| 30 March |
Speaker: Nina Uraltseva (Steklov Institute)
Title:Regularity in free boundary problems
Abstract:
In this talk we discuss the methods, developed in the last decades, for
study the regularity for some problems with free boundaries. These methods
include direct qualitative study of local properties of solutions, various
monotonicity formulas, and blow-up technique which allows to reduce the
analysis of the local properties of solutions to the study of global
solutions.
|
4 April
(Note special date) |
Speaker: Michael Eichmair (MIT)
Title: Isoperimetric structure of initial data sets
Abstract:
I will present recent joint work with Jan Metzger. A basic
question in mathematical relativity is how geometric properties of an
asymptotically flat manifold (or initial data set) encode information
about
the physical properties of the space time that it is embedded in. For
example, the square root of the area of the outermost minimal surface of
an initial data with non-negative scalar curvature provides a lower bound
for the "mass" of its associated space time, as was conjectured by Penrose
and proven by Bray and Huisken-Ilmanen. Other special surfaces that have
been studied in this context include stable constant mean curvature
surfaces and isoperimetric surfaces. I will explain why positive mass
works to the effect that large stable constant mean curvature surfaces are
always isoperimetric. This answers a question of Bray's and complements
the results by Huisken-Yau and Qing-Tian on the "global uniqueness problem
for stable CMC surfaces" in initial data sets with positive scalar
curvature.
|
| 6 April |
Speaker: Yuanqi Wang (Wisconsin)
Title:On four-dimensional anti-self-dual gradient Ricci solitons
Abstract:
Classification of 4-dim gradient Ricci solitons is important to the study
of 4-dim Ricci flow with surgeries. My talk will be based on our
classification of anti-self-dual gradient shrinking Ricci solitons and our
results on anti-self-dual steady Ricci
solitons. This is highly related to the analyticity of Ricci solitons. I
will also discuss something on anti-self-dual Ricci flows.
|
| 13 April |
Speaker: Bing Wang (Princeton)
Title: On the conditions to extend Ricci flow
Abstract:
We show that at a finite singularity of Ricci flow, Ricci curvature
must blowup at least at the rate of type I. Also, the $|Rm||R|$ behaves
like $|Ric|$, it also blows up at least at type I. This statements
are based on some new estimates among scalar, Ricci and Riemannian curvature
tensors.
As applications, we can also show some gap theorems of complete
shrinking Ricci solitons.
|
| 20 April |
Speaker: No Seminar (Karel deLeeuw Lecture)
Title:
Abstract:
|
| 27 April |
Speaker: Ulrich Menne (Golm)
Title:Properties of singular submanifolds
Abstract:
The talk concerns a class of singular submanifolds having a
distributional mean curvature, i.e. integral varifolds of locally
bounded first variation. The main result shows that those objects admit
an approximate second fundamental form. Moreover, a Harnack inequality
and, in codimension one, an isoperimetric inequality of higher order
will be presented under the hypothesis of critical integrability of the
mean curvature.
|
| 4 May |
Speaker: No Seminar
Title:
Abstract:
|
| 11 May |
Speaker: Nestor Guillen (MSRI)
Title: An overview of non-local mean curvature
Abstract:
While the connection between stochastic processes and fully non-linear
integro-differential elliptic equations remains strong in the
non-local (fractional order) case, connections with geometry are less
abundant (i.e. currently there is no known analogue of Monge-Ampere
and Gauss curvature). Recently, Caffarelli and Souganidis came across
an integro-differential invariant for embedded hypersurfaces whose
principal part is a fractional power of the Laplacian, this
pseudo-differential invariant measures the average deviation of the
hypersurface from a plane at all scales, not just infinitesimally. I
will present recent results on this non-local mean curvature,
including integro-differential analogues of the regularity theory of
minimal surfaces (including Almgren's theory) as well as Aleksandrov's
moving plane method.
|
| 18 May |
Speaker: Eric Bahuaud (Stanford)
Title:Ricci flow of asymptotically hyperbolic metrics
Abstract:
Abstract: In this talk I'll discuss recent work showing that the
normalized Ricci flow preserves the set of smooth conformally compact
asymptotically hyperbolic metrics for a short time. I'll also discuss
stability of the flow near certain perturbations of the hyperbolic metric.
|
25 May
Special time: 5:00pm |
Speaker: Brian White (Stanford)
Title:The Besicovitch-Federer Structure Theorem and Related Matters
Abstract:
A highly non-technical talk describing the Besicovitch-Federer structure theorem for k-dimensional sets in R^n and some related results.
|
31 May
(Note Special Date) |
Speaker: Niall O'Murchadha (University of Cork, Ireland)
Title: Scaling the extrinsic curvature in the gravitational initial data:
Generating trapped surfaces.
Abstract:
Abstract: In the standard conformal method of constructing initial data
for the Einstein equations, the `free' data is chosen to be a base
Riemannian 3-metric and a divergenceless, tracefree tensor (TT) relative
to this metric. One then gets a non-linear elliptic equation (the
Lichnerowicz equation) for a conformal factor. If I take a TT tensor and
multiply it by a constant, it is still TT. One can show that the
conformal factor monotonically increases as this constant is increased.
In particular, the ADM mass of the solution becomes unboundedly large.
This forces the appearance of trapped surfaces in the intial data. This
will be a `low-tech' talk, showing how one can prove interesting results
using nothing but the maximum principle.
|
| 1 June |
Speaker: Francisco Martin (Granada)
Title: Properly embedded minimal surfaces in H^2xR with nontrivial topology
Abstract:
We prove that any non simply connected planar domain can be properly and minimally embedded in $\mathbb{H}^2\times\mathbb{R}$.
The examples that we produce are vertical bi-graphs, and they are obtained from the conjugate surface of a Jenkins-Serrin graph.
All these examples have parabolic conformal type and, if the number of ends is finite, they also have finite total curvature.
This is a joint work with M. Magdalena RodrÃguez.
|