This project concerns problems in Atiyah-Singer index theory and several problems in geometry that are related to Dirac operators and/or singular spaces. Atiyah-Singer index theory is one of the landmark results of mathematics that unifies several important developments and results in mathematics and has found many remarkable applications in mathematics and physics. The problems include the study of geometric invariants for singular spaces, the use of Dirac operators in the study of stability problems for Einstein metrics, and the study of the heat kernel and the Bergman kernel as well as their relation with the "best" metrics. Part of the project involves graduate students.
Analytic torsion has found many interesting connections and significant applications, in Seiberg-Witten theory, hyperbolic geometry, and mirror symmetry. The Cheeger-Muller theorem has played an important role in all of these. The PI, along with his collaborator and graduate students, would like to prove the Ray-Singer conjecture/Cheeger-Muller theorem for manifolds with conical singularities which should help us understand more complicated singularities. One of the most important and extensively studied areas of geometry is the study of canonical metrics and it is important to understand the stability issue associated with variational problems. The PI and his student will seek better understanding of the stability of Einstein metrics with positive scalar curvature by exploring the connection with cones and conical singularity. Bergman kernels and conical singularity have been essential ingredients in the recent spectacular solutions of the Yau-Tian-Donaldson conjecture. This proposal aims for the solution of the Ray-Singer conjecture for manifolds with conical singularity, better understanding of the variational structure of the total scalar curvature functional, the Bergman kernel, and the Ricci flow on noncompact manifolds. It also explores the connection with positive mass theorems.
