1 April
(Starts at 4:10PM)

Speaker: John Lott (UC Berkeley)

Title: Locally collapsed 3-manifolds

Abstract: The endgame in Perelman's proof of the geometrization conjecture involves a purely Riemannian statement : a 3-manifold that is locally volume-collapsed, with respect to a lower curvature bound, is a graph manifold. I will present a proof of this statement. This is joint work with Bruce Kleiner.

Speaker: Sergiu Klainerman (Princeton)

Title: Black Hole Uniqueness Results Without Analyticity

Friday 10 April
(Special Seminar
4-5PM in 380D )

Speaker: Camillo de Lellis (University of Zurich)

Title: Higher Integrability for Area-Minimizing Currents

Abstract: The approximation of area-minimizing currents (having small excess) with graphs of harmonic functions is one of the pillars of the regularity
theory. Due to branching phenomena, in codimension larger than 1 we are forced to consider approximations by multiple-valued functions. Chapter 3 of Almgren’s big regularity paper is dedicated to this issue. In this talk I will present a new approach to the main approximation theorem of Almgren, focusing on an easy derivation of the key a-priori estimate of Almgren’s proof, which can be interpreted as a “higher integrability result”.

Speaker: Nicholas Alikakos (University of Athens)

Title: Entire Solutions of Equivariant Elliptic Systems with Variational Structure

Abstract: We discuss the system ∆u - grad W(u) = 0, where u is a vector field on n-dimensional Euclidean space and W is a scalar function on n-dimensional Euclidean space that possesses several global minima and is symmetric under a general finite reflection group G. The solutions we study derive their interest from Geometry (geometric evolution, minimal surfaces) and Phase Transitions.

Friday 17 April
(Special Seminar
4-5PM in 380D )

Speaker: Neshan Wickramasekera (Cambridge)

Title: Regularity of stable codimension-1 integral varifolds

Abstract: This is a continuation of the speaker's seminar given on Oct 15, 2009 (see abstract below). After a quick re-statement of the main regularity and compactness results, the proofs of the key estimates will be discussed in some detail.

Friday 24 April
(Special Seminar
4-5PM in 380D )

Speaker: Mu-Tao Wang (Columbia)

Title: Mean curvature fows of Lagrangian graphs and isotopy problems

Abstract: We discuss two types of Lagrangian graphs: graphs of symplectomorphisms of symplectic manifolds and graphs of one-forms in cotangent bundles. In each case, we present global existence and convergence results of the mean curvature flow and applications to isotopy problems.

Speaker: Ailana Fraser (UBC)

Title: Conformal geometry, minimal surfaces, and the Dirichlet-to-Neumann map

Abstract:

13 May

Speaker: Lizhen Ji

Title: Coarse Schottky problem and equivariant cell decomposition of Teichmuller space

Abstract: In this talk, I will explain some similar results and interaction between locally symmetric spaces and moduli spaces of curves.

For example, let A_g be the moduli space of principally polarized abelian varieties of dimension g, the quotient of the Siegel upper space by Sp(2g, Z), and M_g be the moduli space of projective curves of genus g. Then there is a Jacobian map J: M_g \to A_g, by associating to each curve its Jacobian.

The celebrated Schottky problem is to characterize the image J(M_g). Buser and Sarnak viwed A_g as a complete metric space and showed that J(M_g) lies in a very small neighborhood of the boundary of A_g as g goes to infinity. Motivated by this, Farb formulated the coarse Schottky problem: determine the image of J(M_g) in the asymptotic cone (or tangent space at infinity) C_\infty(A_g) of A_g, as defined by Gromov in large scale geometry.

In a joint work with Enrico Leuzinger, we showed that J(M_g) is d-dense in A_g for some constant d and hence its image in the asymptotic cone C_\infty(A_g) is equal to the whole cone.

Another example is that the symmetric space SL(n, R)/SO(n) admits several important equivariant cell decompositions with respect to the arithmetic group SL(n, Z) and hence a cell decomposition of the locally symmetric space SL(n, Z)\SL(n, R)/SO(n). One such decomposition comes from the Minkowski reduction of quadratic forms (or marked lattices). We generalize the Minkowski reduction to marked hyperbolic Riemann surfaces and obtain an equivariant cell decomposition of the Teichmuller space T_g with respect to the mapping class groups Mod_g.

If time permits, I will also discuss other results on similarities between the two classes of spaces.

Speaker: Ryan Hynd (UC Berkeley)

Title: Delaunay Surfaces in S^3

Abstract: We will characterize the constant mean curvature surfaces in S^3 having a symmetry generalizing rotational symmetry. In this sense, these surfaces are the Delaunay surfaces of S^3.

Speaker: Joel Hass (UC Davis)

Title: Harmonic Maps and the Space of Morse Functions

Abstract: We discuss an approach to studying the space of real valued Morse functions on a 3-manifold. The complexity of such a function can be measured by the number of critical points. Different Morse functions on a manifold can be connected by a path of functions that are Morse except at finitely many times. The complexity of the Morse functions on such a path is connected to the "Stabilization Conjecture" of 3-manifold theory. We give counter-examples to this conjecture by using harmonic maps to study the space of Morse functions on a 3-manifold. This is joint work with Abigail Thompson and William Thurston.

Speaker: Leo Tzou (Stanford)

Title: Calder\'on's Problem for Schr\"odinger Operators on Riemann Surfaces

Abstract: impedance tomography is a promising technique for non-invasive form of medical imaging. Mathematically, it amounts to solving the Calder\'on problem where one tries to recover the coefficients of an elliptic PDE from boundary measurements of the Dirichlet-Neumann map.

In this talk we will consider Calder\'on problem for the operator $\Delta_g + V$ for a fixed two dimensional Riemannian manifold (M,g). More precisely, we ask whether one can recover the scalar potential $V$ from the Dirichlet-Neumann map. We will discuss a geometrical method for showing how this can be accomplished. We will also consider some extensions of this result, namely, whether one can still recover the same information by making measruements on only a small subset of the boundary. Time permitting, we will also show that the inverse scattering problem on asymptotically hyperbolic and asymptotically Euclidean manifolds can be seen as a consequence of this result.

This is joint work with Colin Guillarmou (CNRS, Nice)

Speaker: Yanir Rubinstein (Johns Hopkins)

Title: On global well-posedness of the one-dimensional Schrodinger map flow

Abstract: In joint work with I. Rodnianski and G. Staffilani we establish the global well-posedness of the initial value problem for the Schrodinger map flow for maps from the real line into Kahler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding.

Speaker: Catherine Williams (Stanford)

Title: Asymptotic behavior of marginally trapped tubes

Abstract: In recent years, the physical and mathematical properties of a class of spacetime hypersurfaces known as marginally trapped tubes (MTTs) have been widely investigated. This talk will focus on the asymptotic behavior of such MTTs in relation to traditional black hole event horizons. In particular, in the special case of spherical symmetry, we give conditions near the event horizon of a black hole spacetime which guarantee the existence and 'nice' asymptotic behavior of an MTT, for any matter model satisfying a natural energy condition. We will also discuss related results for a certain a type of matter model, the Higgs field (a self-gravitating nonlinear scalar field).

Speaker: Eric Bahuaud (MSRI and U. of Washington)

Title: Conformal compactification of asymptotically locally hyperbolic metrics

Abstract: Conformally compact metrics provide a model for asymptotically hyperbolic (AH) geometry and are of interest to geometers and physicists alike. Another natural model for AH geometry are 'asymptotically locally hyperbolic metrics' which for a complete metric on a noncompact manifold require that sectional curvature decays to -1 at an appropriate rate outside of an appropriate compact set. In this talk I will describe recent work on the conformal compactification of these metrics. I will explain how the rate of decay of curvature influences the regularity of a conformal compactification of the metric. I will also explain how further regularity is obtained when the metric is Einstein.

Speaker: Tanya Christiansen (Missouri)

Title: The distribution of resonances for Schrödinger operators

Abstract: Resonances are complex numbers analogous to eigenvalues for a class of operators on noncompact domains. Physically, they may correspond to decaying waves, with the real part giving the frequency and the imaginary part the rate of decay. After an introduction to resonances, we will concentrate on the problem of understanding the growth of the resonance counting function for Schrödinger operators. While in the one-dimensional case its asymptotics are known, its behavior in higher dimensions appears quite subtle.

Speaker: Felix Schulze (Freie Universität Berlin)

Title: Foliations of asymptotically flat manifolds by surfaces of Willmore type.

Abstract: In this talk we show the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Hawking mass. This is joint work with T. Lamm and J. Metzger.

Speaker: Lydia Bieri (Harvard)

Title: An Extension of the Stability Theorem of the Minkowski Space in General Relativity

Abstract: The talk addresses the global, nonlinear stability of solutions of the Einstein equations in General Relativity. In particular, it deals with the initial value problem for the Einstein vacuum equations, generalizing the results of D. Christodoulou and S. Klainerman in 'The global nonlinear stability of the Minkowski space'. Every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. We consider the Cauchy problem with more general, asymptotically flat initial data. This yields a spacetime curvature which is not bounded in $L^{\infty}$ any more. The main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime.

Speaker: Quinglan Xia (UC Davis)

Title: Ramified optimal transport and their applications

Abstract: The transportation problem can be formulated as the problem of finding the optimal way to transport a given measure into another with the same mass. In mathematics, there are at least two different but very important types of optimal transportation: Monge-Kantorovich problem and ramified transportation. In this talk, I will give a brief introduction to the theory of ramified optimal transportation. In terms of applied mathematics, optimal transport paths are used to model many "tree shaped" branching structures, which are commonly found in many living and nonliving systems. Trees, river channel networks, blood vessels, lungs, electrical power supply systems, draining and irrigation systems are just some examples. After briefly describing some basic properties (e.g. existence, regularity) as well as numerical simulation of optimal transport paths, I will use this theory to explain the dynamic formation of tree leaves. On the other hand, these tree shaped optimal transport paths can be viewed as geodesics between probability measures. They provide great examples for studying geodesic problems in quasi-metric spaces, where the distance functions satisfied a relaxed triangle inequality: d(x,y) <= K(d(x,z)+d(z,y)). Then, I will introduce a new concept "dimensional distance" on the space of probability measures. With respect to this new metric, the dimension of a probability measure is just the distance of the measure to any atomic measure. In particular, measures concentrated on self-similar fractals (e.g. Cantor set, fat Cantor sets) will be of great interest to us.

Speaker: Weiyong He (University of British Columbia)

Title: The Space of Volume Forms

Abstract: The space of volume forms is of fundamental interest in many subjects, such as in optimal mass transportation theory. Recently, Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. In this talk we will try to solve this equation and a perturbation equation. These two equations are relevant to many interesting problems, such as Namh's equation and some free boundary equations. This is a joint work with X. Chen.

Speaker: Peter Topping (Warwick)

Title: Canonical Solitons for Ricci Flow

Abstract: I will describe what canonical solitons are, their properties, and how they were discovered. I will also sketch how the three different types (shrinking, expanding and steady) interact with different areas of Ricci flow theory.

Speaker: Pascal Romon (Marne-la-Valée and Stanford)

Title: Neutral Kähler structure on the tangent bundle and Lagrangian surfaces

Abstract: Recently Guilfoyle and Klingenberg have constructed a new metric on the tangent bundle $TM$ of any Kähler manifold $M$, starting with the canonical symplectic form of $T^* M$. That metric is actually a pseudo-metric of signature $(n,n)$, and we will present its construction. We will then investigate the Lagrangian submanifolds of $T^* M$ in the case where $M$ is a Riemann surface, and in particular the minimal ones with respect to this pseudo-metric, which lead to surprising results.

Speaker: Neshan Wickramasekera (Cambdrige and UCSD)

Title: Partial regularity for a class of minimal hypersurfaces

Abstract: Consider the class of minimal hypersurfaces ($n$-dimensional stationary integral varifolds) $V$ of the open unit ball in ${\mathbf R}^{n+1}$, having finite mass and satisfying the following two properties: (a) the regular part of $V$ (the part where the varifold is a smooth embedded submanifold) is orientable and stable and (b) no (interior) singular point of $V$ has a neighborhood in which the support of $V$ is the union of a finite number of embedded $C^{1, \alpha}$ hypersurfaces-with-boundary which meet along a common $(n-1)$-dimensional $C^{1, \alpha}$ submanifold. (No assumption is made on the size of the singular set, but notice e.g. vanishing of the $(n-1)$-dimensional Hausdorff measure of the singular set implies the structural condition (b).)

The main result concerning the varifolds in this class is that they are, in the interior, embedded real analytic hypersurfaces away from a closed singular set of co-dimension at least 7 (which is empty if $n \leq 6$ and discrete if $n=7$). This work generalizes the 1981 partial regularty theory of R. Schoen and L. Simon for embedded stable minimal hypersurfaces by replacing their initial regularity hypothesis (which says that the singular set has finite $(n-2)$-dimensional Hausdorff measure) by the (more ''checkable'') structural condition (b). The talk will focus on this result, some applications of it, and what can be said in the absence of condition (b).

Speaker: Anda Degeratu (Albert Einstein Institute and MSRI)

Title: Crepant resolutions of Calabi-Yau orbifolds — an analytical perspective

Abstract: A Calabi-Yau orbifold is locally modeled on $C^n/G$ with $G$ a finite subgroup of $SU(n)$. If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (of which orbifolds and their resolutions of singularities are examples), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with R. Mazzeo.

Speaker: Mihai Tohaneanu (UC Berkeley)

Title: Local energy decay on Schwarzchild and Kerr backgrounds

Abstract: Understanding the decay of linear waves is crucial in dealing with the problem of stability of the Kerr space time. I will talk about one way to measure this decay, namely local energy estimates, from which one can deduce many other useful estimates (uniform energy bounds, pointwise bounds, Strichartz estimates etc). This is joint work with J. Marzuola, J. Metcalfe and D. Tataru (for Schwarzschild) and D. Tataru (for Kerr).

12 November

Speaker: Mark Haskins (Imperial College and Stanford)

Title: Singular special Lagrangian n-folds

Abstract: We discuss recent progress on understanding singular special Lagrangian n-folds. Our focus will be on joint work with N. Kapouleas using gluing methods to construct a wide variety of special Lagrangian cones in every dimension three and greater.

Speaker: Dean Baskin (Stanford)

Title: The wave equation on asymptotically de Sitter spaces

Abstract: The de Sitter space is one of the simplest exact solutions of the vacuum Einstein equations. In this talk, I will describe the structure of the forward fundamental solutions of the wave and Klein-Gordon equations on a class of asymptotically de Sitter-like Lorentz manifolds. This class includes perturbations of de Sitter space and so it may help us examine the question of stability of the de Sitter solution.

Speaker: Adrian Butscher (Stanford University)

Title: Hamiltonian Stationary Tori in Kähler Manifolds

Abstract: Let M be a Kähler manifold. If an existence condition is satisfied at some point p in M, then a small, Hamiltonian stationary and Lagrangian torus can be constructed in a neighbourhood of p. The existence condition in question is that a certain functional on the unitary frame bundle of M has a nondegenerate critical point. This result is a Hamiltonian stationary Lagrangian analogue of a famous result by R. Ye, which states that a small constant mean curvature sphere can be constructed in a neighbourood of a point p in a Riemanian manifold M when p is a non-degenerate critical point of the scalar curvature of M.