Georgia Institute of Technology College of Sciences

We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers. This is joint work with Asa Levi.

The Jacobian variety of a smooth complex curve is a complex torus that admits two different algebraic descriptions. The Jacobian can be described as the Picard variety, which is the moduli space of line bundles, or it can be described as the Albanese variey, which is the universal abelian variety that contains the curve. I will talk about how to extend a family of Jacobians varieties by adding degenerate fibers. Corresponding to the two different descriptions of the Jacobian are two different extensions of the Jacobian: the Neron model and the relative moduli space of stable sheaves. I will explain what these two extensions are and then prove that they are equivalent. This equivalence has surprising consequences for both the Neron model and the moduli space of stable sheaves.

The three objects in the title come together in the study of ergodic properties of geodesic flows on flat surfaces. I will go over how these three things are intimately related, state some classical results about the unique ergodicity of translation flows and present new results which generalize much of the classical theory and also apply to non-compact (infinite genus) surfaces.

Many real-world graphs are large and growing. This naturally raises the question of a suitable concept of graph convergence. For graphs with average degree proportional to the number of vertices (dense graphs), this question is by now quite well-studied. But real world graphs tend to be sparse, in the sense that the average or even the maximal degree is bounded by some reasonably small constant. In this talk, I study several notions of convergence for graphs of bounded degree and show that, in contrast to dense graphs, where various a priori different notions of convergence are equivalent, the corresponding notions are not equivalent for sparse graphs. I then describe a notion of convergence formulated in terms of a large deviation principle which implies all previous notions of convergence.

First-passage percolation is a model of a random metric on a infinite network. It deals with a collection of points which can be reached within a given time from a fixed starting point, when the network of roads is given, but the passage times of the road are random. It was introduced back in the 60's but most of its fundamental questions are still open. In this talk, we will overview some recent advances in this model focusing on the existence, fluctuation and geometry of its geodesics. Based on joint works with M. Damron and J. Hanson.

The number of permutations with specified descent set is maximized on the (classical) alternating permutations, which are counted by the Euler numbers. Asymptotics for the Euler numbers are well-studied. A counterpart of the descent statistic is the exceedance set statistic which is known to be maximized on a single run of consecutive exceedances. An exact enumeration is known, but the asymptotics have not been studied. We provide asymptotics using multivariate analytic combinatorics (providing a uniformity that goes beyond the range of a basic central limit theorem). This answers a question of E. Clark and R. Ehrenborg. As further applications we also discuss generalized pattern avoidance, and the number of orbit types (n-cycles) that admit a stretching pair (a certificate for "turbulence" in the context of combinatorial dynamics). This includes joint work with R. Ferraz de Andrade and B. Nagle and J. N. Cooper.