The Southeast Geometry Seminar (SGS) is a semiannual series of one day events sponsored by:
Abstract: We will discuss the geometry of the curve complex associated to a closed orientable surface, and its relevance to the theory of Heegaard splittings of 3-manifolds. This discussion will include a definition of the distance of a Heegaard splitting. Our main result is that for a fixed pseudo-Anosov map φ defined on the boundary of a particular handlebody (and satisfying a certain necessary technical condition), the distance of the Heegaard splitting obtained by gluing two copies of the handlebody using the iterate φn grows linearly in n. This is joint work with Saul Schleimer.
Abstract: In this talk, we discuss recent numerical experiments in the simulation of the process by which knotted tubes pull tight. We will cover theoretical and computational results on this topic, and include some animations of the tightening process.
Abstract: The talk concerns Legendrian submanifolds of 1-jet spaces, and in particular their contact homology. Legendrian contact homology is a part of Symplectic Field Theory. It is a framework for finding isotopy invariants of Legendrian submanifolds of contact manifolds by "counting" rigid (pseudo-)holomorphic disks. Contact homology has proved very useful: For Legendrian submanifolds of dimension 1, the Riemann mapping theorem can be used to give a combinatorial and computable description of the theory and many contact geometric results has been derived. Contact homology of Legendrian submanifolds of higher dimensions is a very rich theory. Apart from applications intrinsic to contact geometry, it has been used to derive powerful invariants of knots in 3-space. However computing Legendrian contact homology from first principles is hard: finding pseudo-holomorphic disks involves solving a non-linear first order partial differential equation and this is an infinite dimensional problem.
In the talk I will describe how to reduce the computation of Legendrian contact homology in 1-jet spaces to a finite dimensional problem in Morse theory.
Abstract: A sequence fi of functions on a domain D in the plane is said to be equi-Poisson if the mapping (fi, fi+1) from D to R2 is area-preserving for all i. Recently, the equi-Poisson sequences with period n (i.e., fi = fi+n for all i) have been shown to be related to problems in cluster algebras and other mathematical structures. In this talk, I will explain the basics of the theory, construct some examples, and prove some classification theorems for n-periodic sequences for low values of n.
Abstract: Many examples are known of special Lagrangian (SL) cones with an isolated singularity in the origin. Only very few, sporadic, examples of these are known to admit SL desingularizations. I will present a "prescribed boundary problem" which helps understand the topological aspects of this question and identifies obstructions to the existence of Lagrangian desingularizations. I will give various examples, then show that all obstructions vanish when working with SL cones in 2 and 3 dimensions. Finally I will produce "soft" (Maslov-zero) desingularizations of these cones using the Lagrangian h-principle. This work is joint with Mark Haskins (Imperial College). .
Abstract: I will discuss joint work with Annalisa Calini on finite-gap solutions of the vortex filament flow, an evolution equation for curves in three-dimensional Euclidean space which is a geometric counterpart of the cubic focussing nonlinear Schrodinger equation (NLS). The goal of our work is to build a correspondence between the geometric/ topological properties of solutions of the flow and the algebro-geometric data (e.g., the Floquet spectrum) for periodic finite-gap solutions of NLS. Motivated by the geometry of elastic rod centerlines, we prove that spectra that are symmetric about the origin lead to evolving filaments that are periodically planar. We also use isoperiodic deformations to construct solutions, in the form of iterated cable knots, whose knot type may be determined from the spectrum.