Probability, Differential Geometry and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
Shrinking gradient Kähler-Ricci solitons are a generalization of Kähler-Einstein metrics of positive scalar curvature. These arise naturally in the study of the Kähler-Ricci flow, for which they are known to model finite-time Type I singularities by work of Naber and Enders-Müller-Topping. I will present some recent work on the classification of shrinking gradient Kähler-Ricci solitons on complex surfaces. In particular, we classify all non-compact examples with bounded scalar curvature. Together with a recent result of Li-Wang, this completes the classification in the non-compact case. Combining further with previous work of Tian, Wang, Zhu, and others in the compact case gives the complete classification. This is joint work with R. Bamler, R. Conlon, and A. Deruelle.
Watch online on the 7th at 3 PM (UT-6) via this Zoom link.
In dimensions four and higher, the Ricci flow may encounter singularities modelled on cones with nonnegative scalar curvature. It may be possible to resolve such singularities and continue the flow using expanding Ricci solitons asymptotic to these cones, if they exist. I will discuss joint work with Richard Bamler in which we develop a degree theory for four-dimensional asymptotically conical expanding Ricci solitons, which in particular implies the existence of expanders asymptotic to a large class of cones.
Watch online on the 3rd at 3 PM (UT-5) via this Zoom link.
The classical grim reaper curves are known in differential geometry as the only translating solitons of the mean curvature flow in the plane but have also recently been discovered to be critical points of an entropy-like energy functional, thus garnering attention for their unexpected connections to thermodynamics, information theory and Hamilton-type entropy. We use a variational approach to studying the critical points of this entropy energy-like functional in all Riemannian and pseudo-Riemannian 2-space forms using the concept of p-elasticae, the first variation formula and properties of the Riemann curvature tensor. We show that there are periodic solutions of the Euler-Lagrange equations associated with this functional in certain ambient spaces and obtain closed periodic solutions where they may exist.
Abstract pdf Dr. Luan Hoang hosts our talented, engaging guest speaker.