Schedule
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Each session consists of an introductory lecture (40 min.) and
an advanced lecture (40 min.), with a short break in between (10 min.).
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We plan to have a welcome lunch on Jan. 9 (12:00 -) and a banquet
on Jan. 11 (18:00 -).
Jan. 9 (Tue)
Brane Gravity and Higher Dimensional Black Holes
Tetsuya Shiromizu (Tokyo Institute of Technology)
According to the recent progress of superstring theory, our
universe may be described by a thin wall in higher dimensional
spacetimes. Moreover it has turned out that there is a possibility
to produce higher dimensional black holes in accelerators. In my
talk, we will review these topics.
Shrinking and Expanding Solutions of Lagrangian Mean Curvature Flow
Yng-Ing Lee (Taiwan University, Taiwan)
The existence of Lagrangian minimal submanifolds (or in particular
special Lagrangian submanifold) is an important and fundamental
problem. After years' study, the problem is still wildly open. One way
to obtain minimal submanifolds is using mean curvature flow. Lagrangian
condition is preserved along (smooth) mean curvature flow. Thus one can
use it to study the existence of Lagrangian minimal submanifolds.
However, complicated singularities may develop after short time of the
flow. The study of singularities plays an essential role in carrying
out the project. In these two lectures, we will first introduce some
basic definitions and properties, and briefly summarize related results
and backgrounds. Then we report a joint work with Mu-Tao Wang in this
direction.
Jan. 10 (Wed)
A Parabolic Free Boundary Problem with Bernoulli Type Condition on the
Free Boundary
Georg Weiss (University of Tokyo)
We report on new results concerning the parabolic free boundary problem
u -
t u = 0 in
{u>0},
| u| = 1 on
{u>0}.
Concerning regularity we focus on a joint result with John
Andersson (Max Planck Institute for Mathematics in the Sciences,
Leipzig, Germany). In a second part we discuss properties of the
singular set; that part is related to the analysis of
caloric/harmonic measures.
Let us describe the result with J. Andersson in a few words:
For a realistic class of solutions, containing for example all
limits of the singular perturbation problem
u -
t u =
(u) as
0,
we prove that one-sided flatness of the free boundary implies regularity.
In particular, we show that the topological free boundary
{u>0} can be decomposed into an open regular set
(relative to
{u>0}) which is locally a surface with
Hoelder-continuous space normal, and a closed singular set.
Our result extends the main theorem in the paper by H. W. Alt - L. A.
Caffarelli (1981) to more general solutions as well as the
time-dependent case. Our proof uses methods developed in H. W. Alt - L. A.
Caffarelli (1981), however we replace the core of that paper, which
relies on non-positive mean curvature at singular points, by an
argument based on scaling discrepancies, which promises to be
applicable to more general free boundary or free discontinuity
problems.
On an Elementary Metric Mollification Method and
Its Application to Closed Manifolds with Boundary
Pengzi Miao (Monash University, Australia)
In the first talk, I will present an elementary method to
approximate a metric which is C0 "bent" along a hypersurface.
Then I will explain how one can apply it in the setting of asymptotically
flat manifolds and closed manifolds with negative scalar curvature.
In the second talk, I will report a recent result of Agol, Storm and
Thurston, which gives an inequality between the volume of a compact
hyperbolic 3-manifold with minimal surface boundary and the simplicial
volume of its doubling. The proof makes uses of the above metric
approximation method and the celebrated work of Perelman on geometrization.
Various Mathematical Questions Arising in Phase Separation Problem
Yoshihiro Tonegawa (Hokkaido University)
The Modica-Mortola functional of phase transitions is well-known to
Gamma converge to the hypersurface area functional as the small
parameter tends to zero. It is now known that the control of mean
curvature-like quantity is sufficient for the convergence, and more
subtle questions can be asked about its property. In two lectures I
describe the state of understanding of the functional and the related
problems such as Allen-Cahn equation with or without coupled terms,
Cahn-Hilliard equation and a functional motivated by the large
deviation theory. Some of the questions are of technical and others of
intrinsic geometric nature in the setting of geometric measure theory.
Jan. 11 (Thu)
Analysis of Physical Systems Involving Multiple Spatial Scales:
the Examples of Liquid Crystal Elastomers and of Superhydrophobic Surfaces
Antonio DeSimone (SISSA International School for Advanced Studies,
Italy)
Liquid crystal elastomers are solids which combine
the optical properties of liquid crystals with the
mechanical properties of rubbery solids.
They display phase transformations, material instabilities,
and microstructures
in a similar way to shape-memory alloys.
The richness of the underlying material symmetries
makes the mathematical analysis of this system particularly rewarding.
Recent progress, ranging from analytical relaxation results
to numerical simulations of the macroscopic mechanical
response will be reviewed.
The other topic is wetting on rough surfaces.
We will study the impact that roughness has on the hydrophobic
behavior of a surface by means of homogenization techniques.
Dynamical Horizons in General Relativity
Marc Mars (Universidad de Salamanca, Spain)
First talk:
Einstein's theory of gravity is formulated on a four-dimensional
manifold endowed with a Lorentzian metric that satisfies the Einstein
equations. The initial data formulation and evolution equations
of the vacuum Einstein field equations will be reviewed
and short time existence and uniqueness results will be mentioned.
Long time existence is one of the most important open problems in
General Relativity. The singularity theorems and the cosmic
censorship hypothesis will be briefly discussed.
Second talk:
The singularity theorems require strong gravitational fields.
One important way of implementing this is by requiring the presence
of trapped surfaces in spacetime. The borderline case of marginally trapped
surfaces is particularly important as they can serve as local boundaries
of black holes. The evolution in time of marginally trapped surfaces
leads to the so-called dynamical horizons. In this talk I will review the most
important results involving such objects, including general properties,
a local existence theorem and a recent compactness theorem due to Andersson and Metzger.
On Quasi-Local Mass and Its Positivity
Mu-Tao Wang (Columbia University, USA)
The quasi-local mass is a quantity associated with a
compact space-like region in the space-time. It is
expected that this information can be derived from the boundary,
݃ which is a two-dimensional space-like surface.
By Throne's hoop conjecture, the quasi-local mass is supposed to
be closely related to the formation of black holes in the enclosed
region. We shall discuss some recent developments in this
direction which include Shi-Tam's proof of the positivity of the
Brown-York mass, Liu-Yau's quasi-local mass and its positivity,
and a generalization by Wang-Yau to a quasi-local energy-momentum
space-like vector. The construction relies on the solutions of
some canonically defined elliptic and parabolic equations
associated to the geometry of the surface, and the application of
the positive mass theorem.
Jan. 12 (Fri)
Applications of Gluing Constructions in General Relativity
Daniel Pollack (University of Washington, USA)
Gluing constructions are one of the basic tools in geometric
analysis and have been applied in a wide variety of problems. Over the
last seven years they have been introduced as a tool in general
relativity. We will present an overview of these applications,
highlighting new analytic aspects for gluing which have first appeared
in the context of general relativity. We will also indicate areas of
work in progress concerning black holes in higher dimensions.
Defects in Nonconvex Variational Problems
Shankar Venkataramani (University of Arizona, USA)
Defects and singularities are ubiquitous in systems that come up in
condensed matter physics and material science. Using examples from thin
elastic sheets, and also roll patterns in Rayleigh-Benard convection,
we will explore the mechanisms underlying the formation of
singularities, emphasizing the role of geometry in the analysis of
these problems. We will also discuss some new avenues of mathematical
("theoretical") research that are motivated by these "practical"
problems.
The Yamabe Constants of Infinite Coverings and a Positive Mass Theorem
Kazuo Akutagawa (Tokyo University of Science)
The Yamabe constant Y(M, C) of a given closed conformal manifold
(M, C) is defined by the infimum of
the normalized total-scalar-curvature functional E
among all metrics in C.
The study of the second variation of this functional E led
O. Kobayashi and Schoen to independently introduce a natural
differential-topological invariant Y(M), which is obtained by
taking the supremum of Y(M, C) over the space of all conformal classes.
This invariant Y(M) is called the Yamabe invariant of M.
For the study of the Yamabe invariant, the relationship between Y(M, C)
and those of its conformal coverings is important, the case
when Y(M, C) > 0 particularly. When Y(M, C) ≤ 0, by
the uniqueness of unit-volume constant scalar curvature metrics in C,
the desired relation is clear.
When Y(M, C) > 0, such a uniqueness does not hold.
However, Aubin proved that Y(M, C) is strictly less than the Yamabe
constant of any of its non-trivial finite conformal coverings,
called Aubin's Lemma.
In this talk, we generalize this lemma to the one for the Yamabe constant of
any (M, C) of its infinite conformal coverings,
under a certain topological condition on the relation between 1(M)
and 1(M).
For the proof of this, we also establish a version of positive mass theorem
for a specific class of asymptotically flat manifolds with singularities.
Last modified: Dec. 25, 2006
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