Georgia Institute of Technology College of Sciences

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Gamma is an element of the free abelian group on Gamma. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. A rank-determining set of a metric graph Gamma is defined to be a subset A of Gamma such that the rank of a divisor D on Gamma is always equal to the rank of D restricted on A. I will present an algorithm to derive the reduced divisor from any effective divisor in the same linear system, and show constructively that there exist finite rank-determining sets. Based on this discovery, we can compute the rank of an arbitrary divisor on any metric graph. In addition, I will discuss the properties of rank-determining sets in general and formulate a criterion for rank-determining sets.

Modulation spaces are a class of Banach spaces which provide a quantitative time-frequency analysis of functions via the Short-Time Fourier Transform. The modulation spaces are the "right" spaces for time-frequency analysis andthey occur in many problems in the same way that Besov Spaces are attached to wavelet theory and issues of smoothness. In this seminar, I will talk about embeddings of modulation Spaces into BMO or VMO (the space of functions of bounded or vanishing mean oscillation, respectively ). Membership in VMO is central to the Balian-Low Theorem, which is a cornerstone of time-frequency analysis.

In linear algebra classes we learn that a symmetic matrix with real entries has real eigenvalues. But many times we deal with nonsymmetric matrices that we want them to have real eigenvalues and be stable under a small perturbation. In the 1930's totally positive matrices were discovered in mechanical problems of vibtrations, then lost for over 50 years. They were rediscovered in the 1990's as esoteric objects in quantum groups and crystal bases. In the 2000's these matrices appeared in relation to Teichmuller space and its quantization. I plan to give a high school introduction to totally positive matrices.

It is known that, relative to any fixed vertex q of a finite graph, there exists a unique q-reduced divisor (G-Parking function based at q) in each linear equivalence class of divisors. In this talk, I will give an efficient algorithm for finding such reduced divisors. Using this, I will give an explicit and efficient bijection between the Jacobian group and the set of spanning trees of the graph. Then I will outline some applications of the main results, including a new approach to the Random Spanning Tree problem, efficient computation of the group law in the critical and sandpile group, efficient algorithm for the chip-firing game of Baker and Norine, and the relation to the Riemann-Roch theory on finite graphs.