Georgia Institute of Technology College of Sciences

Many real life problems rely on understanding the solutions to a system of polynomial equations. In this talk, I will outline some of these applications and how tools from algebraic geometry can provide answers to relevant engineering questions.

The brain performs efficient, adaptable, and robust computations of noisy sensory information in changing environments. While artificial neural networks have achieved remarkable successes in recent years, the brain's computational capacity is yet to be matched. To understand mechanisms underlying the exquisite computational efficiency and flexibility of the brain, complex architecture and dynamics of the biological neural networks should be studied. In this talk, I will give a broad overview of recent research projects from my group, that investigate links between neural coding and network structures using data-driven modeling.

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”.  Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations.  After an informal introduction to dispersive equations, I will illustrate how to understand the long-time behavior solutions to dispersive waves via various results I obtained over the years.

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.