Hello everybody! Next Wednesday we will have another meeting of KSE Mathematics Seminar with a lecture from a visiting researcher:
“Counting problems over finite fields and dynamics on algebraic groups”
Speaker: Jakub Byszewski (Jagiellonian University)
🗓When: July 8, 16:30–17:30
📍Where: KSE Dragon Capital Building, 3 M. Shpaka St., Room 3.05
Abstract:
Two elementary counting problems over finite fields --- counting invertible matrices of a given size and counting irreducible polynomials of a given degree --- can be interpreted dynamically: they amount to counting fixed points or periodic orbits of Frobenius on algebraic groups such as the general linear group and the additive group. In this talk, we will discuss what happens when these examples are replaced by an arbitrary endomorphism of an algebraic group over the algebraic closure of a finite field.
We will study the number of periodic points, the associated dynamical zeta function, and related questions inspired by classical dynamics. For semisimple groups, classical work of Steinberg shows that the answer is essentially governed by cohomology. For general algebraic groups, however, this purely cohomological picture acquires an additional p-adic distortion. We will illustrate this phenomenon using examples involving tori, abelian varieties, and vector groups.
We will introduce a class of dynamical systems that contains all of these examples, as well as further examples coming from dynamics on compact abelian groups and cellular automata. In this setting, we will state a strong rational/transcendental dichotomy for the associated zeta functions and discuss whether there is a variant of the prime number theorem for periodic orbits.
The talk will emphasize examples and will be aimed at a general mathematical audience. It is based on joint work with Gunther Cornelissen and Marc Houben
