Georgia Institute of Technology College of Sciences

Piecewise linear Fermi-Ulam pingpongs. We consider a particle moving freely between two periodically moving infinitely heavy walls. We assume that one wall is fixed and the second one moves with piecewise linear velocities. We study the question about existence and abundance of accelerating orbits for that model. This is a joint work with Jacopo de Simoi

The heart is an electro-mechanical system in which, under normal conditions, electrical waves propagate in a coordinated manner to initiate an efficient contraction. In pathologic states, propagation can destabilize and exhibit period-doubling bifurcations that can result in both quasiperiodic and spatiotemporally chaotic oscillations. In turn, these oscillations can lead to single or multiple rapidly rotating spiral or scroll waves that generate complex spatiotemporal patterns of activation that inhibit contraction and can be lethal if untreated. Despite much study, little is known about the actual mechanisms that initiate, perpetuate, and terminate reentrant waves in cardiac tissue. In this talk, I will discuss experimental and theoretical approaches to understanding the dynamics of cardiac arrhythmias. Then I will show how state-of-the-art voltage-sensitive fluorescent dyes can be used to image the electrical waves present in cardiac tissue, leading to new insights about their underlying dynamics. I will establish a relationship between the response of cardiac tissue to an electric field and the spatial distribution of heterogeneities in the scale-free coronary vascular structure. I will discuss how in response to a pulsed electric field E, these heterogeneities serve as nucleation sites for the generation of intramural electrical waves with a source density ?(E) and a characteristic time constant ? for tissue excitation that obeys a power law. These intramural wave sources permit targeting of electrical turbulence near the cores of the vortices of electrical activity that drive complex fibrillatory dynamics. Therefore, rapid synchronization of cardiac tissue and termination of fibrillation can be achieved with a series of low-energy pulses. I will finish with results showing the efficacy and clinical application of this novel low energy mechanism in vitro and in vivo. e

(algebraic statistics reading seminar)

In the lecture I will describe how several questions in geometric combinatorics translate into questions about graphs and hypergraphs. 1. Borsuk's problem. 2. Tverberg theorem and Tverberg's type problems. Tverberg's theorem asserts that (r-1)(d+1)+1 points in d-space can be divided into r parts whose convex hull intersect. I will discuss situations where less points admit such a partition and connections with graph theory. (For more background, look at this MO question Tverberg partitions with less than (r-1)(d+1)+1 points<http://mathoverflow.net/questions/88718/tverberg-partitions-with-less-than-r-1d11-points> ) 3. Helly type theorems and conditions on induced subgraphs and sub-hypergraphs. I will explain the origin to the following conjecture of Meshulam and me: There is an absolute upper bound for the chromatic number of graphs with no induced cycles of length divisible by 3. 4. Embedding of 2-dimensional complexes and high dimensional minors. I will discuss the following conjecture: A 2-dimensional simplicial complex with E edges and F 2-dimensional faces that can be embedded into 4-space satisfies F < 4e. (For more background see my post *F ≤ 4E*<http://gilkalai.wordpress.com/2013/02/01/f-4e/> )