Fall 2018 Texas Geometry and Topology Conference Abstracts

Dynamical systems arising from the classification of geometric structures on manifolds, Bill Goldman (Maryland)

The classification of locally homogeneous geometric structures on manifolds leads to interesting dynamical systems. This talk describes some examples of this classification (some with trivial dynamics and others with chaotic dynamics) and how this leads to questions about automorphism groups of affine cubic surfaces in 3-space.

Harmonic surfaces and simple loops, Vlad Markovic(Caltech)

Fix a suitable (non-elementary)  map f from a closed surface S into a closed hyperbolic 3-manifold M, and consider the moduli space of harmonic maps homotopic to f and with respect to varying  metrics on S and M (the metrics on M are assumed to be negatively curved). We show that the set of such metrics for which the corresponding harmonic map is in Whitney's general position is an open, dense, and connected subset of this moduli space. One application of this result is the proof of the special case of the Simple Loop conjecture when M is hyperbolic.

Spherical cone metrics, Rafe Mazzeo(Stanford)

An old problem, accessible from several different parts of mathematics, is to understand metrics on the 2-sphere with constant positive Gauss curvature with conic singularities at a prescribed set of points, and with prescribed cone angles.  This turns out to be considerably more subtle than the flat or hyperbolic cases, and there are many known constraints on the cone angles.  The situation is well understood when all cone angles are less than 2\pi, but for larger cone angles, uniqueness fails, and many new phenomena emerge. I will discuss recent progress, obtained jointly with Xuwen Zhu, which indicates the central role of clustering of cone points in this problem.

Inscribing rectangles in curves, Rich Schwartz(Brown)

The notorious Square Peg Problem asks if every Jordan loop has 4 points which make the vertices of the square.  In this talk I will show some software which explores this problem and tries to approach it by looking at inscribed rectangles instead. I'll illustrate the proof of my result that all but at most 4 points of any Jordan loop are vertices of inscribed rectangles, and I'll also discuss an integral formula for paths of inscribed rectangles and its consequences.

Topological uniqueness of self-expanders of low entropy, Lu Wang (Wisconsin)

Self-expanders are a special class of solutions to the mean curvature flow. In this talk, I will show, for any regular cone with small entropy, all self-expanders that are asymptotic to the cone are in the same isotopy class. This is joint work with J. Bernstein.

Translating Solutions of Mean Curvature Flow, Brian White (Stanford)

I will discuss surfaces that move by translation under mean curvature flow: why they are interesting, what we know about classifying them, and interesting new families of examples. The new results are joint work with David Hoffman, Tom Ilmanen, and Francisco Martin.

Minimal area metrics on Riemann surfaces with crossing bands of geodesics, Barton Zwiebach (MIT)

We examine a minimal area problem on Riemann surfaces closely related to `extremal length’ problems and to systolic geometry. While metrics with single bands of geodesics are well understood, those where bands cross are not.  With the help of calibrations, the problem can be formulated as a convex program. An equivalent dual program involves maximization and helps get some general results.  The programs are amenable to numerical solution and give metrics for a number of interesting surfaces.