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.

An isoradial graph is one which can be embedded into the plane such that each face is inscribable in a circle of common radius. We consider the superposition of an isoradial graph, and its interior dual graph, approximating a simply-connected domain, and prove that the height function associated to the dimer configurations is conformally invariant in the scaling limit, and has the same distribution as a Gaussian Free Field.

Chemical polymers are long chains of molecules built up from many individual monomers. Examples are plastics (like polyester and PVC), biopolymers (like cellulose, DNA, and starch) and rubber. By some estimates over 60% of research in the chemical industry is related to polymers. The complex shapes and seemingly random dynamics inherent in polymer chains make them natural candidates for mathematical modelling. The probability and statistical physics literature abounds with polymer models, and while most are simple to understand they are notoriously difficult to analyze. In this talk I will describe the general flavor of polymer models and then speak more in depth on my own recent results for two specific models. The first is the self-avoiding walk in two dimensions, which has recently become amenable to study thanks to the invention of the Schramm-Loewner Evolution. Joint work with Hugo-Duminil Copin shows that a specific feature of the self-avoiding walk, called the bridge decomposition, carries over to its conjectured scaling limit, the SLE(8/3) process. The second model is for directed polymers in dimension 1+1. Recent joint work with Kostya Khanin and Jeremy Quastel shows that this model can be fully understood when one considers the polymer in the previously undetected "intermediate" disorder regime. This work ultimately leads to the construction of a new type of diffusion process, similar to but actually very different from Brownian motion.

I will discuss regularity of fully nonlinear elliptic equations when the usual uniform upper bound on the ellipticity is replaced by bound on its $L^d$ norm, where $d$ is the dimension of the ambient space. Our estimates refine the classical theory and require several new ideas that we believe are of independent interest. As an application, we prove homogenization for a class of stationary ergodic strictly elliptic equations.