SCIENTIFIC PROGRAMS AND ACTIVITIES

2:00 - 3:00 p.m.
Jeff Viaclovsky (Wisconsin)
Critical metrics on connected sums of Einstein four-manifolds (slides)

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.

3:30 - 4:30 p.m.
Jeff Streets (University of California - Irvine)
On the singularity formation in fourth-order curvature flows (slides)

The L2 norm of the Riemannian curvature tensor is a natural energy to associate to a Riemannian manifold, especially in dimension 4. A natural path for understanding the structure of this functional and its minimizers is via its gradient flow, the "L2 flow." This is a quasi-linear fourth order parabolic equation for a Riemannian metric, which one might hope shares behavior in common with the Yang-Mills flow. We verify this idea by exhibiting structural results for finite time singularities of this flow resembling results on Yang-Mills flow. We also exhibit a new short-time existence statement for the flow exhibiting a lower bound for the existence time purely in terms of a measure of the volume growth of the initial data. As corollaries we establish new compactness and diffeomorphism finiteness theorems for four-manifolds generalizing known results to ones with have effectively minimal hypotheses/dependencies. These results all rely on a new technique for controlling the growth of distances along a geometric flow, which is especially well-suited to the L2 flow.

The first lecture will start at 2:00 p.m. . and the second lecture at 3:30 p.m.

Albert Chau (UBC)
The Kahler Ricci flow on complete Kahler manifolds with unbounded curvature

Consider the Kahler Ricci flow on a complete non-compact Kahler manifold (M, g). A classical result of W.-X. Shi says that the flow has a short time solution if g has bounded curvature. This talk addresses the problem of finding a solution when g has unbounded curvature. I will begin by discussing some a priori estimates and general existence results for the flow. I will then discuss applications in the case g is a U(n) invariant Kahler metric on C^n. The talk is based on joint work with L.F. Tam and K.F Li.

Tommy Murphy (McMaster University)
Spectral geometry of toric Einstein Manifolds

The eigenvalues of the Laplacian encode fundamental geometric information about a Riemannian metric. As an example of their importance, I will discuss how they arose in work of Cao, Hamilton and Illmanan concerning stability of Einstein manifolds. I will then outline joint work with Stuart Hall which allows us to make progress on these problems for Einstein metrics with large symmetry groups. We calculate bounds on the first non-zero eigenvalue for certain Hermitian-Einstein four manifolds. Similar ideas allow us estimate to the spectral gap (the distance between the first and second non-zero
eigenvalues) for any toric Kaehler-Einstein manifold M in terms of the polytope associated to M.

2:00 p.m.
Mihalis Dafermos (Princeton and Cambridge)
The stability of the schwarzschild solution to gravitational perturbations

3:30 p.m.
Grigorios Fournodavlos (Toronto)
Singular Ricci Solutions and their Stability under the Ricci Flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions $n$+1 ≥ 3, and all have a point singularity where the curvature blows up. Their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow "pushes away" from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. We prove a local well-posedness result for the Ricci flow in a natural class of initial data close to the singular ones, which in particular implies that the "opening up" of the singularity persists for the perturbations as well.
** joint with Spyros Alexakis and Dezhong Chen.

Please note the first lecture will take place from 2:00-3:00 p.m. and the second lecture will take place from 3:30-4:30 p.m.

Monday, Jan. 20
Room 230
Fields Institute

2:00 p.m.
Sergiu Klainerman, Princeton
Are Black Holes real

I will give a survey on some of the most basic issues inthe mathematical theory of Black Holes: Rigidity, Stability and collapse.

3:30 p.m.
Reza Seyyedali, Waterloo
Extremal metrics on ruled manifolds

Consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We show that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kahler metrics on the projectivization of the dual vector bundle. This is joint work with Zhiqin Lu.

Richard Bamler, Stanford
There are finitely many surgeries in Perelman's Ricci flow

Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.
In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $C t^{-1}$ for large $t$. Using this curvature bound it is possible to give a more precise picture of the long-time behavior of the flow.

Arick Shao, Toronto
Unique continuation from infinity for linear waves

We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. The results are nearly optimal in many respects, and they can be considered as analogues of uniqueness from infinity results for second-order elliptic operators. This work is partly motivated by questions in general relativity.
This is joint work with Spyros Alexakis and Volker Schlue.

Please note the first talk is schedule to start at 2:00 p.m. and the second talk will start at 3:30 p.m.

Nov. 15
*2:00 p.m
Room 230

* note revised time

Speakers:
2:00 p.m Pengfei Guan, McGill

3:30 p.m. Ben Weinkove, Northwestern University
The Monge-Ampere equation for (n-1)-plurisubharmonic functions

A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of every n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.