Georgia Institute of Technology College of Sciences

I will first discuss the motivation and background information necessary to study the subjects of branched covers and of contact geometry. In particular we will give some examples and constructions of topological branched covers as well as present the fundamental theorems in this area. But little is understood about the general constructions, and even less about how branched covers behave in the setting of contact geometry, which is the focus of my research. The remainder of the talk will focus on the results I have thus far and current projects.

Critical care is a branch of medicine concerned with the provision of life support or organ support systems in patients who are critically ill and require intensive monitoring. Such monitoring allows us to collect massive amounts of data (usually at the level of organ dynamics, such as electrocardiogram, but recently also at the level of genes). In my talk I’ll show several examples of how ideas from nonlinear dynamics and statistical physics can be applied for the analysis of these data in order to understand and eventually predict physiologic status of critically ill patients: (1) Heart beats, respiration and blood pressure variations can be viewed as a dynamics of a system of coupled nonlinear oscillators (heart, lungs, vessels). From this perspective, a live support devise (e.g. mechanical ventilator used to support breathing) acts as an external driving force on one of the oscillators (lungs). I’ll show that mechanical ventilator entrances the dynamics of whole cardiovascular system and leads to phase synchronization between respiration and heart beats. (2) Then I’ll discuss how fluctuation-dissipation theorem can be used in order to predict heart rate relaxation after a stress (e.g. treadmill exercise test) from the heart rate fluctuations during the stress. (3) Finally, I’ll demonstrate that phase space dynamics of leukocyte gene expression during critical illness and recovery has an attractor state, associated with immunological health.

An interesting problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. A particular interest is so called physical vacuum which naturally arises in physical problems. The main difficulty lies in the fact that the physical systems become degenerate along the boundary. I'll present the well- posedness result of 3D compressible Euler equations for polytropic gases. This is a joint work with Nader Masmoudi.

Pseudo-Anosov mapping classes on surfaces have a rich structure, uncovered by William Thurston in the 1980's. We will discuss the 1995 Bestvina-Handel algorithmic proof of Thurston's theorem, and in particular the "transition matrix" T that their algorithm computes. We study the Bestvina-Handel proof carefully, and show that the dilatation is the largest real root of a particular polynomial divisor P(x) of the characteristic polynomial C(x) = | xI-T |. While C(x) is in general not an invariant of the mapping class, we prove that P(x) is. The polynomial P(x) contains the minimum polynomial M(x) of the dilatation as a divisor, however it does not in general coincide with M(x).In this talk we will review the background and describe the mathematics that underlies the new invariant. This represents joint work with Peter Brinkmann and Keiko Kawamuro.