Nature optimizes shape, and geometry reveals the rigid structures that emerge at these extremes. To understand these structures, my research program operates at the interface of geometry, topology, and analysis. I will begin this talk by discussing Hamilton’s Ricci flow and our recent breakthroughs in identifying and classifying singularity models. Guided by symmetry principles, we uncover a surprising connection between symplectic mechanics and complex geometry to address fundamental questions in the field.

Our work above contributes to a global program aimed at generalizing Perelman’s Ricci-flow-with-surgery framework, which resolved the Poincaré Conjecture, to higher dimensions. As an application, I will explain how the Ricci flow deforms a closed Riemannian manifold into a round sphere under a natural curvature assumption in any dimension. That establishes our complete resolution of a landmark conjecture proposed by S. Nishikawa. Next, I will highlight our work on the variational stability of critical points via the Morse index, where we introduce an innovative spectral technique to decouple and analyze multiple physical constraints. Finally, I will outline my recent efforts translating pure geometric theories to applied frontiers, highlighting a project on mesh transformations for tensor networks supported by the U.S. Department of Energy and Los Alamos National Laboratory.

This presentation is archived in the Mediasite catalog and available at the following url ...
  mediacast.ttu.edu/Mediasite/Play/235ab27e6ed04f808bb054f9304b3e2e1d