Georgia Institute of Technology College of Sciences

A signed graph is a pair (G, \Sigma) where G is a graph and \Sigma is a subset of the edges of G. A cycle C in G is even (resp. odd) if E(C) \cap \Sigma is even (resp. odd). A blocking pair in a signed graph is a pair of vertices {x, y} such that every odd cycle in (G, \Sigma) intersects at least one of the vertices x and y. Blocking pairs arise in a natural way in the study of even cycle matroids on signed graphs as well as signed graphs with no odd K_5 minor. In this article, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of characterizing signed graphs with no odd K_5 minor and determing when two signed graphs represent the same even cycle matroid. This is joint work with Irene Pivotto and Paul Wollan.

Euler's celebrated pentagonal numbers theorem is one themost fundamental in the theory of partitions and q-hypergeometric series.The recurrence formula that it yields is what MacMahon used to compute atable of values of the partition function to verify the deep Hardy-Ramanujanformula. On seeing this table, Ramanujan wrote down his spectacular partition congruences. The author recently proved two new companions to Euler'stheorem in which the role of the pentagonal numbers is replaced by the squares.These companions are deeper in the sense that lacunarity can be achievedeven with the introduction of a parameter. One of these companions isdeduced from a partial theta identity in Ramanujan's Lost Notebook and theother from a q-hypergeometric identity of George Andrews. We will explainconnections between our companions and various classical results such asthe Jacobi triple product identity for theta functions and the partitiontheorems of Sylvester and Fine. The talk will be accessible to non-experts.

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.