Georgia Institute of Technology College of Sciences

Ryser (1951) provided the conditions under which any $r\times s$ Latin rectangle can be extended to an $n\times n$ Latin square. In this talk, we provide various generalizations of this result in higher dimensions. We also proof an analogue of Ryser’s theorem for symmetric latin cubes.

There are two standard ways to associate a principally polarized abelian variety (ppav) to a smooth algebraic curve X of genus g. The Jacobian variety Jac(X) is a ppav of dimension g. An etale double cover X’->X determines the Prym variety Prym(X’/X), which is a ppav of dimension g-1. These two objects are related by Recillas’ trigonal construction: given an etale double cover X’->X of a trigonal curve X, we can construct a tetragonal curve Y such that Prym(X’/X) is isomorphic to Jac(Y).

I will talk about a tropical version of the trigonal construction, where algebraic curves are replaced by metric graphs and ppavs by real tori with integral structure. Given a double cover X’->X of a trigonal graph X, we obtain a tetragonal graph Y such that the tropical Prym variety Prym(X’/X) and the tropical Jacobian Jac(Y) are isomorphic.

This construction has two applications. First, we can use it to compute the second moment of the tropical Prym variety for g up to 4, and conjecturally for all g, which has arithmetic applications. Second, the tropical trigonal construction provides an explicit resolution of the Prym—Torelli map in genus 4.

Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior.  We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.  By analyzing the Gibbs distribution of various fixed-magnetization models from equilibrium statistical mechanics, we show that particles moving stochastically according to local affinities can solve various useful collective tasks. Finally, we will briefly introduce new tools that may prove fruitful in nonequilibrium settings as well.

Work of Levin and Przytycki shows that if two non-special rational
functions f and g of degree $> 1 $over $\mathbb{C}$ share the same set of
preperiodic points, there are $m$, $n$, and $r$ such that $f^m g^n = f^r$.
In other words, $f$ and $g$ nearly commute.  One might ask if there are
other sorts of relations non-special rational functions $f$ and $g$ over $\mathbb{C}$
might satisfy when they do not share the same set of preperiodic
points.  We will present a recent proof of Beaumont that shows that
they may not, that if f and g do not share the same set of preperiodic
points, then they generate a free semi-group under composition.  The
proof builds on work of Bell, Huang, Peng, and the speaker, and uses a
ping-pong lemma similar to the one used by Tits in his proof of the
Tits alternative for finitely generated linear groups.