Georgia Institute of Technology College of Sciences

Last week we saw combinatorial reconstruction. This time we are going to explain a new approach to Scenery Reconstruction. This new approach could allow us to prove that being able to distinguish sceneries implies reconstructability.

In this talk, I will introduce the class of logconcave and s-concave functions, illustrate their properties, and explain their connections to convex geometry. I will present a simple and unified approach for proving probabilistic inequalities for logconcave and s-concave densities on the real line. Lastly I will use these techniques to prove two important theorems in convex geometry: Grunbaum's theorem, every halfspace cut through the centroid of a convex body contains at least a 1/e volume fraction of the body, and the Milman-Pajor inequality, a convex body in R^n is sandwiched between its inertial ellipsoid and a factor n scaling of it. Joint work with Santosh Vempala.

The expression dynamics of interacting genes depends on the topology of the regulatory network, the quantitative nature of feedbacks and interactions between DNA, RNA and proteins, and the biochemical state of the intracellular and surrounding environment. In this talk we show that dynamics of a gene regulatory network can also depend sensitively on the copy number of genes and promoters. Genetic regulatory networks include an overrepresentation of subgraphs commonly known as network motifs. We consider positive feedback, bistable feedback, and toggle switch motifs and show that variation in gene copy number can cause a sequence of saddle-node bifurcations in the corresponding differential equations models, which leads to multiple orders of magnitude change in gene expression. A similar analysis of a 3-gene motif with successive inhibition (the ``repressilator'') reveals that changes in gene copy number can also cause a Hopf bifurcation, thus leading to a qualitative switch in system behavior among oscillatory and equilibrium dynamics. Importantly, we show that these bifurcations exist over a wide range of parameter values, thus reinforcing our claim that copy number is a key control parameter in the expression dynamics of regulatory networks.

A plasma is a completed ionized gas. In many applications such as in nuclear fusion or astrophysical phenomena, the plasma has very high temperature and low density, thus collisions can be ignored. The standard kinetic models for a collisionless plasma are the Vlasov- Maxwell and Vlasov-Poisson systems. The Vlasov-Poisson system is also used to model galaxy dynamics, where a star plays the role of a particle. There exists infinitely many equilibria for Vlasov models and their stability is a very important issue in physics. I will describe some of my works on stability and instability of various Vlasov equilibria.