
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
201415
Fields
Geometric Analysis Colloquium
at the Fields Institute, Stewart Library
Organizing
Committee:
Spyros Alexakis (Toronto), Walter Craig (Fields &
McMaster)
Spiro Karigiannis (Waterloo), McKenzie Wang (McMaster)



UPCOMING SEMINARS 
May 8  9, 2015
Friday May 8: 2  4:30 p.m.
Saturday May 9: 10  3:00 p.m.
Stewart Library

Friday May 8
2:003:00: 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 nonnegative
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.
3:304:30: Tristan Collins (Harvard University), Convergence
of the Jflow on toric varieties
I will discuss the convergence of the Jflow, which is the gradient
flow of Donaldson's Jfunctional. It is known that the Jflow does
not converge in general  a notion of algebrogeometric stability
has been proposed by LejmiSzekelyhidi 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
10:1511:15: 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 logsubharmonicity 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.
11:3012:30: Heather Macbeth (Princeton University), KaehlerEinstein
metrics and higher alphainvariants
I will describe a condition on the Bergman metrics of a Fano manifold
M, which guarantees the existence of a KaehlerEinstein metric on
M. I will also discuss a conjectural relationship between this condition
and M's higher alphainvariants \alpha_{m,k}(M), analogous to a
1991 theorem of Tian for \alpha_{m,2}(M).
2:003:00: 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 EinsteinScalar Field
System
Abstract

Tuesday January 27,
2015
Stewart Library 
Cecile Huneau (Ecole Normale Supérieure)
Stability in exponential time of Minkowski Spacetime with a translation
spacelike Killing field
In the presence of a translation spacelike 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 spacetime at timelike infinity.

Friday Oct 17, 2014

2:003:00 – 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.
3:304:30  Brett Kotschwar (Arizona State University)
Uniqueness and uniquecontinuation 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 kharmonicmap heattype
flow) in order to overcome the diffeomorphisminvariancebased degeneracy
of the original flow. We also describe a short, quantitative proof
of the backward uniqueness of certain secondorder curvature flows
based on the consideration of a simple energy/frequencytype quantity.

Saturday Oct 18, 2014 
10:1511:15 — Shengda Hu (Wilfrid Laurier University)
Generalized holomorphic bundles and KobayashiHitchin correspondence
We discuss an analogue of the HermitianEinstein equations for generalized
Kaehler manifolds. We also introduce a notion of stability for generalized
holomorphic bundles on generalized Kaehler manifolds, and establish
a KobayashiHitchintype correspondence between stable bundles and
solutions of the generalized HermitianEinstein equations.
11:3012:30 — 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.
2:003:00 — 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 2convex 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.


