Georgia Institute of Technology College of Sciences

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.

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.

As the term "big data'' appears more and more frequently in our daily life and research activities, it changes our knowledge of how large the scale of the data can be and challenges the application of numerical analysis for performing statistical calculations. In this talk, I will focus on two basic statistics problems sampling a multivariate normal distribution and maximum likelihood estimation and illustrate the scalability issue that dense numerical linear algebra techniques are facing. The large-scale challenge motivates us to develop scalable methods for dense matrices, commonly seen in statistical analysis. I will present several recent developments on the computations of matrix functions and on the solution of a linear system of equations, where the matrices therein are large-scale, fully dense, but structured. The driving ideas of these developments are the exploration of the structures and the use of fast matrix-vector multiplications to reduce the general quadratic cost in storage and cubic cost in computation. "Big data'' offers a fresh opportunity for numerical analysts to develop algorithms with a central goal of scalability in mind. It also brings in a new stream of requests to high performance computing for highly parallel codes accompanied with the development of numerical algorithms. Scalable and parallelizable methods are key for convincing statisticians and practitioners to apply the powerful statistical theories on large-scale data that they currently feel uncomfortable to handle.

The Weak Structure Theorem of Robertson and Seymour is the cornerstone of many of the algorithmic applications of graph minors techniques. The theorem states that any graph which has both large tree-width and excludes a fixed size clique minor contains a large, nearly planar subgraph. In this talk, we will discuss a new proof of this result which is significantly simpler than the original proof of Robertson and Seymour. As a testament to the simplicity of the proof, one can extract explicit constants to the bounds given in the theorem. We will assume no previous knowledge about graph minors or tree-width. This is joint work with Ken Kawarabayashi and Robin Thomas