2:00-3:00: Ovidiu Munteanu (University of Connecticut), The geometry of Ricci solitons

I will present recent development about the structure of four dimensional shrinking Ricci solitons. I will show how some basic information about scalar curvature allows us to better understand such solitons. For example, assuming the scalar curvature is bounded, these manifolds must have their curvature operator bounded in norm and non-negative at infinity. Furthermore, if the scalar curvature converges to zero, then they must be asymptotically conical. Some generalizations in higher dimension will also be discussed. This talk is based on joint work with Jiaping Wang.

3:30-4:30: Tristan Collins (Harvard University), Convergence of the J-flow on toric varieties

I will discuss the convergence of the J-flow, which is the gradient flow of Donaldson's J-functional. It is known that the J-flow does not converge in general -- a notion of algebro-geometric stability has been proposed by Lejmi-Szekelyhidi which is conjectured to be equivalent to the convergence of the flow. I will discuss a proof of this conjecture on toric varieties. This work is joint with G. Szekelyhidi.

Saturday May 9

10:15-11:15: Long Li (McMaster University), On the convexity of the Mabuchi energy functional along (singular) geodesics

It is conjectured by X.X. Chen that the Mabuchi energy functional is convex along the geodesic connecting two Kaehler metrics, during his study in uniqueness of cscK metrics. Now we can give an affirmative answer to this question in joint work with X.X. Chen and Mihai Paun. The first breakthrough in this subject is the work by Berman and Berndtsson last year, where they proved the weak convexity of the Mabuchi energy functional based on the log-subharmonicity of Bergman kernels. Our work is somewhat a "global version" of Bergman kernels approximation, and also completes the conjecture by proving the continuity of the Mabuchi energy functional along the geodesic. Finally, we will discuss generalization of this convexity to the conic case.

11:30-12:30: Heather Macbeth (Princeton University), Kaehler-Einstein metrics and higher alpha-invariants

I will describe a condition on the Bergman metrics of a Fano manifold M, which guarantees the existence of a Kaehler-Einstein metric on M. I will also discuss a conjectural relationship between this condition and M's higher alpha-invariants \alpha_{m,k}(M), analogous to a 1991 theorem of Tian for \alpha_{m,2}(M).

2:00-3:00: Marcus Khuri (Stony Brook University), Geometric Inequalities in General Relativity

We will give a survey of geometric inequalities in general relativity, and then focus on contributions from angular momentum and charge. In particular, we will introduce inequalities which give new criteria for the formation of black holes.

PAST SEMINARS

Thursday March 12, 2015
2:00pm

Stewart Library

Jared Speck (Massachusetts Institute of Technology)
Stable Big Bang Formation in Solutions to the Einstein-Scalar Field System

Abstract

Tuesday January 27, 2015
Stewart Library

Cecile Huneau (Ecole Normale Supérieure)
Stability in exponential time of Minkowski Space-time with a translation space-like Killing field

In the presence of a translation space-like Killing field the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar field. We work in generalised wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty is due to the weak decay of free solutions to the wave equation in 2 dimensions. To prove long time existence of solutions, we have to rely on the particular structure of Einstein equations in wave coordinates. We also have to carefully choose the behaviour of our metric in the exterior region to enforce convergence to Minkowski space-time at time-like infinity.

Friday Oct 17, 2014

2:00-3:00 – Jason Lotay (University College London)
Coupled flows, convexity and Lagrangians

The idea of coupling two geometric flows has previously been primarily motivated by analytic considerations. In the symplectic setting, we provide a geometric motivation for a new coupling of a submanifold flow with a flow of the ambient structure, which also enjoys good analytic properties. In addition, we exhibit a natural functional whose gradient flow agrees with our submanifold flow in the Kaehler setting and which is related to calibrated geometry. We also obtain a surprising convexity result for our functional which could provide a useful tool in studying certain minimal Lagrangians. This is joint work with T. Pacini.

3:30-4:30 -- Brett Kotschwar (Arizona State University)
Uniqueness and unique-continuation for geometric flows via energy methods

We describe a short, direct method to prove the uniqueness of solutions to curvature flows of all orders, including the Ricci flow, the L^2 curvature flow, and other flows related to the ambient obstruction tensor. Our approach, an alternative to the DeTurck trick, is based on the consideration of simple energy quantities defined in terms of the actual solutions to the equations, and allows one to avoid the step -- itself nontrivial in the noncompact setting -- of solving an auxiliary parabolic equation (e.g., a k-harmonic-map heat-type flow) in order to overcome the diffeomorphism-invariance-based degeneracy of the original flow. We also describe a short, quantitative proof of the backward uniqueness of certain second-order curvature flows based on the consideration of a simple energy/frequency-type quantity.

Saturday Oct 18, 2014

10:15-11:15 — Shengda Hu (Wilfrid Laurier University)
Generalized holomorphic bundles and Kobayashi-Hitchin correspondence

We discuss an analogue of the Hermitian-Einstein equations for generalized Kaehler manifolds. We also introduce a notion of stability for generalized holomorphic bundles on generalized Kaehler manifolds, and establish a Kobayashi-Hitchin-type correspondence between stable bundles and solutions of the generalized Hermitian-Einstein equations.

11:30-12:30 — Jonathan Luk (Massachusetts Institute of Technology)
Stability of the Kerr Cauchy horizon and the strong cosmic censorship conjecture in general relativity

The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present recent results proving that the Cauchy horizon is $C^0$ stable and discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

2:00-3:00 — Robert Haslhofer (Courant Institute, New York University)
Mean curvature flow with surgery

We give a new proof for the existence of mean curvature flow with surgery for 2-convex hypersurfaces. Our proof works in all dimensions, including mean convex surfaces in R^3. We also derive a priori estimates for a more general class of flows. This is joint work with Bruce Kleiner.