Georgia Institute of Technology College of Sciences

This talk aims to give a glimpse into the theory of divisors on graphs in tropical geometry, and its recent application in bijective combinatorics. I will start by introducing basic notions and results of the subject. Then I will mention some of its connections with other fields in math. Finally I will talk about my own work on how tropical geometry leads to an unexpectedly simple class of bijections between spanning trees of a graph and its sandpile group.

We present new methods for segmentation of large datasets with graph based structure. The method combines ideas from classical nonlinear PDE-based image segmentation with fast and accessible linear algebra methods for computing information about the spectrum of the graph Laplacian. The goal of the algorithms is to solve semi-supervised and unsupervised graph cut optimization problems. I will present results for image processing applications such as image labeling and hyperspectral video segmentation, and results from machine learning and community detection in social networks, including modularity optimization posed as a graph total variation minimization problem.

PDEs (such as Navier-Stokes) are in principle infinite-dimensional dynamical systems. However, recent studies support conjecture that the turbulent solutions of spatially extended dissipative systems evolve within an `inertial' manifold spanned by a finite number of 'entangled' modes, dynamically isolated from the residual set of isolated, transient degrees of freedom. We provide numerical evidence that this finite-dimensional manifold on which the long-time dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for Kuramoto-Sivashinsky system, and find it to be equal to the `'physical dimension' computed previously via the hyperbolicity properties of covariant Lyapunov vectors. (with Xiong Ding, H. Chate, E. Siminos and K. A. Takeuchi)

Accounting for Heterogenous Interactions in the Spread Infections, Failures, and Behaviors_ The scaled SIS (susceptible-infected-susceptible) network process that we introduced extends traditional birth-death process by accounting for heterogeneous interactions between individuals. An edge in the network represents contacts between two individuals, potentially leading to contagion of a susceptible by an infective. The inclusion of the network structure introduces combinatorial complexity, making such processes difficult to analyze. The scaled SIS process has a closed-form equilibrium distribution of the Gibbs form. The network structure and the infection and healing rates determine susceptibility to infection or failures. We study this at steady-state for three scales: 1) characterizing susceptibility of individuals, 2) characterizing susceptibility of communities, 3) characterizing susceptibility of the entire population. We show that the heterogeneity of the network structure results in some individuals being more likely to be infected than others, but not necessarily the individuals with the most number of interactions (i.e., degree). We also show that "densely connected" subgraphs are more vulnerable to infections and determine when network structures include these more vulnerable communities.