of GL(n) should always be irreducible.

In this talk, I will prove a version of this conjecture for low weight, genus 2 Siegel modular forms. These two-dimensional analogues of weight 1 modular forms are, conjecturally, the automorphic objects that correspond to abelian surfaces.