Schedule for: 19w5115 - Women in Geometry 2

We discuss harmonic maps from two dimensional domains including a harmonic maps approach to uniformization and finding minimal surfaces.

I will outline the main techniques for studying flows of immersed, codi- mension 1, manifolds in Rn evolving with curvature dependent speed. As the field of such flows is very large, I will exemplify these tools on a few of the most known curvature flows on closed convex hypersurfaces.

I will discuss the known results of (1) existence of periodic geodesics on Riemannian manifolds; (2) volume and diameter upper bounds for the length of the shortest periodic geodesics. I will also talk about various known techniques that potentially can help establishing the existence and the upper bounds.

I will consider the problem of prescribing the Gaussian curvature (under pointwise conformal change of the metric) on surfaces with conical singularities. This question has been first raised by Troyanov and it is a generalization of the Kazdan-Warner problem for regular surfaces, known as the Nirenberg problem on the sphere. Answer this question amounts to solve a singular differential problem on the surface. This Liouville-type equation has been studied first in the case $K > 0$, where $K$ denotes the Gaussian curvature we want to prescribe. I will review classical results on the regular case, more recent results obtained for positive functions $K$ in presence of singularities and I will present some new results in the sign-changing singular case.

We will examine the question of whether or not an orbifold (with singular points) can be isospectral to a manifold. The answer is known to be “no” in the case of two-dimensional manifolds/orbifolds with non-negative Euler characteristic. We propose to use heat invariants for functions and the analogous invariants for one- forms to distinguish between manifolds and orbifolds in the three-dimensional case.

The space $S^3 \subset \mathbb{C}^2$, comes naturally equipped a contact structure: a geometry which gives strong connections between low-dimensional topology and symplectic geometry. We study Legendrian knots– knots which are compatible with the contact structure – and Lagrangian cobordisms between them in $\mathbb{C}^2$.

Conserved quantities – mass, linear momentum, center of mass, and angular momentum – are important in various areas of physics as in many cases they provide an essential characterization of a physical system. The concepts of mass and linear momentum are by now well-established in General Relativity, and recently there has been a lot of progress in understanding the center of mass of isolated systems. However, we believe that the somewhat more mysterious notion of angular momentum deserves more attention. In this talk we will discuss the key geometric ideas behind this notion and outline the questions that our team plans to address.