Activities This Week

The conjecture of Loxton and var der Poorten is a criterion for a formal power series to be the expansion at 0 of a rational function, and is related to a famous theorem of Cobham in the theory of finite automata. It was proved by Adamczewski and Bell in 2013. Recently, Schafke and Singer found a novel approach that lead also to a simple conceptual proof of Cobham’s theorem. We shall explain these results and the cohomological machinery behind them, and discuss what is missing from the picture to establish an elliptic analogue.

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.