Date/Time
Date(s) - 21/03/2025
3:30 pm - 4:30 pm

Speaker: Dr. Brett Nasserden (McMaster University)

Location: Hamilton Hall, Room 305

Title: Dynamical systems in the intersection of arithmetic, combinatorics, and geometry: A linear algebraic approach

Abstract: Dynamical systems arise in algebraic geometry when studying how rational maps between a variety and itself change under composition. To illustrate the concept, consider smooth projective curves: On the projective line one can define a map locally defined by f(x)=x^2+c, where c is a complex number. One is interested in the properties of iterates such as f(f(x)), f(f(f(x))), and so on and how the behavior depends on c. Alternatively, for an elliptic curve E, consider the map g:E?E defined by g(p)=2p using the group structure on E. Studying iterations of g is equivalent to analyzing the behavior of multiplication by powers of 2 on E. Completing the trichotomy, curves of genus at least 2 only admit finite-order automorphisms.

In the first case, the map is ramified at infinity and non-isomorphic; in the second case, the map is unramified but still non-isomorphic (due to 2-torsion points!), while in the third case, all surjective maps are isomorphic and of finite order. In other words, the properties of surjective maps of curves reflect the trichotomy of curves: genus 0 (ramified, non-isomorphic surjective maps), genus 1 (unramified, non-isomorphic surjective maps), and genus at least 2 (no non-isomorphic surjective maps).

In higher dimensions, understanding surjective self-maps remains an interesting open problem, with connections to the minimal model program and applications to arithmetic geometry. Everywhere-defined maps on projective spaces are given by collections of homogeneous polynomials of the same degree with no common zeros. A key question in this area is whether one can construct interesting non-constant families of such maps parametrized by the points of a smooth complex variety X. In this talk, I will discuss progress on this question when X is an elliptic curve or a toric variety, with applications to a conjecture of Kawaguchi and Silverman.

 

Coffee will be served in the same room, HH 305 at 3:00pm. All are welcome.