Seminar
Speaker: Dr. Ramesh Mete (IIT Bombay)
Host: Saikat Mazumdar
Title: Reading seminar on the Yamabe flow
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, September 10
Venue: Room 113
Abstract: In the 1980s, Hamilton proposed a heat flow approach to the Yamabe problem (uniformization theorem in dimension >2). The goal is to start from a given initial metric and deform it to a metric with constant scalar curvature by means of an evolution equation. Hamilton showed that the flow exists for all time t > 0; however, convergence turns out to be highly nontrivial. 
For conformally flat metrics, Ye [1994] showed that the flow converges to a metric of constant scalar curvature. Subsequently, Schwetlick-Struwe [2003] showed that the flow converged in dimensions 2<n<6 under the assumption that the Yamabe energy of the initial metric was "not too large". The energy assumption was then removed by Brendle [2005] to establish unconditional convergence of the flow in low dimensions 2<n<6. Convergence in dimensions n>5 was also shown by Brendle [2007] under a technical hypothesis on the conformal class. 
Our aim in particular is to discuss the work of Schwetlick-Struwe and Brendle in low dimensions.

Mathematics Colloquium
Speaker: Manas Rachh, IIT Bombay
Title: Complex scattering makes for simple numerics
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, September 10
Venue: Ramanujan Hall
Abstract: Scattering problems involving unbounded interfaces occur frequently in physics and engineering settings. Due to this prevalence, there exist many numerical methods for solving such problems. Unfortunately, the complicated behavior of solutions in the vicinity of infinite interfaces can make it challenging to derive explicit error bounds for these methods. Many of these methods also require a large computational domain, and so require a large number of discretization points to accurately solve the problem.
In this talk, we will present a class of decomposable scattering problems. For this class of problems, the PDE domain can be decomposed into a collection of simple subdomains. The fundamental solutions for these simple regions can then be used to reduce the scattering problem into an integral equation on the interfaces between these subdomains. These integral equations can then be analytically continued into the complex plane, where they can be safely truncated with controllable accuracy. We demonstrate this procedure for the example of two dielectric waveguides meeting at an interface. For this problem, we show that the fundamental solutions and densities decay exponentially in the complex plane, and so the analytically continued integral equation can be truncated with exponential accuracy.