Georgia Institute of Technology College of Sciences

I will discuss an ongoing project to reconstruct a gene network from time-series data from a mammalian signaling pathway. The data is generated from gene knockouts and the techniques involve computational algebra. Specifically, one creates an pseudomonomial "ideal of non-disposable sets" and applies a analogue of Stanley-Reisner theory and Alexander duality to it. Of course, things never work as well in practice, due to issue such as noise, discretization, and scalability, and so I will discuss some of these challenges and current progress.

A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points.  Stable polynomials have close relations to matroids and hyperbolic programming.  We will discuss a generalization of stability to algebraic varieties of codimension larger than one.  They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.

We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by the Stokes or the Navier-Stokes equations, in a two dimensional open bounded and connected domain.  We consider the velocity field steered by a control input that acts tangentially on the boundary of the domain through the  Navier slip boundary conditions. This is motivated by mixing  within a cavity or vessel  by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip boundary control  that optimizes mixing at a given final time. Non-dissipative scalars, both passive and active, governed by the transport equation will be discussed.  In the absence of diffusion, transport and mixing occur due to pure advection.  This essentially leads to a nonlinear control problem of a semi-dissipative system. We shall provide a rigorous proof of the existence of an optimal controller, derive the first-order necessary conditions for optimality, and present some preliminary results on the numerical implementation.

I will talk about algorithms (with unlimited computational power) which adaptively probe pairs of vertices of a graph to learn the presence or absence of edges and whose goal is to output a large clique. I will focus on the case of the random graph G(n,1/2), in which case the size of the largest clique is roughly 2\log(n). Our main result shows that if the number of pairs queried is linear in n and adaptivity is restricted to finitely many rounds, then the largest clique cannot be found; more precisely, no algorithm can find a clique larger than c\log(n) where c < 2 is an explicit constant. I will also discuss this question in the planted clique model. This is based on joint works with Uriel Feige, David Gamarnik, Joe Neeman, Benjamin Schiffer, and Prasad Tetali.