# Seminars & Events for Joint Princeton Rutgers Geometric PDEs Seminar

##### TBA - Viaclovsky

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. I will discuss a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on CP^2, and the product metric on S^2 x S^2. Using these metrics in various gluing configurations, critical metrics are found on connected sums for a specific Riemannian functional, which depends on the global geometry of the factors. This is joint work with Matt Gursky.** **

##### TBA - Sesum

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. We will discuss conformally flat complete Yamabe flow and show

that in some cases we can give the precise description of singularity profiles close to the extinction time of the solution. We will also talk about a construction of new compact ancient solutions to the Yamabe flow. This is a joint work with Daskalopoulos, King and Manuel del Pino.** **

##### The logarithmic Minkowski problem

The logarithmic Minkowski problem asks for necessary and sufficient conditions in order that a nonnegative finite Borel measure in (n-1)-dimensional projective space be the cone-volume measure of the unit ball of an n-dimensional Banach space. The solution to this problem is presented. Its relation to conjectured geometric inequalities that are stronger than the classical Brunn-Minkowski inequality will be explained.

##### On the Reality of Black Holes

##### Talk #1: Long time behavior of forced 2D SQG equations

We prove the absence of anomalous dissipation of energy for the forced critical surface quasi-geostrophic equation (SQG) in {\mathbb {R}}^2 and the existence of a compact finite dimensional golbal attractor in {\mathbb T}^2. The absence of anomalous dissipation can be proved for rather rough forces, and employs methods that are suitable for situations when uniform bounds for the dissipation are not available. For the finite dimensionality of the attractor in the space-periodic case, the global regularity of the forced critical SQG equation needs to be revisited, with a new and final proof. We show that the system looses infinite dimensional information, by obtaining strong long time bounds that are independent of initial data. This is joint work with A. Tarfulea and V. Vicol.

##### The linear stability of the Schwarzschild solution under gravitational perturbations in general relativity

I will discuss joint work with G. Holzegel and I. Rodnianski showing the linear stability of the celebrated Schwarzschild black hole solution in general relativity.

##### Large N asymptotics of Optimal partitions of Dirichlet eigenvalues

In this talk, we will discuss the following problem: Given a bounded domain $\Omega$ in R^n, and a positive energy N, one divides $\Omega$ into N subdomains, $\Omega_j, j= 1, 2,..., N$. We consider the so-called optimal partitions that give the least possible value for the sum of the first Dirichelet eigenvalues on these sumdomains among all admissible partitions of $\Omega$. For given N the problem has been studied by various authors. I shall discuss some recent progress and conjectures on the analysis on asymptotic behavior these optimal partitions as N tends to infinite.

##### Ricci flow through singularities

It has been a long-standing problem in geometric analysis to find a good definition of generalized solutions to the Ricci flow equation that would formalize the heuristic idea of flowing through singularities. I will discuss a notion in the 3-d case that has good analytical properties, enabling one to prove existence and compactness of solutions, as well as a number of structural results. It may also be used to partly address a question of Perelman concerning the convergence of Ricci flow with surgery to a canonical flow through singularities.