Georgia Institute of Technology College of Sciences

Tropical geometry provides a combinatorial approach for studying geometric objects by reducing them to graphs and polytopes. In recent years, tropical techniques have been applied in numerous areas such as optimization, number theory, phylogenetic trees in biology, and auction systems in economics. My talk will focus on geometric counting problems and their tropical counterpart. By considering these combinatorial gadgets, we gain newinsights into old problems, and tools for approaching new problems.

After briefly describing my research interests, I’ll speak on two results that involve points moving around on surfaces. The first result shows how to “hear the shape of a billiard table.” A point bouncing around a polygon encodes a sequence of edges. We show how to recover geometric information about the table from the collection of all such bounce sequences. This is joint work with Calderon, Coles, Davis, and Oliveira. The second result answers the question, “Given n distinct points in a closed ball, when can a new point be added in a continuous fashion?” We answer this question for all values of n and for all dimensions. Our results generalize the Brouwer fixed point theorem, which gives a negative answer when n=1. This is joint work with Chen and Gadish.

In this talk, we describe transforming a theoretical algorithm from structural graph theory into open-source software being actively used for real-world data analysis in computational biology. Specifically, we apply the r-dominating set algorithm for graph classes of bounded expansion in the setting of metagenome analysis. We discuss algorithmic improvements required for a practical implementation, alongside exciting preliminary biological results (on real data!). Finally, we include a brief retrospective on the collaboration process. No prior knowledge in metagenomics or structural graph theory is assumed. Based on joint work with T. Brown, D. Moritz, M. O’Brien, F. Reidl and T. Reiter.

The goal of this lecture is to explain and motivate the connection between Aubry-Mather theory (Dynamical Systems), and viscosity solutions of the Hamilton-Jacobi equation (PDE).This connection is the content of weak KAM Theory.The talk should be accessible to the “generic” mathematician. No a priori knowledge of any of the two subjects is assumed.The set-up of this theory is classical mechanical systems, in its Lagrangian formulation to take advantage of the action principle. This is the natural setting for Celestial Mechanics. Today it is also the setting for motions of satellites in the solar system.Hamilton found a reformulation of Lagrangian mechanics in terms of position and momentum instead of position and speed. In this formulation appears the Hamilton-Jacobi equation. Although this is a partial differential equation, its solutions allow to find solutions of the Hamiltonian (or Lagrangian) systems which are, in fact, governed by an ordinary differential equation.KAM (Kolmogorov-Arnold-Moser) theorem addressed at its beginning (Kolomogorov) the problem of stability of the solar system. It came as a surprise, since Poincare ́’s earlier work pointed to instability. In fact, some initial conditions lead to instability (Poincare ́) and some others lead to stability(Kolomogorov).Aubry-Mather theory finds some more substantial stable motion that survives outside the region where KAM theorem applies.The KAM theorem also provides global differentiable solutions to the Hamilton-Jacobi equation.It is known that the Hamilton-Jacobi equation usually does not have smooth global solutions. Lions & Crandall developed a theory of weak solutions of the Hamilton-Jacobi equation.Weak KAM theory explains how the Aubry-Mather sets can be obtained from the points where weak solutions of the Hamilton-Jacobi equation are differentiable.