Mathematics Colloquium
Time and Date: 4-5 pm Wednesday, 21 September 2022
Speaker: Prof. Saikat Mazumdar, IIT Bombay
Title: Searching conformally for metrics with constant curvature.
Abstract: I will start by surveying the Yamabe problem, which asks to find a (conformal) metric with constant scalar curvature on a compact Riemannian manifold. This amounts to solving a nonlinear PDE involving the Laplacian. The solution to the Yamabe problem highlighted the role played by local and global geometry of the manifold and the unexpected connection to the positive mass theorem of general relativity.
I will first discuss the case of compact surfaces, and introduce some tools and techniques from Calculus of Variations, Nonlinear Analysis on Manifolds, and PDEs. In the remaining time, I will discuss the higher-order version of the Yamabe problem: “Given a compact Riemannian manifold, does there exists a conformal metric with constant Q-curvature”?
Note: The speakers make an effort to make Mathematics Colloquia accessible to the general audience.