Informal Geometric Analysis Seminar

Department of Mathematics

University of Maryland

Fall 2015 - Spring 2016

Date: Thursdays at 4:00pm.
Room: Mathematics Building 2300.

Organized by: T. Darvas, Y.A. Rubinstein.

The aim of this seminar is to attract graduate students to Geometric Analysis, through learning and research talks. All talks should be accessible to beginning graduate students who might have background either in PDE or in geometry, but not necessarily in both.

Previous years: 2012-2013, 2013-2014, 2014-2015.

  • September 17, Jim Isenberg (University of Oregon),
    Some Open Problems in Mathematical Relativity
    Einsteinís theory of General Relativity (GR), which celebrates its centenary this year, provides a beautiful mathematical model of large scale gravitational phenomena in astrophysics and cosmology. It also provides mathematicians with wonderfully challenging mathematical questions to explore. After introducing GR and its initial value problem, I discuss a few of the outstanding questions in mathematical relativity which are currently being studied. Depending on time, I will discuss
    1) Parametrization and construction of initial data sets which satisfy the Einstein constraint equations
    2) Strong cosmic censorship and the long time behavior of spacetime solutions of Einsteinís equations
    3) Stability of the Kerr black hole solution.

  • September 24, Jim Isenberg (University of Oregon),
    Some Open Problems in Mathematical Relativity: Parametrization and construction of initial data sets

  • October 1, Jacob Bernstein (Johns Hopkins University)
    Title: Properties of hypersurfaces of low entropy
    Abstract: The entropy is a quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the geometric complexity of a hypersurface of Euclidean space.It is closely related to the mean curvature flow.On the one hand, the entropy controls the dynamics of the flow.On the other hand, the mean curvature flow may be used to study the entropy.In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy really are simple.

  • October 8, Amitai Yuval (Hebrew University),
    Title: Geodesics of positive Lagrangians and almost Calabi-Yau reduced spaces
    Abstract: The space of positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of all Lagrangian submanifolds. A Hamiltonian isotopy class of positive Lagrangians admits a natural Riemannian metric, which gives rise to a notion of geodesics. The geodesic equation is a fully non-linear degenerate elliptic PDE, and it is not known yet whether the initial value problem and boundary problem have solutions in general.
    We will talk about Hamiltonian classes of positive Lagrangians which are invariant under a Lie group Hamiltonian action. Such a Hamiltonian class is isometric to the corresponding class in the symplectic reduced space, which has a natural almost Calabi-Yau structure. We will show that when the symplectic reduced space is of real dimension 2, both the initial value problem and boundary problem have unique solutions. As examples, we will discuss Hamiltonian classes of symmetric positive Lagrangians in toric Calabi-Yau manifolds and Milnor fibers. As time permits, we will show as an application that in these cases, the Riemannian metric induces a metric space structure on every Hamiltonian isotopy class, and that the obtained metric spaces can be embedded isometrically in L^2 spaces.

  • October 15, Amitai Yuval (Hebrew University),
    Title: Geodesics of positive Lagrangians and almost Calabi-Yau reduced spaces (continued)

  • October 22, Hans Joachim Hein (UMD)
    Title: Calabi-Yau cones
    A Riemannian cone is a warped product space C = (0,infty) x L with metric g_C = dr^2 + r^2*g_L, where r denotes the standard coordinate on (0,infty) and (L, g_L) is some given closed Riemannian manifold called the link or cross-section of the cone. We say that (C, g_C) is a Calabi-Yau cone if the metric g_C is Ricci-flat Kahler. I will try to explain why people care about such cones and what you can do with them.

  • October 29, Martin Li (Hong Kong)
    Title: Simonsí identity and its consequences
    Abstract: In a celebrated work of J. Simons in 1968, he discovered a fundamental identity about the Laplacian of the second fundamental form of a minimal submanifold. The identity (and its inequality form) gives curvature estimates for stable minimal hypersurfaces, which is closed related to the classical Bernstein theorem and regularity theory of minimal hypersurfaces. On the other hand, when the ambient space is homogeneous like the round sphere, the identity gives nice rigidity results about its minimal submanifolds. We will discuss some old and new results in this aspect and also indicate how this could be related to the study of free boundary minimal surfaces.

  • November 5, David Hoffman (Stanford)
    Title: Limiting behavior of sequences of embedded minimal disks
    : we prove that it is possible to get families of catenoids as limit leaves of a limit lamination of embedded minimal disks.We can also produce sequences whose curvature blows up on any specified closed subset of the real line. Our method allows us to give another counterexample to the general Calabi-Yau conjecture for hyperbolic space, producing a complete and embedded---but not properly embedded---simply connected minimal surface on either side of any area-minimizing catenoid in hyperbolic space. This is joint work with Brian White.

  • November 12, John Loftin (Rutgers),
    Title: "Affine Spheres and the Real Monge-Ampere Equation"
    Affine differential geometry is the study of differential invariants of hypersurfaces in R^{n+1} which are invariant under volume-preserving affine actions on R^{n+1}.We'll define and discuss some of the basic objects in the theory (affine spheres, affine maximal hypersurfaces), and their relation to real Monge-Ampere equations.We'll focus on the case of hyperbolic affine spheres, and discuss some issues in existence and regularity of solutions due to Cheng-Yau.

  • November 19, Xin Dong (UMD)
    Title: Korn's Inequality and a Theorem on Geometric Rigidity

  • December 3, Yakov Shlapentokh-Rothman (Princeton)
    Title: Introduction to the Black-Hole "No Hair Conjecture"
    We will introduce and motivate the notion of a black-hole in general relativity and explain the famous "no hair conjecture." Next, we will present the classic Carter-Robinson theory which establishes the conjecture for asymptotically flat, axisymmetric black-holes with no matter. If time permits, we will end with a discussion of recent work (joint with Otis Chodosh) that shows that this conjecture dramatically fails when one adds in even the very simple matter model of a massive scalar field.

  • December 8, Vamsi Pingali (JHU)
    Title: A generalised Monge-Ampere equation
    A fully nonlinear PDE of the Monge-Ampere type will be introduced in this talk. Places where it pops up (both locally and globally) will be mentioned. A few results (existence and a priori estimates) - both existing and new will be discussed

  • March 4, Dmitry Jacobson (McGill) (note special time)
    Title: Nodal sets in conformal geometry.
    we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension at least 3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n>=3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. This is joint work with Yaiza Canzani, Rod Gover and Raphael Ponge. If time permits, we shall discuss related results for operators on graphs.

Fall 2016: Herman Gluck (Oct 6).