The talks are held on Wednesdays from 4:00 to 5:00 PM in Building 2, Room 143 (unless otherwise noted).
Wednesday February 20: Martin Man-chun Li (University of British Columbia)
Title: Eigenvalue estimates and a compactness theorem for embedded minimal surfaces in the unit ball
Abstract:
Recently, A. Fraser and R. Schoen introduced an extremal eigenvalue problem for surfaces with boundary. They discovered an interesting relationship between extremal metrics and minimal surfaces in the unit ball satisfying a free boundary condition.In this talk, we will begin with some known results on the Steklov eigenvalue problem for the Dirichlet-to-Neumann map on compact manifolds with boundary. After giving some examples of minimal surfaces in the unit ball, we will prove a lower bound on the first Steklov eigenvalue for those which are embedded. We will then use this to show that the space of embedded minimal surfaces in the unit ball, with fixed topological type, is compact in the smooth topology. All these results apply if the ambient space is any compact 3-manifolds with nonnegative Ricci curvature and strictly convex boundary. In the end, we will mention some open problems in this direction. This is joint work with A. Fraser.
Wednesday March 20: Camillo De Lellis (University of Zurich)
Title: Regularity theory for area-minimizing currents
Abstract: It was established by Almgren at the beginning of the eighties that area-minimizing $n$-dimensional currents in Riemannian manifolds are regular up to a singular set of dimension at most $n-2$. To reach this
goal Almgren developed an entirely new regularity theory, which occupies a very large monograph, published posthumously.
This talk is based on a series of joint works with Emanuele Spadaro, where we give alternative proofs to all Almgren's main steps, resulting into a much more manageable approach to his entire theory.
Wednesday April 3: Reto Mueller (Imperial College London)
Title: Dynamical stability and instability of Ricci-flat metrics
Abstract: Let M be a compact manifold. A Ricci-flat metric on M is a
Riemannian metric with vanishing Ricci curvature. Ricci-flat metrics
are fairly hard to construct, and their properties are of great interest.
They are the critical points of the Einstein-Hilbert functional, the fixed
points of Hamilton's Ricci flow and the critical points of Perelman's
lambda-functional.
In this talk, we are concerned with the stability properties of Ricci-flat
metrics under Ricci flow. We will explain the following stability and
instability results. If a Ricci-flat metric is a local maximizer of lambda,
then every Ricci flow starting close to it exists for all times and
converges (modulo diffeomorphisms) to a nearby Ricci-flat metric.
If a Ricci-flat metric is not a local maximizer of lambda, then there
exists a nontrivial ancient Ricci flow emerging from it. This is joint
work with Robert Haslhofer.
Fall 2012
The talks are held on Wednesdays from 4:00 to 5:00 PM in Building 2, Room 143 (unless otherwise noted).
Wednesday October 24: Joseph Lauer (MIT)
Title: A New Length Estimate for Curve Shortening Flow and Low Regularity Initial Data for Closed Curves
Abstract: In this talk I'll introduce a crude geometric quantity that controls the length of a smooth curve as it evolves by curve shortening flow. The estimates obtained do not depend on the initial length of the curve, and we show that they can be used to control the evolution of any locally-connected compact set in the plane as it evolves by the related level set flow.
Wednesday November 14: Alessio Figalli (University of Texas at Austin)
Title: Closing Aubry Sets
Abstract: Given a Hamiltonian H on a compact manifold, one can define the "projected
Aubry set" associated to H: this is a subset of M which "captures" many important features of the Hamiltonian dynamics, and it is strongly related to the regularity of viscosity (sub)solutions to the
Hamilton-Jacobi equation.
The Mane conjecture in C^k topology states that, for a generic potential V(generic w.r.t. the C^k topology), the Aubry set associated to H+V is either a fixed point or a periodic orbit.
In this talk I will describe how, given a Hamiltonian which possesses a sufficiently smooth viscosity (sub)solution to the Hamilton-Jacobi equation, for any \epsilon >0 there exists a potential V_\epsilon, whose C^2-norm is bounded by \epsilon, such that the Aubry set associated to H+V_\epsilon is either a fixed point of a periodic orbit. This represents
a first step through the solution of the Mane Conjecture in C^2 topology.
Moreover, we will see how these techniques allow to solve the Mane
conjecture in C^1 topology. This is a joint work with Ludovic Rifford.
BROWN-MIT GEOMETRIC ANALYSIS SEMINAR Wednesday December 5: Andre Neves (Imperial College London) and Igor Rodnianski (MIT) at 3PM in Building 4, Room 149 Andre Neves' Title: Min-max Theory and the Willmore Conjecture
Igor Rodnianski's Title: Propagation and Interaction of Impulsive Gravitational Waves
Spring 2012
The talks are held on Wednesdays from 4:00 to 5:00 PM in Building 2, Room 105 (unless otherwise noted).
Wednesday February 15: Christina Sormani (CUNY)
Title: Properties of the Intrinsic Flat Convergence
Abstract: The Intrinsic Flat distance between Riemannian manifolds has been applied to study the stability of the Positive Mass Theorem, the rectifiability of Gromov-Hausdorff limits of Riemannian manifolds, and smooth convergence away from singular sets. In this talk, we will present properties of Riemannian manifolds which are conserved under Intrinsic Flat convergence. The initial notion of the Intrinsic Flat distance and lower semicontinuity of mass and the continuity of filling volumes of balls is joint work with Stefan Wenger. The speaker will also present more recent work including an Arzela-Ascoli Theorem, a notion called the Sliced Filling Volume and, if time allows, the Tetrahedral Property.
JOINT SEMINAR With PDE/ANALYSIS SEMINAR - Wednesday February 22: Jonathan Luk (Princeton)
Title: Impulsive Gravitational Waves
Abstract: We consider spacetimes satisfying the vacuum Einstein equations with impulsive Gravitational waves without symmetry assumptions. These are spacetimes such that the Riemann curvature tensor has a delta singularity across a null hypersurface. We prove local existence and uniqueness for the characteristic initial value problem with initial data that has a delta singularity in the curvature tensor. A precise description of the propagation of singularity is also given. The proof introduces a new type of energy estimates for the vacuum Einstein equations, allowing the L2 norm of some components of the curvature tensor to be infinite. The new estimate allows us to prove local existence and uniqueness for a general class of initial data which is non-regular along a null direction. We will also discuss extensions of this theorem, which can be applied to understand colliding impulsive gravitational waves and the formation of trapped surface. This is joint work with I. Rodnianski.
JOINT SEMINAR With PDE/ANALYSIS SEMINAR - Tuesday March 6: Tobias Lamm (University of Frankfurt): 4PM in 2-131 Title: Rigidity results for conformal immersions in R^n
Abstract: By a classical result of Codazzi every closed, totally umbilic surface in $\R^n$ is a round sphere. De Lellis and Muller proved a rigidity statement corresponding to this result. More precisely, they showed that for every closed surface in $\R^3$, whose traceless second fundamental form is "small" in $L^2$, there exists a conformal parametrization whose distance to a standard parametrization of a round sphere is small in $W^{2,2}$. In a recent joint work with H. Nguyen (Warwick) we were able to extend
this result to arbitrary codimensions. Moreover, we obtained related rigidity results for inversions of the catenoid and Enneper`s minimal surface. In my talk I will review the analytic preliminaries (i.e. the results of Muller-Sverak and Kuwert-Li) and I will sketch the proof of the above mentioned results.
Wednesday March 14: Dmitri Burago (Penn State)
Title: Boundary Rigidity and Minimal Surfaces: A Survey
Abstract: A Riemannian manifold with boundary is said to be boundary rigid if its metric
is uniquely determined by the boundary distance function, that is the restriction
of the distance function to the boundary. Loosely speaking, this means that
the Riemannian metric can be recovered from measuring distances between
boundary points only. The goal is to show that certain classes of metrics are
boundary rigid (and, ideally, to suggest a procedure for recovering the metric).
To visualize that, imagine that one wants to find out what the Earth is made of.
More generally, one wants to find out what is inside a solid body made of
different materials (in other words, properties of the medium change from point
to point). The speed of sound depends on the material. One can "tap" at some
points of the surface of the body and "listen when the sound gets to other points".
The question is if this information is enough to determine what is inside.
This problem has been extensively studied from PDE viewpoint: the distance
between boundary points can be interpreted as a "travel time" for a solution of
the wave equation. Hence this becomes a classic Inverse Problem when we
have some information about solutions of a certain PDE and want to recover its
coefficients. For instance such problems naturally arise in geophysics (when
we want to find out what is inside the Earth by sending sound waves), medical
imaging etc.
In a on-going joint project with S. Ivanov we suggest an alternative geometric
approach to this problem. In our earlier work, using this approach we were able
to show boundary rigidity for metrics close to flat ones (in all dimensions), thus
giving the first open class of boundary rigid metrics beyond two dimensions. We
were now able to extend this result to include metrics close to a hyperbolic one.
The approach is grown up from another long-term project of studying surface
area functionals in normed spaces, which we have been working on it for more
than ten years. There are a number of related issues regarding area-minimizing
surfaces in Riemannian manifold. The talk gives a non-technical survey of ideas
involved. It assumes no background in inverse problems and is supposed to be
accessible to a general math audience (in other words, we will sweep technical
details under the carpet).
Wednesday March 21: Lu Wang (Johns Hopkins)
Title: Rigidity of Self-shrinkers of Mean Curvature Flow
Abstract: Recently, using the desingularization technique,
Kapouleas-Kleene-Moller and independently Nguyen has successfully
constructed a new family of smooth complete embedded self-shrinkers
asymptotic to cones. These are the first non-rotationally symmetric
examples after planes, spheres, cylinders and Angenent's torus.
In this talk, we report some new rigidity (at infinity) theorems of
self-shrinkers. The results are two folds with emphasis on the
asymptotically cylindrical case. First, we show the uniqueness of smooth
properly embedded self-shrinkers asymptotic to any given regular cone in
Euclidean space. Second, we discuss the optimal condition on the
asymptotics of self-shrinkers, so that the uniqueness of self-shrinkers
asymptotic to generalized shrinking cylinders holds true. This gives a
partial affirmative answer to the cylinder rigidity conjecture. The point
of our theorems is that we do not require completeness of self-shrinkers.
One of the main ingredients of the proofs is the (anisotropic) Carleman
estimates inspired by the work of Escauriaza-Seregin-Sverak.
Among applications, we obtain some non-existence results of
self-shrinkers. Namely, except hyperplanes, there do not exist any other
smooth complete properly embedded self-shrinkers with ends asymptotic to
rotationally symmetric cones.
SPECIAL DATE AND LOCATION: Friday April 13: Benjamin Schmidt (Michigan State) 3PM in 2-131
Title: Three-manifolds of Constant Vector Curvature
Abstract: A Riemannian manifold M has constant vector curvature k if every
tangent vector to M belongs to a tangent plane of curvature k. When all
sectional curvatures of M are additionally either bounded above or below by k,
M is said to have extremal curvature k. I will discuss classification results
for finite volume three-manifolds of constant vector curvature k and extremal
curvature k. The talk will be based on joint work with Jon Wolfson.
Wednesday April 18: Jim Isenberg (University of Oregon)
Title: Asymptotic Behavior of Degenerate Neckpinches in Ricci Flow
Abstract: We discuss the detailed nature of the geometry of rotationally symmetric degenerate neckpinch singularities which develop in the course of Ricci flow.
Wednesday May 9: Thomas Mettler (MSRI) ~~ SPECIAL TIME: 3PM ~ ROOM: 56-144 ~~
Title: Weyl Metrisability for Projective Surfaces
Abstract: The existence problem for Riemannian metrics on a surface with prescribed unparametrised geodesics was first studied by R. Liouville. He observed that the problem can be formulated as a linear first order PDE system which in general will only admit trivial solutions. The necessary and sufficient conditions for local existence of nontrival solutions were found only recently by Bryant, Dunajski and Eastwood. Surprisingly the conditions are rather complicated. However if one looks for Weyl structures on surfaces with prescribed unparametrised geodesics the situation is different. In this talk I will use techniques from complex geometry to show that the corresponding PDE system always admits local solutions. I will also show that the Weyl structures on the 2-sphere whose geodesics are the great circles, are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane. If time permits, I will explain that the Weyl metrisability problem for projective surfaces has a natural analogue in all even dimensions.
Fall 2011
The talks are held on Wednesdays from 4:00 to 5:00 PM in Building 2, Room 139 (unless otherwise noted).
Wednesday Sepetember 21: Sean Timothy Paul (University of Wisconsin , Madison) and Nikola Kamburov (MIT)
Speaker Sean Timothy Paul (University of Wisconsin, Madison) 3:00PM
Title: Hyperdiscriminants, Orbit Closures, and Lower Bounds on K-energy Maps
Abstract: One of the main problems in complex geometry is to detect the existence of "canonical" Kahler metrics in a given Kahler class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. In this case the presence of positive curvature makes this problem extremely difficult and has led to a striking series of conjectures--the "standard conjectures"-- which relate the existence of these special metrics (which are solutions to the complex Monge-Ampere equation, a fully non-linear elliptic p.d.e . ) to the algebraic geometry of the pluri-anticanonical images of the manifold. Yau speculated that the relevant algebraic geometry would be related (somehow) to Mumford's Geometric Invariant Theory. Eventually it was conjectured that K-energy bounds along Bergman potentials could be deduced from an appropriate notion of "semi-stability". Recently this conjecture has been completely justified by the speaker, building upon work of Gang Tian and Gelfand-Kapranov-Zelevinsky and Weyman-Zelevinsky. It is the aim of this talk to outline progress on the standard conjectures and to discuss the entire theory in the context of complex algebraic groups and dominance of rational representations of such groups.
Speaker: Nikola Kamburov (MIT) 4:00PM
Title: On a Free Boundary Variant of a Conjecture of De Giorgi
Abstract: Drawing out a fascinating connection between Bernstein's
problem, on the one hand, and the study of global, bounded and monotone
solutions to the semilinear elliptic equation $\Delta u = u3 - u$ in
$R^n$, on the other, a famous conjecture of De Giorgi states that
the level sets of such solutions are hyperplanes, at least in dimension
$n\leq 8$. The conjecture was verified for $n\leq 8$ by Savin. Recently,
Del Pino, Kowalczyk and Wei constructed a counterexample in dimension
$n=9$, using an intricate fixed point argument. In this talk, I would
like to discuss the construction of such a counterexample for an
appealing free boundary variant of De Giorgi's conjecture. Our approach
uses the more transparent means of the method of barriers.
Wednesday September 28: Dan Lee (CUNY)
Title: The Spacetime Positive Mass Theorem
Abstract: I will discuss the proof of the spacetime positive mass
theorem, recently proved in joint work with Eichmair, Huang, and
Schoen. For dimensions less than $8$, we show that an asymptotically
flat initial data set satisfying the dominant energy condition must
have $E \ge |P|$, where $(E,P)$ is the ADM energy-momentum. Our proof
follows the basic idea of the proof of the time-symmetric case, with
marginally outer trapped hypersurfaces (MOTS) taking the place of
minimal hypersurfaces, with some added complications.
Wednesday October 5: Robert Haslhofer (ETH Zurich)
Title: Singularities in 4d Ricci Flow
Abstract: In this talk, we discuss the formation of singularities in
higher-dimensional Ricci flow without pointwise curvature assumptions. We
show that the space of singularity models with bounded entropy and locally
bounded energy is orbifold-compact in arbitrary dimensions. In dimension
four, a delicate localized Gauss-Bonnet estimate even allows us to drop the
assumption on energy in favor of (essentially) an upper bound for the Euler
characteristic. We will also see how these results are part of a larger
project exploring high curvature regions in 4d Ricci flow. This is all
joint work with Reto Mueller.
Wednesday October 12: Niels Martin-Moller (MIT)
Title: Gluing of Self-shrinkers for Mean Curvature Flow: Complete, Embedded Examples w/ Genus
Abstract: For mean curvature flow in any ambient Riemannian 3-manifold, the
self-similar surfaces in R^3 form the basic "atoms" of the singularity
theory. However, the only known complete, embedded examples are: 1)
Flat planes, round cylinders, round spheres and 2) One (not round)
torus found by Angenent in 1989.
We will discuss gluing constructions that yield new complete,
embedded, self-shrinking surfaces of large genus g in R^3 (as expected
from numerics by Tom Ilmanen in 1995) by fusing known examples (plane
and sphere) while introducing genus. The analysis for non-compact ends
is complicated by the unbounded geometry, and PDEs of
Ornstein-Uhlenbeck type need to be understood well via Liouville-type
results, which in turn enable compactifying parts of the problem, in
order to construct resolvents and close the non-linear PDE system (w/
N. Kapouleas & S. Kleene).
Wednesday October 19: Lan-Hsuan Huang (Columbia)
Title: Hypersurfaces with Nonnegative Scalar Curvature
Abstract: Since the time of Gauss, geometers have been interested in
the interplay between the intrinsic metric structure of hypersurfaces
and their extrinsic geometry from the ambient space. For example, a
result of Sacksteder tells us that if a complete hypersurface has
non-negative sectional curvature, then its second fundamental form in
Euclidean space must be positive semi-definite.
In a recent joint work with Damin Wu, we study hypersurfaces under a
much weaker curvature condition. We prove that a hypersurface with
nonnegative scalar curvature which is either closed or complete of
finite many regular ends must be weakly mean convex. This result is
optimal in the sense that the scalar curvature cannot be replaced by
other k-th mean curvatures. The result and argument have applications
to the mean curvature flow, positive mass theorem, and rigidity
theorems.
Wednesday October 26: John Head (Courant)
Title: Mean Curvature Flow of Two-Convex Hypersurfaces
Abstract: We discuss mean curvature flow of two-convex hypersurfaces in
Euclidean space: we explain the relationship between the
Huisken-Sinestrari surgery program and the well-known weak solution, and
we show how this theory can be used to prove regularity estimates for
mean curvature flow.
Wednesday November 2: Peter Herbrich (University of Cambridge)
Title: On Inaudible Properties of Broken Drums - Isospectral Domains
with Mixed Boundary Conditions
Abstract: Since Kac raised the question "Can one hear the shape of a drum?",
various families of non-smooth counterexamples have been constructed
using the transplantation method which is based on a group-theoretic
technique by Sunada. We apply the transplantation method to domains
with mixed boundary conditions which can be interpreted as broken
drums. The method is translated into graph theory which allowed for a
computer-aided search for transplantable pairs, a classification in
terms of induced representations, and the development of tools with
which new pairs can be generated from given ones.
The talk finishes with a presentation of various new pairs among which
there are 10 versions of the Gordon-Webb-Wolpert drums with mixed
boundary conditions. In the end, we discuss inaudible properties and
show the first example of a connected drum that sounds disconnected
and of a broken drum that sounds unbroken, that is, a planar domain
with mixed boundary conditions that is isospectral to a domain with
Dirichlet boundary conditions. Above all, the latter example shows
that an orbifold can be Dirichlet isospectral to a manifold.
Wednesday November 9: Eveline Legendre (MIT and University of Toulouse)
Title: The Toric Futaki Invariant on Quadrilaterals
Abstract: I will first explain why, in the toric framework, Kahler and Sasaki metrics of constant scalar curvature corresponds to solutions of the same PDE problem on convex polytopes. Then, I will show that (the toric version of) the Futaki invariant is the only obstruction to the existence of such a solution whenever the polytope is a quadrilateral. Finally, I will explain how the Futaki invariant is related to a certain isoperimetric quotient in the Sasaki case and why this leads to (the first known) examples of non-isometric Sasaki metrics of constant scalar curvature, compatible with the same contact structure.
JOINT SEMINAR With PDE/ANALYSIS SEMINAR: Tuesday November 15: Monika Ludwig (Polytechnic Institute of New York & Vienna University of Technology): 4PM in 2-142
Title: Valuations on Sobolov Spaces
Abstract: Let $F$ be a space of real valued functions, for example, the Sobolev
space ${W^{1,1}}({\mathbb R} ^n)$ and let ${\mathbb A} $ be an abelian
semi-group. A function ${\operatorname{z}}: F\to {\mathbb A} $ is called
a {\em valuation} if
$${\operatorname{z}}(f\vee g) + {\operatorname{z}}(f\wedge g)=
{\operatorname{z}}(f) + {\operatorname{z}}(g)$$
for all $f,g\in F$, where $f\vee g$ denotes the pointwise maximum and
$f\wedge g$ the pointwise minimum of $f$ and $g$. If the abelian
semi-group is given as the set of convex bodies (compact convex sets) in
${\mathbb R} ^n$ with Minkowski addition (defined by $K+L=\{x+y: x\in K,
y\in L\}$), we talk about Minkowski valuations.
After a brief excursion to the history of valuations in geometry, we
give a complete classification of affinely contravariant Minkowski
valuations on ${W^{1,1}}({\mathbb R} ^n)$ and show that every such
valuation ${\operatorname{z}}$ is given as
$${\operatorname{z}} (f) = c\, \Pi \langle f \rangle$$
for all $f\in{W^{1,1}}({\mathbb R} ^n)$ with a suitable constant $c\ge
0$. Here the convex body $\Pi \langle f \rangle$ is defined via its
support function as
$$h(\Pi \langle f \rangle, v)=\frac12\int_{{\mathbb R} ^n} |v\cdot
\nabla f(x)|\,dx.$$
We discuss the connection of the convex bodies $\Pi\langle f\rangle$ and
$\langle f \rangle$ (both introduced by Lutwak, Yang \& Zhang) with the
optimal norm in the sharp Sobolev inequality for a general norm, the
affine Sobolev-Zhang inequality and the solution of the functional
Minkowski problem.
SPECIAL SEMINAR: Tuesday November 22: Yanir Rubinstein (Stanford University): 3PM in 2-132
Title: Kahler-Einstein Metrics Singular along a Divisor
Abstract: The simplest example of a Kahler-Einstein (KE) metric is a football. A European football corresponds to a smooth KE metric, while an American one corresponds to a KE metric with conical singularities. The existence of smooth KE metrics on compact Kahler manifolds was proven in the 70's by Aubin and Yau for nonpositive curvature, and in the early 90's by Tian for positive curvature, under some assumptions.
In the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is bent at some angle along a complex hypersurface), motivated by applications to algebraic geometry. More recently, Donaldsonsuggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk we will describe a proof of Tian's conjecture in the case the divisor is smooth, as well as a proof of the first of Donaldson's conjectures, obtained recently in joint work with T. Jeffres and R. Mazzeo.
Wednesday November 30: Carla Cederbaum (Duke)
Title: The Newtonian Limit of Geometrostatics
Abstract: Geometrostatics is an important subdomain of Einstein's General
Relativity. It describes the mathematical and physical properties of
static isolated relativistic systems such as stars, galaxies or black
holes. For example, geometrostatic systems have a well-defined ADM-mass
(Chrusciel, Bartnik) and (if this is nonzero) also a center of mass
(Huisken-Yau, Metzger) induced by a CMC-foliation at infinity. We will
present surface integral formulae for these physical properties in
general geometrostatic systems. Together with an asymptotic analysis,
these can be used to prove that ADM-mass and center of mass 'converge'
to the Newtonian mass and center of mass in the Newtonian limit
$c\to\infty$ (using Ehler's frame theory). We will discuss geometric
similarities of geometrostatic and classical static Newtonian systems
along the way.
Wednesday December 7: Daniele Valtorta (University of Milan)
Title: Geometric Analysis Seminar: Sharp Lower Bounds for the First Eigenvalue of the P-laplacian with Ricci
Curvature Assumptions
Abstract: In this seminar, we will speak of sharp lower bounds for the first eigenvalue of the p-laplacian on a Riemannian manifold with a Ricci lower bound and some rigidity theorems. After a brief review of the results available for the usual laplacian, we will discuss how it is possible to generalize them for the generic p-case. In particular, we will focus on two techniques. The first one is based on Schoen-Yau type gradient estimates and in the usual case relies on the Bochner formula. As we will see it is possible to generalize this important theorem using the linearized p-laplacian. Another technique is based on Levy-Gromov isoperimetric ineaqualities and permits easily to obtain a generalization of the well-known Obata's theorem. Recent improvements have been made to isoperimetric inequalities by Milman, especially when the lower bound of the Ricci curvature is negative.
This work has been done in collaboration with Aaron Naber.
SPECIAL DATE AND LOCATION: Friday December 9: Valentino Tosatti (Columbia) 4PM Location: 2-105
Title: Collapsing of Abelian Fibred Calabi-Yaus and Hyperkahler Mirror Symmetry
Abstract: We will address the problem of understanding the collapsing
of Ricci-flat Kahler metric on abelian fibred projective Calabi-Yau
manifolds. We will then explain an application of these results to the
Strominger-Yau-Zaslow picture of mirror symmetry for some hyperkahler
manifolds. Joint work with Mark Gross and Yuguang Zhang.
Spring 2011
The talks are held on Wednesdays from 4:00 to 5:00 PM in Building 2, Room 139 (unless otherwise noted).
Thursday February 3: Jacob Bernstein (Stanford University) Location: 2-135 at 4:00 PM
Title: A Variational Characterization of the Catenoid
Abstract: We show that the catenoid is the unique surface of least area (suitably understood) within a geometrically natural class of minimal surfaces. The proof relies on a techniques involving the Weierstrass representation used by Osserman and Schiffer to show the sharp isoperimetric inequality for minimal annuli. An alternate approach that avoids the Weierstrass representation will also be discussed. This latter approach depends on a conjectural sharp eigenvalue estimate for a geometric operator and has interesting connections with spectral theory. This is joint work with C. Breiner.
Wednesday February 9: Luca Martinazzi (Centro De Giorgi, Pisa)
Title: The relaxed energy for harmonic maps from B^3 into S^2
Abstract: I will discuss the relaxed Dirichlet energy for maps from B^3
into S^2 (definitions, meaning, etc... ), treating the regularity theory
in particular. This field contains several long standing open questions. I
will present a recent result which gives us a better understanding of the
minimizers of this energy, and casts even more open questions.
Wednesday February 16: Tom Ilmanen (ETH Zurich)
Title: Initial Time Singularities in Mean Curvature Flow
Abstract: Let M_0 be a closed subset of R^n+1 that is a smooth hypersurface
except for a finite number of isolated singular points. Suppose that
M_0 is asymptotic to a regular cone near each singular point.
Can we flow M_0 by mean curvature?
Theorem (n<7): there exists a smooth mean curvature evolution starting
at M_0 and defined for a short time 0
Such an initial M_0 might arise as the limit of a smooth mean
curvature evolution defined earlier than t=0. Thus, the result allows
us to flow through singularities in some cases.
We use a monotonicity formula that complements the monotonicity
formula of Huisken. The method applies to other geometric heat flows
as well.
Wednesday February 23: Lars Andersson (Max Planck Institute for Gravitational Physics)
Title: The Black Hole Stability Problem
Abstract: The problem of nonlinear stability for the Kerr model of a rotating black hole is one of the central problems in general relativity. The analysis of linear fields of spins 0,1,2 on the Kerr spacetime is an important model problem for full nonlinear stability. In this talk, I will present recent work with Pieter Blue which makes use of the hidden symmetry related to the Carter constant to circumvent these difficulties, and give a "physical space" approach to estimates for
the wave equation, including energy bounds, trapping, and dispersive estimates.
Wednesday March 2: Christina Sormani (CUNY GC and Lehman College)
Title: The Intrinsic Flat Distance between Oriented Riemannian Manifolds
Abstract: We introduce the intrinsic flat distance between compact oriented Riemannian manifolds with finite volume. The limits
of sequences with uniform upper bounds on their volume and diameter are integral current spaces: countably H^m rectifiable metric
spaces with boundary. When the sequence has a uniform lower bound on Ricci curvature and volume, then by work of Cheeger-Colding,
we see that the Gromov-Hausdorff and Intrinsic Flat limits agree. In general, Intrinsic Flat Limits are subsets of the GH limits.
Intrinsic Flat limits may exist when there is no GH limit as it is a weaker notion of convergence. This is joint work with
S. Wenger and is based on work of Ambrosio-Kirchheim. See http://comet.lehman.cuny.edu/intrinsicflat.html for relevant
papers/preprints.
Wednesday March 9: Igor Rodnianski (Princeton University)
Title: On Formation of Trapped Surfaces in General Relativity
Wednesday April 13: Ulrich Menne (Albert Einstein Institute, Golm)
Title: Existence of an Approximate Second Fundamental Form for Singular Submanifolds with Generalized Mean Curvature
Abstract: The objects considered are integral varifolds with locally bounded first variation. It is established that these are
rectifiable of second order. The proof relies on a new differentiability criterion for functions in Lebesgue spaces phrased in terms of approximability by harmonic functions. Throughout the talk, concepts from geometric measure theory will be illustrated by
many examples.
Wednesday April 20 : Rob Neel (Lehigh University) : SPECIAL TIME 3PM
Title: Some Geometric Aspects of Brownian Motion on a Minimal Surface
Abstract: We demonstrate how stochastic methods can be applied to minimal surface theory via a concrete example requiring relatively few probabilistic technicalities. In particular, after explaining how the parabolicity and area growth of minimal ends have been previously studied using universal superharmonic functions, we describe an alternative approach, yielding stronger results, based on studying Brownian motion on the surface. We attempt to provide intuitive explanations of the stochastic techniques involved.
Wednesday May 4: Bruce Kleiner (New York University)
Title: Asymptotic Geometry, Finite Generation of Fundamental Groups, and Harmonic Functions.
Wednesday May 11: Stephen Kleene (MIT)
Title: Complete Embedded MCF Self Shrinkers
Abstract: (Joint work w/ Nicos Kapouleas and Niels Martin Moller): We present a new family of complete embedded self shrinking surfaces in $R^3$, extending the list beyond the plane, sphere, cylinder, and Angenent's Torus. They are the first and only such surfaces known outside the class of surfaces of revolution. The surfaces are constructed as highly symmetric singular perturbations of the sphere intersecting a plane through a great circle.