Georgia Institute of Technology College of Sciences

Antibiotics have greatly reduced morbidity and mortality from infectious diseases. Although antibiotic resistance is not a new problem, it breadth now constitutes asignificant threat to human health. One strategy to help combat resistance is to find novel ways of using obsolete antibiotics. For strains of E. coli and P. aeruginosa, pairs of antibiotics have been found where evolution of resistance to one increases, sometimes significantly, sensitivity to the other. These researchers have proposed cycling such pairs to treat infections. Similar strategies are being investigated to treat cancer. Using systems of ODEs, we model several possible treatment protocols using pairs and triples of such antibiotics, and investigate the speed of ascent of multiply resistant mutants. Rapid ascent would doom this strategy. This is joint work with Klas Udekwu (Stockholm University).

The Weeks manifold W is a closed orientable hyperbolic 3-manifold with the smallest volume. Understanding contact structures on hyperbolic 3-manifolds is one of problems in contact topology. Stipsicz previously showed that there are 4 non-isotopic tight contact structures on the Weeks manifold. In this talk, we will exhibit 7 non-isotopic tight contact structures on W with non-vanishing Ozsvath-Szabo invariants.

A sparse bound is a novel method to bound a bilinear form. Such a bound gives effortless weighted inequalities, which are also easy to quantify. The range of forms which admit a sparse bound is broad. This short survey of the subject will include the case of spherical averages, which has a remarkably easy proof.

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will continue our discussion on the operations we use for characterizing feasible (G, a0, a1, a2, b1, b2). If time permits, we will also discuss useful structures for obtaining that characterization, such as frame, ideal frame, and framework. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

This talk concerns to spectral gap of random regular graphs. First, we prove that almost all bipartite biregular graphs are almost Ramanujan by providing a tight upper bound for the non trivial eigenvalues of its adjacency operator, proving Alon's Conjecture for this family of graphs. Also, we use a spectral algorithm to recover hidden communities in a random network model we call regular stochastic block model. Our proofs rely on a technique introduced recently by Massoullie, which we developed for random regular graphs.

In this talk, we study solvers for geometrically embedded graph structured block linear systems. The general form of such systems, PSD-Graph-Structured Block Matrices (PGSBM), arise in scientific computing, linear elasticity, the inner loop of interior point algorithms for linear programming, and can be viewed as extensions of graph Laplacians into multiple labels at each graph vertex. Linear elasticity problems, more commonly referred to as trusses, describe forces on a geometrically embedded object.We present an asymptotically faster algorithm for solving linear systems in well-shaped 3-D trusses. Our algorithm utilizes the geometric structures to combine nested dissection and support theory, which are both well studied techniques for solving linear systems. We decompose a well-shaped 3-D truss into balanced regions with small boundaries, run Gaussian elimination to eliminate the interior vertices, and then solve the remaining linear system by preconditioning with the boundaries.On the other hand, we prove that the geometric structures are ``necessary`` for designing fast solvers. Specifically, solving linear systems in general trusses is as hard as solving general linear systems over the real. Furthermore, we give some other PGSBM linear systems for which fast solvers imply fast solvers for general linear systems.Based on the joint works with Robert Schwieterman and Rasmus Kyng.

In this series of talks, I will introduce basic concepts and results in singularity theory of smooth and holomorphic maps. In the first talk, I will present a gentle introduction to the elements of singularity theory and give a proof of the well-known Morse Lemma that illustrates key geometric and algebraic principles of singularity theory.

We will introduce Arnold diffusion in Mather's setting. There are many advances toward proof of this. In particular, we will study an approach based on recent work of Marian-Gidea and Jean-Pierre Marco.

The original concept ofdimension for posets was formulatedby Dushnik and Miller in 1941 and hasbeen studied extensively in the literature.Over the years, a number of variant formsof dimension have been proposed withvarying degrees of interest and application.However, in the recent past, two variantshave received extensive attention. Theyare Boolean dimension and local dimension.This is the first of two talks on these twoconcepts, with the second talk givenby Heather Smith. In this talk, wewill introduce the two parameters and providemotivation for their study. We will alsogive some concrete examples andprove some basic inequalities.This is joint work with a GeorgiaTech team in which my colleaguesare Fidel Barrera-Cruz, Tom Prag,Heather Smith and Libby Taylor.