Georgia Institute of Technology College of Sciences

Volume preserving maps naturally arise in the study of many natural phenomena including incompressible fluid-flows, magnetic field-line flows, granular mixing, and celestial mechanics. Codimension one invariant tori play a fundamental role in the dynamics of these maps as they form boundaries to transport; orbits that begin on one side cannot cross to the other. In this talk I will present a Fourier-based, quasi-Newton scheme to compute the invariant tori of three-dimensional volume-preserving maps. I will further show how this method can be used to predict the perturbation threshold for their destruction and study the mechanics of their breakup.

We will discuss how to define two invariants of knots using sutured Heegaard Floer homology, contact structures and limiting processes. These invariants turn out to be a reformulation of the plus and minus versions of knot Heegaard Floer homology and thus give a``sutured interpretation'' of these invariants and point to a deep connection between Heegaard Floer theory and contact geometry. If time permits we will also discuss the possibility of defining invariants of non-compact manifolds and of contact structures on such manifolds.

Equilibrium computation is among the most significant additions to the theory of algorithms and computational complexity in the last decade - it has its own character, quite distinct from the computability of optimization problems. Our contribution to this evolving theory can be summarized in the following sentence: Natural equilibrium computation problems tend to exhibit striking dichotomies. The dichotomy for Nash equilibrium, showing a qualitative difference between 2-Nash and k- Nash for k > 2, has been known for some time. We establish a dichotomy for market equilibrium. For this purpose. we need to define the notion of Leontief-free functions which help capture the joint utility of a bundle of goods that are substitutes, e.g., bread and bagels. We note that when goods are complements, e.g., bread and butter, the classical Leontief function does a splendid job. Surprisingly enough, for the former case, utility functions had been defined only for special cases in economics, e.g., CES utility function. We were led to our notion from the high vantage point provided by an algorithmic approach to market equilibria. Note: Joint work with Jugal Garg and Ruta Mehta.

In 2011, Rothvoß showed that there exists a 0/1 polytope such that any higher-dimensional polytope projecting to it must have a subexponential number of facets, i.e., its linear extension complexity is subexponential. The question as to whether there exists a 0/1 polytope having high PSD extension complexity was left open, i.e. is there a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a subexponential sized semidefinite cone and an affine space? We answer this question in the affirmative using a new technique to rescale semidefinite factorizations