Georgia Institute of Technology College of Sciences

In a recent work with June Huh, we proved the log-concavity of the characteristic polynomial of a realizable matroid by relating its coefficients to intersection numbers on an algebraic variety and applying an algebraic geometric inequality. This extended earlier work of Huh which resolved a long-standing conjecture in graph theory. In this talk, we rephrase the problem in terms of more familiar algebraic geometry, outline the proof, and discuss an approach to extending this proof to all matroids. Our approach suggests a general theory of positivity in tropical geometry.

We will discuss the details of the Markov bases chapter not covered in the previous talks.(Algebraic statistics reading seminar)

Electrical stimulation of cardiac cells causes an action potential wave to propagate through myocardial tissue, resulting in muscular contraction and pumping blood through the body. Approximately two thirds of unexpected, sudden cardiac deaths, presumably due to ventricular arrhythmias, occur without recognition of cardiac disease. While conduction failure has been linked to arrhythmia, the major players in conduction have yet to be well established. Additionally, recent experimental studies have shown that ephaptic coupling, or field effects, occurring in microdomains may be another method of communication between cardiac cells, bringing into question the classic understanding that action potential propagation occurs primarily through gap junctions. In this talk, I will introduce the mechanisms behind cardiac conduction, give an overview of previously studied models, and present and discuss results from a new model for the electrical activity in cardiac cells with simplifications that afford more efficient numerical simulation, yet capture complex cellular geometry and spatial inhomogeneities that are critical to ephaptic coupling.

The online matching problem has received significant attention in recent years because of its connections to allocation problems in internet advertising, crowd sourcing, etc. In these real-world applications, the typical goal is not to maximize the number of allocations; rather it is to maximize the number of “successful” allocations, where success of an allocation is governed by a stochastic event that comes after the allocation. These applications motivate us to introduce stochastic rewards in the online matching problem. In this talk, I will formally define this problem, point out its connections to previously studied allocation problems, give a deterministic algorithm that is close to optimal in its competitive ratio, and describe some directions of future research in this line of work. (Based on joint work with Aranyak Mehta.)

We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem whichgeneralizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function $u(x,t)$, $$0\in \frac{\partial}{\partial t}\partial_u \Psi(x/\ve,x,u)+\nabla_x\cdot \nabla_\eta\psi(x/\ve,x,t,u,\nabla u) - f(x/\ve,x,t, u), $$ on a bounded domain $\Om\subset \R^n$, $t\in(0,T)$, together with initial-boundary conditions, where $\Psi(z,x,\cdot)$ is strictly convex and $\psi(z,x,t,u,\cdot)$ is a $C^1$ convex function, both with quadratic growth,satisfying some additional technical hypotheses. As functions of the oscillatory variable, $\Psi(\cdot,x,u),\psi(\cdot,x,t,u,\eta)$ and $f(\cdot,x,t,u)$ belong to the generalized Besicovitch space $\BB^2$ associated with an arbitrary ergodic algebra $\AA$. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to usual $L^2$ convergence in the Cartesian product $\Om\X\Pi$, where $\Pi$ is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with uniformly bounded sequences in $L^2$.

I will review recent progress concerning nonparametric estimation of log-concave densities and related families in $R^1$ and $R^d$. In the case of $R^1$, I will present limit theory for the estimators at fixed points at which the population density has a non-zero second derivative and for the resulting natural mode estimator under a corresponding hypothesis. In the case of $R^d$ with $d\ge 2$ will briefly discuss some recent progress and sketch a variety of open problems.

A conjecture of Ivanov asserts that finite index subgroups of the mapping class group of higher genus surfaces have trivial rational homology. Putman and Wieland use what they call higher Prym representations, which are extensions of the representation induced by the action of the mapping class group on homology, to better understand the conjecture. In particular, they prove that if Ivanov's conjecture is true for some genus g surface, it is true for all higher genus surfaces. On the other hand, they also prove that if there is a counterexample to Ivanov's conjecture, it is of a specific form.

In this talk, we will characterize the compact operators on Bergman spaces of the ball and polydisc. The main result we will discuss shows that an operator on the Bergman space is compact if and only if its Berezin transform vanishes on the boundary and additionally this operator belongs to the Toeplitz algebra. We additionally will comment about how to extend these results to bounded symmetric domains, and for "Bergman-type" function spaces.

Statistical modeling amounts to specifying a set of candidates for what the probability distribution of an observed random quantity might be. Many models used in practice are of an algebraic nature in thatthey are defined in terms of a polynomial parametrization. The goal of this talk is to exemplify how techniques from computational algebraic geometry may be used to solve statistical problems thatconcern algebraic models. The focus will be on applications in hypothesis testing and parameter identification, for which we will survey some of the known results and open problems.

Every subcubic triangle-free graph on n vertices contains an independent set of size at least 5n/14 (Staton'79). We strengthen this result by showing that all such graphs have fractional chromatic number at most 14/5, thus confirming a conjecture by Heckman and Thomas. (Joint work with J.-S. Sereni and J. Volec)

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

We will begin by formulating the Riemann-Roch theorem for graphs, due to the speaker and Norine. We will then describe some refinements and applications. Refinements include a Riemann-Roch theorem for tropical curves, proved by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for metrized complexes of curves, proved by Amini and the speaker. Applications include a new proof by Cools-Draisma-Payne-Robeva of the Brill-Noether theorem in algebraic geometry, a generalization by Amini and the speaker of the Eisenbud-Harris theory of limit linear series, and a new Chabauty-Coleman style bound for the number of rational points on an algebraic curve over the rationals, proved recently by Katz and Zureick-Brown.

For a given finite graph G of minimum degree at least k, let G_{p} be a random subgraph of G obtained by taking each edge independently with probability p. We prove that (i) if p \ge \omega/k for a function \omega=\omega(k) that tends to infinity as k does, then G_p asymptotically almost surely contains a cycle (and thus a path) of length at least (1-o(1))k, and (ii) if p \ge (1+o(1))\ln k/k, then G_p asymptotically almost surely contains a path of length at least k. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking G to be the complete graph on k+1 vertices. Joint w/ Michael Krivelevich (Tel Aviv), Benny Sudakov (UCLA).

Abstract: In this talk we will survey some recent developments in the scattering theory of complete, infinite-volume manifolds with ends modeled on quotients of hyperbolic space. The theory of scattering resonances for such spaces is in many ways parallel to the classical case of eigenvalues on a compact Riemann surface. However, it is only relatively recently that progress has been made in understanding the distribution of these resonances. We will give some introduction to the theory of resonances in this context and try to sketch this recent progress. We will also discuss some interesting outstanding conjectures and present numerical evidence related to these.

The purpose of the GSC Symposium is to provide an opportunity for professors, postdocs, and graduate students in the Atlanta area to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. Certainly, the symposium is open to whole mathematics and computer sciences communities. The previous meetings were held at Emory University (2009), Georgia Institute of Technology (2010), Emory University (2011) and University of Georgia (2012). The 2013 GSC Symposium will be held at the Georgia State University campus and is organized by Alexandra Smirnova and Vladimir Bondarenko in the Department of Mathematics and Statistics, Georgia State. The following researchers have agreed to give invited plenary lectures: Hao Gao, Department of Mathematics and Computer Science, Department of Radiology and Imaging Sciences, Emory University; Guillermo Goldsztein, School of Mathematics, Georgia Institute of Technology; Yi Jiang, Department of Mathematics and Statistics, Georgia State University; Caner Kazanci, Department of Mathematics, University of Georgia; Brani Vidakovic, College of Engineering, Georgia Institute of Technology. There will be poster sessions. Anyone attending this symposium may present a poster. We especially encourage graduate students and postdocs to use this opportunity displaying their research results. Please register at the Symposium website.