Georgia Institute of Technology College of Sciences

We show that an embedding of a (small) ball into a contact manifold is contact if and only if it preserves the (modified) shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back (to a closed one-form) of a contact form by a coisotropic embedding of a fixed manifold (of maximal dimension) and of a given homotopy type. The proof is based on displacement information about (non)-Lagrangian submanifolds that comes from J-holomorphic curve methods (and gives topological invariants), and the construction of a coisotropic torus whose image (under a given embedding that is not contact) admits a transverse contact vector field (i.e. a convex surface in dimension 3). The definition of shape preserving does not involve derivatives and is preserved by uniform convergence (on compact subsets). As a consequence, we prove C^0-rigidity of contact embeddings (and diffeomorphisms). The underlying ideas are adaptations of symplectic techniques to contact manifolds that, in contrast to symplectic capacities, work well in the contact setting; the heart of the proof however uses purely contact topological methods.

The boundary method is a new algorithm for solving semi-discrete transport problems involving a variety of ground cost functions. By reformulating a transport problem as an optimal coupling problem, one can construct a partition of its continuous space whose boundaries allow accurate determination of the transport map and its associated Wasserstein distance. The boundary method approximates region boundaries using the general auction algorithm, controlling problem size with a multigrid discard approach. This talk describes numerical and mathematical results obtained when the ground cost is a convex combination of lp norms, and shares preliminary work involving other ground cost functions.

In this introductory talk we present some basic results in ergodic theory, due to Poincare, von Neumann, and Birkhoff. We will also discuss many examples of dynamical systems where the theory can be applied. As motivation for a broad audience, we will go over the connection of the theory with some classical problems in statistical mechanics (part 2 of 3).

We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

Quantum topology is a collection of ideas and techniques for studying knots and manifolds using ideas coming from quantum mechanics and quantum field theory. We present a gentle introduction to this topic via Kauffman bracket skein algebras of surfaces, an algebraic object that relates "quantum information" about knots embedded in the surface to the representation theory of the fundamental group of the surface. In general, skein algebras are difficult to compute. We associate to every triangulation of the surface a simple algebra called a "quantum torus" into which the skein algebra embeds. In joint work with Thang Le, we make use of this embedding to give a simple proof of a difficult theorem.

We will review the definition of the Chekanov-Eliashberg differentialgraded algebra for Legendrian knots in R^3 and look at examples tounderstand a few of the invariants that come from Legendrian contacthomology.

An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound. Though they arise in many applications, there are only a few known methods for constructing ETFs. One of the most popular classes of ETFs, called harmonic ETFs, is constructed using the structure of finite abelian groups. In this talk we will discuss a broad generalization of harmonic ETFs. This generalization allows us to construct ETFs using many different structures in the place of abelian groups, including nonabelian groups, Gelfand pairs of finite groups, and more. We apply this theory to construct an infinite family of ETFs using the group schemes associated with certain Suzuki 2-groups. Notably, this is the first known infinite family of equiangular lines arising from nonabelian groups.

The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. The Talbot effect, named after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed.

The Jacobian (or sandpile group) of a graph is a well-studied group associated with the graph, known to biject with the set of spanning trees of the graph via a number of classical combinatorial mappings. The algebraic definition of a Jacobian extends to regular matroids, but without the notion of vertices, many such combinatorial bijections fail to generalize. In this talk, I will discuss how orientations provide a way to overcome such obstacle. We give a novel, effectively computable bijection scheme between the Jacobian and the set of bases of a regular matroid, in which polyhedral geometry plays an important role; along the way we also obtain new enumerative results related to the Tutte polynomial. This is joint work with Spencer Backman and Matt Baker.

An expander polynomial in F_p, the finite field with p elements, is a polynomial f(x_1,...,x_n) such that there exists an absolute c>0 with the property that for every set A in F_p (of cardinality not particularly close to p) the cardinality of f(A,...,A) = {f(a_1,...,a_n) : a in A} is at least |A|^{1+c}. Given an expander polynomial, a very interesting question is to determine a threshold T so that |A|> T implies that |f(A,...,A)| contains, say, half the elements of F_p and so is about as large as it can be. For a large number of "natural appearing" expander polynomials like f(x,y,z) = xy+z and f(x,y,z) = x(y+z), the best known threshold is T= p^{2/3}. What is interesting is that there are several proofs of this threshold of very different “depth” and complexity. We will discuss why for the expander polynomial f(x,y,z,w) = (x-y)(z-w), where f(A,A,A,A) consists of the product of differences of elements of A, one may take T = p^{5/8}. We will also discuss the more complicated setting where A is a subset of a not necessarily prime order finite field.

In this talk I am going to present a way to study numerically the complex domains of invariant Tori for the dissipative standar map. The numerical approach is based on a Nash Moser method. This is work in progress jointly with R. Calleja.