Georgia Institute of Technology College of Sciences

Given two complex polynomials, we can try to mathematically paste them together to obtain a rational function through a procedure known as mating the polynomials. In this talk, we will begin by trying to understand the "shape" of complex polynomials in general. We will then discuss the mating of two quadratic polynomials: we explore examples where the mating does exist, and examples where it does not. There will be lots of movies and exploration in this talk.

For the sake of puzzle-lovers worldwide, IBM Research offers a monthly mathematical challenge known as Ponder This. Every month, a new challenge is posted together with the solution for the previous month's riddle. Prof. Oded Margalit has served as the Ponder This puzzlemaster for the last decade. In this talk, he’ll survey some of most interesting riddles posted over the years, and tell some anecdotes about various challenges and regular solvers, such as one person who sent in his solution from an intensive care unit. Several challenges have led to conference and journal papers, such as a PRL paper born from a riddle on random walks, and an ITA 2014 paper on a water hose model (using quantum entanglement to break location-based encryption). Other monthly challenges have riffed on games such as 2048, Kakuro, an infinite chess game, the probability of backgammon ending with a double, Fischer Random Chess, and more. Other challenges have been more purely mathematic, focusing on minimal hash functions, combinatorial test design, or finding a natural number n such that round ((1+2 cos(20))^n) is divisible by 10^9. The talk will present a still-open question about a permutation-firing cannon. The talk will be self contained.

The regularity properties of solutions to linear partial differential equations in domains depend on the structure of the equation, the degree of smoothness of the coefficients of the equation, and of the boundary of the domain. Quantifying this dependence is a classical problem, and modern techniques can answer some of these questions with remarkable precision. For both physical and theoretical reasons, it is important to consider partial differential equations with non-smooth coefficients. We’ll discuss how some classical tools in harmonic and complex analysis have played a central role in answering questions in this subject at the interface of harmonic analysis and PDE.

Evolution of random systems as well as dynamical systems with chaotic (stochastic) behavior traditionally (and seemingly naturally) is described by studying only asymptotic in time (when time tends to infinity) their properties. The corresponding results are formulated in the form of various limit theorems (CLT, large deviations, etc). Likewise basically all the main notions (entropy, Lyapunov exponents, etc) involve either taking limit when time goes to infinity or averaging over an infinite time interval. Recently a series of results was obtained demonstrating that finite time predictions for such systems are possible. So far the results are on the intersection of dynamical systems, probability and combinatorics. However, this area suggests some new analytical, statistical and geometric problems to name a few, as well as opens up possibility to obtain new types of results in various applications. I will describe the results on (extremely) simple examples which will make this talk quite accessible.