Speaker:  Zhizhang Xie,Texas A&M

 https://artsci.tamu.edu/mathematics/contact/profiles/zhizhang-xie.html).

Title: On Gromov’s Dihedral Rigidity Conjecture of Scalar Curvature

Abstract:
In this talk, I will discuss my joint work with Jinmin Wang and Guoliang Yu on a new index theorem that applies to manifolds with certain types of singularities, such as corners or more general polyhedral boundaries. Roughly speaking, this index theorem links geometric data—such as curvature and dihedral angles—to topological and analytical invariants.

As an application, we prove Gromov’s dihedral rigidity conjecture. This conjecture compares scalar curvature, mean curvature, and dihedral angles for manifolds whose boundaries resemble polyhedra, and it has deep implications for understanding geometric structures that arise in both mathematics and physics.

Building on further developments of this index theorem, we also solve another conjecture of Gromov, known as the flat corner domination conjecture. As a consequence, we obtain a positive solution to a long-standing problem in discrete geometry—the Stoker conjecture.