Georgia Institute of Technology College of Sciences

Complex fluids are abundant in our daily life. Unlike traditional solids, liquids and the diluted solutions, the model equations for complex fluids continue to evolve with the new experimental evidences and emerging applications. Most of these important properties are due to the coupling and competition between effects from different scales or even from different physical origins/principles. The energetic variational approaches (EnVarA), motivated by the seminal works of Onsager and Rayleigh, are designed to study such systems. In this talk, I will discuss several complex fluid systems, and the associated mathematical issues.

Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as a minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees in a graph form a regular matroid. In this talk, I will give a short historical overview of the tree-counting problem and a related quantity from electrical circuit theory: the effective resistance. I will describe a characterization of effective resistances in terms of a certain polytope and discuss a recent application to discrete notions of curvature on graphs. The talk is based on the article: https://arxiv.org/abs/2410.07756

The Birkhoff Ergodic Theorem describes typical behaviors and averaged quantities with respect to an invariant measure. In this talk, I will focus on "observable" events, equating observability with positive Lebesgue measure. From this observational viewpoint, "typical" means typical with respect to Lebesgue measure. This leads immediately to issues for attractors, where all invariant measures are singular. I will present highlights of developments in smooth ergodic theory that address these questions. The theory of physical and SRB measures applies to dynamical systems that are deterministic as well as random, in finite and infinite dimensions (where observability has to be interpreted differently). This body of ideas argue in favor of convergence of ergodic averages for typical orbits. But the picture is a little more complicated: In the last part of the talk, I will discuss some recent work that shows that in many natural settings (e.g. reaction networks), it is also typical for ergodic averages 
to fluctuate in perpetuity due to heteroclinic-like behavior.

A binary shuffle square is a binary word of even length that can be partitioned into two disjoint, identical subwords. While recognizing shuffle squares is NP-hard, we show that they are surprisingly ubiquitous. We prove that a uniformly random binary word $s$ of length $2n$ is a shuffle square with probability $\frac 12-o(n^{-1/15})$, verifying a conjecture of He, Huang, Nam, and Thaper. In particular, almost every binary word is at most two bit-deletions away from a shuffle square, giving the best possible average case for the “Longest Twin” problem.

By revealing the bits of $s$ sequentially,  we reformulate the problem as a discrete stochastic process. We track the evolution of a “buffer set”, a collection of suffixes produced by the revealed bits. In this setting, there is a simple greedy algorithm which behaves like a SSRW; we define a local optimization which creates a negative bias. We also show that the buffer set is robust enough to absorb small defects, yielding a perfect partition with high probability.