Schedule for: 21w5512 - Algebraic Dynamics and its Connections to Difference and Differential Equations

A buffet dinner is served daily between 5:30pm and 7:30pm in Kinnear Center 105, main floor of the Kinnear Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Many questions in combinatorics, probability and statistical mechanics can be reduced to counting lattice paths (walks) in regions of the plane. A standard approach to counting problems is to consider properties of the associated generating function. These functions have long been well understood for walks in the full plane and in a half plane. Recently much attention has focused on walks in the first quadrant of the plane and has now resulted in a complete characterization of those walks whose generating functions are algebraic, holonomic (solutions of linear differential equations) or at least differentially algebraic (solutions of algebraic differential equations). After an introduction this topic, I will discuss how a Galois theory of linear difference equations allows one to reduce determining these properties to deciding if two points of certain algebraic curves are in the same orbit under certain special automorphisms of these curves and,finally, how one solves this latter problem.

Zhang has proposed dynamical versions of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split' maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related'. In this talk we discuss uniform versions of their results across 1-parameter families of certain split maps and curves. This is joint work with Harry Schmidt

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Please turn your cameras on to take a virtual group photo.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Zariski dense conjecture posed by Zhang, Medvedev-Scanlon, and Amerik-Compana states that for any given rational self-map $\Phi$ of a quasi-projective variety $X$ defined over an algebraically closed field $K$ of characteristic 0, the following dichotomy must hold: Either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ must leave invariant a non-constant rational function $f$, i.e. $f \circ \Phi = f$. We formulate a variant of this conjecture in the case where $K$ is a field of characteristic $p > 0$. We will explain that in the case where the variety $X$ is isotrivial, the presence of the Frobenius endomorphism creates significant difficulties. We will discuss these difficulties in the case of algebraic tori and semiabelian varieties.

In this talk I will describe recent joint work with Omar Leon Sanchez that begins a systematic study of commutative unital rings (usually finitely generated integral algebras over a field) equipped with a binary operation which is a derivation in each argument. This is motivated by Poisson algebra, and the Poisson Dixmier-Moeglin equivalence problem. I will explain some of the subtleties that arise and some of what we achieve.

In recent years, the model theoretic notion of strong minimality has played a significant role in various transcendence results related to nonlinear differential equations. In this talk, we will talk about a new notion called the degree of nonminimality, which can be defined in a general model theoretic setting, but already has had significant application in the differential setting. This talk describes joint works with Matthew Devilbiss and Rahim Moosa.

We prove a dynamical analogue of the Shafarevich conjecture for morphisms $f:\mathbb{P}^N_K\rightarrow \mathbb{P}^N_K$ of degree $d\geq 2$, defined over a number field $K$. This extends previous work of Silverman and others in the case $N=1$. This is joint work with Holly Krieger, Nicole Looper, and Myrto Mavraki.

The dynamical behavior of a rational self-map on projective space is governed by its dynamical degrees, numerical invariants that record the growth rate of iterated preimages of subvarieties. I will review the definition and significance of dynamical degrees and then focus on some recent examples of rational maps for whose dynamical degrees turn out, surprisingly, to be transcendental numbers. This is joint work with Jason Bell, Mattias Jonsson and Holly Krieger.

In this talk I will discuss recent work, joint with D. Blázquez-Sanz, G. Casale, and J. Freitag, towards proving the Ax-Schanuel theorem for uniformizers of geometric structures. One of the goal will be to highlight the role played, in our work/proof, by the model theory of differentially closed fields. In the talk, I will use the case of curves - including Shimura curves and other non-arithmetic hyperbolic curves - to give concrete examples of the general setting.

I will discuss recent work, joint with John Doyle and Bella Tobin, regarding equidistribution in stochastic dynamical systems. We prove that, under some basic assumptions on boundedness, the heights for stochastic families of rational maps on the projective line are Weil heights, associated to a new class of adelic measures called called generalized adelic measures. These generalize earlier notions of Favre-Rivera-Letelier and Mavraki-Ye. I will discuss the proof of an equidistribution theorem for generalized adelic measures, and as an application, we will prove an equidistribution for random backwards orbits in the stochastic dynamical system.

I will discuss recent work from a paper completed by JC Saunders, Dang Khoa Nguyen and myself where we resolve the algebraicity problem for the Artin-Mazur zeta function for surjective endomorphisms of d-dimensional positive characteristic tori. As a result of this work, we provide a complete characterization and explicit formula when the is zeta function is algebraic.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.