Obstructions to the existence of conformally compact Einstein manifolds

Speaker: 

Matthew Gursky

Institution: 

University of Notre Dame

Time: 

Tuesday, March 13, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

In this talk I will describe a singular boundary value problem for Einstein metrics.  This problem arises in the Fefferman-Graham theory of conformal invariants, and in the AdS/CFT correspondence.   After giving a brief overview of some important results and examples, I will present a recent construction of boundary data which cannot admit a solution.  Finally, I will introduce a more general index-theoretic invariant which gives an obstruction to existence in the case of spin manifolds.  This is joint work with Q. Han and S. Stolz.

Convergence of Riemannian manifolds with scale invariant curvature bounds

Speaker: 

Norman Zergaenge

Institution: 

University of Warwick

Time: 

Tuesday, January 30, 2018 - 4:00pm

Host: 

Location: 

RH 306

A key challenge in Riemannian geometry is to find ``best" metrics on compact manifolds. To construct such metrics explicitly one is interested to know if approximation sequences contain subsequences that converge in some sense to a limit manifold.

In this talk we will present convergence results of sequences of closed Riemannian
4-manifolds with almost vanishing L2-norm of a curvature tensor and a non-collapsing bound on the volume of small balls.  For instance we consider a sequence of closed Riemannian 4-manifolds,
whose L2-norm of the Riemannian curvature tensor is uniformly bounded from
above, and whose L2-norm of the traceless Ricci-tensor tends to zero.  Here,
under the assumption of a uniform non-collapsing bound, which is very close
to the euclidean situation, and a uniform diameter bound, we show that there
exists a subsequence which converges in the Gromov-Hausdor sense to an
Einstein manifold.

To prove these results, we use Jeffrey Streets' L2-curvature 
ow. In particular, we use his ``tubular averaging technique" in order to prove fine distance
estimates of this flow which only depend on significant geometric bounds.

Stable Horizons and the Penrose Conjecture

Speaker: 

Henri Roesch

Institution: 

UC Irvine

Time: 

Tuesday, March 6, 2018 - 4:00pm to 5:00pm

Location: 

RH 306

In the first half of the talk, we introduce a new quasi-local mass with interesting properties along null flows off of a 2-sphere in spacetime or, equivalently, foliations of a null cone. We also show how certain, fairly generic, convexity assumptions on the null cone allows for a proof of the Penrose Conjecture. On the Black Hole Horizon, we find that the convexity assumptions become sharp; therefore, the second half of the talk will explore the existence of a class of Black Hole Horizons admitting such convexity. From this, building upon the work of S. Alexakis, we will show that the Schwarzschild Null Cone--the case of equality for the Penrose Conjecture--is also critical in light of recent work on the perturbation of stable, weakly isolated Horizons.

 

 

On Hamiltonian Gromov-Witten theory for symplectic reductions

Speaker: 

Rui Wang

Institution: 

UC Irvine

Time: 

Tuesday, November 7, 2017 - 4:00pm

Location: 

RH 306

In this talk, I will first review our work on defining a new quantum deformation for the (Chen-Ruan) cohomology ring of a symplectic reduction. Then I will explain the relation between this quantum deformation and the well-known quantum cohomology ring. Our construction is based on the study of moduli spaces of symplectic vortices with proper metrics. This is a joint project with B. Chen and B. Wang.

Algebraic Gluing of Holomorphic Discs in K3 Surfaces and Tropical Geometry

Speaker: 

Yu-Shen Lin

Institution: 

Harvard CMSA

Time: 

Monday, October 30, 2017 - 4:00pm

Location: 

RH 340P

We will start from the motivation of the tropical geometry. Then
we will explain how to use Lagrangian Floer theory to establish the
correspondence between the weighted counting of tropical curves to the
counting of holomorphic discs in K3 surfaces. In particular, the result
provides the existence of new holomorphic discs which do not come easily
from direct gluing argument.

 

 

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