Georgia Institute of Technology College of Sciences

An order-n Latin square is an n by n array of n symbols such that each row and column contains each symbol exactly once.  Latin squares were famously studied by Euler in the 1700s, and at present they are still a central object of study in modern extremal and probabilistic combinatorics.  In this talk, I will give some history about Latin squares, share some simple-to-state yet notoriously difficult open problems, and present some of my own research on Latin squares.

For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. By constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying the Batchelor-Wood formula. For an annulus with wall velocities slightly different from the rigid-rotation, we also constructed Prandtl-Batchelor flows, whose leading order terms are rotating shear flows. This is a joint work with Chen Gao, Mingwen Fei and Tao Tao. 

Networks of coupled oscillators are studied in biology, chemistry, physics, and engineering. The Kuramoto model is a simple dynamical system that models the nonlinear interaction among coupled oscillators. It has received widespread attention since it is simple enough to be analyzed rigorously yet complex enough to exhibit interesting emergent behaviors.

One such emergent behavior is the spontaneous synchronization of oscillators into special configurations. In the past decades, rigorous analysis of such synchronization configurations has been the focus of intensive studies.

In this talk, we explore the new insight to this problem provided by an algebraic and tropical approach.

This is an ongoing project. We make use of two dual Lawrence polytopes $P$ and $P*$ of a graph $G$, to study invariants of the graph. The $h$-vector of the graphic (resp. cographic) matroid complex associated to $G$ coincides with the $h^*$-vector of the Lawrence polytope $P$ (resp. $P^*$). In general, the $h$-vector is an invariant defined for an abstract simplicial complex, which encodes the number of faces of different dimensions. The $h^*$-vector, a.k.a. the $\delta$-polynomial, is an invariant defined for a rational polytope, which is obtained by dilating the polytope. By dissecting the Lawrence polytopes, we may study the $h$-vectors associated to the graph $G$ at a finer level. In particular, we understand activities and reduced divisors of the graph $G$ in a more geometric way. I will try to make the talk self-contained.