Georgia Institute of Technology College of Sciences

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a PDE-theoretical framework for studying time-averages of dynamical systems that evolve in both fast and slow scales. Patterns arise upon time-averaging, which in turn affects long term dynamics via nonlinear coupling. We apply this framework to geophysical fluid dynamics in spherical and bounded domains subject to strong Coriolis force and/or Lorentz force.

Let a,b,c >= 2 be integers satisfying 1/a + 1/b + 1/c > 1. Darmon and Granville proved that the generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions; conjecturally something stronger is true: for a,b,c \geq 3 there are no non-trivial solutions and for (a,b,c) = (2,3,n) with n >= 10 the only solutions are the trivial solutions and (+- 3,-2,1) (or (+- 3,-2,+- 1) when n is even). I'll explain how the modular method used to prove Fermat's last theorem adapts to solve generalized Fermat equations and use it to solve the equation x^2 + y^3 = z^10.

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this talk, I will prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring. This is joint work with Gexin Yu.

A few results and two general conjectures regarding analysis of Boolean functions, influence, and threshold phenomena will be presented. Boolean functions are functions of n Boolean variables with values in {0,1}. They are important in combinatorics, theoretical computer science, probability theory, and game theory. Influence. Causality is a topic of great interest everywhere, and if causality is not complicated enough, we can ask what is the influence one event has on another one. Ben-Or and Linial studied influence in the context of collective coin flipping---a problem in theoretical computer science. Fourier analysis. Over the last two decades, Fourier analysis of Boolean functions and related objects played a growing role in discrete mathematics, and theoretical computer science. Threshold phenomena. Threshold phenomena refer to sharp transition in the probability of certain events depending on a parameter p near a critical value. A classic example that goes back to Erdos and Renyi, is the behavior of certain monotone properties of random graphs. Influence of variables on Boolean functions is connected to their Fourier analysis and threshold behavior, as well as to discrete isoperimetry and noise sensitivity. The first Conjecture to be described (with Friedgut) is called the Entropy-Influence Conjecture. (It was featured on Tao's blog.) It gives a far-reaching extension to the KKL theorem, and theorems by Friedgut, Bourgain, and the speaker. The second Conjecture (with Kahn) proposes a far-reaching generalization of results by Friedgut, Bourgain and Hatami.