Speaker: Qin Deng
On a smooth Riemannian manifold, the uniqueness of a geodesic given initial conditions follows from standard ODE theory. In this talk, I will extend a version of this result to the setting of RCD(K,N) spaces, which are metric measure spaces satisfying a synthetic notion of Ricci curvature bounded below first introduced by Sturm-Lott-Villani. Read more.
Speaker: John Harvey
The geometry of space-time used for the theory of relativity is Lorentzian geometry. Gravity is expressed through the curvature of space-time. I will introduce Lorentzian geometry and describe some of its key features. I will explain some physical motivation for generalizing this geometry and demonstrate some of the potentially undesirable phenomena that can arise as we do this. Recently Kunzinger and Sämann proposed Lorentzian length spaces as a synthetic generalization of Lorentzian geometry. Generally, a synthetic geometry is a theory which takes certain theorems from a well-established smooth geometric theory and uses them instead as axioms. I will discuss these spaces and, in particular, ways of defining curvature on them.This will include recent work I have done with Tobias Beran, Lewis Napper and Felix Rott showing that curvature bounds, which are essentially local, can have implications on the global geometry of the space. Read more.
Speaker: Ruma Maity
Nematic liquid crystals represent a transitional state of matter between liquid and crystalline phases that combine the fluidity of liquids with the ordered structure of crystalline solids. These materials are widely utilized in various practical applications, such as display devices, sensors, thermometers, nanoparticle organizations, proteins, and cell membranes. In this talk, we discuss finite element approximation of the nonlinear elliptic partial differential equations associated with the Landau-de Gennes model for nematic liquid crystals. We establish the existence and local uniqueness of the discrete solutions, a priori error estimates, and a posteriori error estimates that steer the adaptive refinement process. Additionally, we explore Ball and Majumdar's modifications of the Landau-de Gennes Q- tensor model that enforces the physically realistic values of the Q tensor eigenvalues. We discuss some numerical experiments that corroborate the theoretical estimates, and adaptive… Read more.