The theory of taking limits of Riemannian manifolds is extensively studied given that certain curvature quantities are uniformly bounded from below. For instance, the theory of Alexandrov spaces is developed to study limit spaces with sectional curvature bounded from below, and similar situations apply to spaces with bounded Ricci curvature via Cheeger-Colding-Naber Theory. However, the corresponding results on scalar curvature are still far from being understood. Recently, Gromov proposed to study spaces with positive scalar curvature in terms of polyhedra and use it to prove the compactness theorem. We will discuss a simpler proof proposed via Ricci flow introduced by Bamler. At the same time, we will motivate the corresponding rigidity conjecture and some related questions about scalar curvature. 

We will discuss more motivations for the dihedral rigidity problem and focus on some relations with general relativity. After that, I will sketch the proof of the conjecture for a large class of polyhedrons in dimension between 3 and 7 modulo the theory of capillary surfaces.  

Dirac Operator is a powerful tool to study positive scalar curvature. Concerning positive scalar curvature on a manifold with boundary, it’s natural to ask how to formulate a valid boundary value problem for Dirac operator (1st order elliptic). In fact, Dirichlet boundary condition, which is natural for Laplacian operator (2nd order elliptic), turns out to be too strong for 1st order elliptic operators. In this talk, we focus on Dirac-type operators, with principal symbols capturing the Clifford relation just like the usual Dirac operator, and discuss some basic materials to get ready for elliptic boundary value problems. 

We would study the metric perturbation technique by Chodosh-Eichmair. From this, one can form a foliation of prescribed mean curvature surface which provides rigidity under scalar curvature assumption. 

Last time, we defined the differential operator on the boundary associated to the given Dirac type operator using the information of the symbols. Recall that such associated boundary operator is a self-adjoint elliptic first order operator defined on a closed manifold (the boundary), so we can make use of its spectrum (L^2 decomposition of sections defined on the boundary) to investigate what kind of boundary condition is natural for the Dirac type operator. Statements will be given and the ideas of the proof will be sketched, and some important examples will be discussed. 

Last time, we studied the hybrid Sobolev spaces using the spectrum of a boundary adapted operator. In particular, the hybrid Sobolev space $\check{H}(A)$ is the image of (extended) trace map on the Dirac-maximal domain, and any boundary condition we will consider is a closed subspace of $\check{H}(A)$. Among all the boundary conditions, a special class called D-elliptic boundary conditions will be the main subject we are discussing. To understand this D-elliptic boundary condition, we will start with the famous APS condition, and try to explore based on that. Under these D-elliptic boundary conditions, nice boundary regularity result is obtained. And we will also discuss many other examples which belong to the class of D-elliptic boundary conditions. 

I will discuss a theorem originally due to Professor Fefferman and Phong, which is about the estimate of the lowest eigenvalue of Schrödinger equations with weak regularity assumptions on the potentials. In particular, we will look at the situations where the potential is small in a Morrey norm and how this is applied to the dihedral rigidity problem in Professor Brendle's work.