Informal Geometric Analysis Archives for Academic Year 2016


TBA

When: Thu, September 17, 2015 - 4:00pm
Where: Math2300
Speaker: Jim Eisenberg (University of Oregon) -
Abstract: TBA

Properties of hypersurfaces of low entropy

When: Thu, October 1, 2015 - 4:00pm
Where: Math2300
Speaker: Jacob Bernstein (JHU) -
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.

Geodesics of positive Lagrangians and almost Calabi-Yau reduced spaces

When: Thu, October 8, 2015 - 4:00pm
Where: MATH2300
Speaker: Amitai Yuval (Jerusalem) -
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.

Geodesics of positive Lagrangians and almost Calabi-Yau reduced spaces (continuation)

When: Thu, October 15, 2015 - 4:00pm
Where: MATH2300
Speaker: Amitai Yuval (Jerusalem) -
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.

Calabi-Yau cones

When: Thu, October 22, 2015 - 4:00pm
Where: MATH2300
Speaker: Hans-Joachim Hein (UMD) -
Abstract: 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.

Simons’ identity and its consequences

When: Thu, October 29, 2015 - 4:00pm
Where: MATH2300
Speaker: Martin Li (Hong Kong) -
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.

Limiting behavior of sequences of embedded minimal disks

When: Thu, November 5, 2015 - 3:30pm
Where: MATH2300
Speaker: David Hoffman (Stanford) -
Abstract: 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.

Affine Spheres and the Real Monge-Ampere Equation

When: Thu, November 12, 2015 - 4:00pm
Where: MATH2300
Speaker: John Loftin (Rutgers) -
Abstract: 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.


Korn's Inequality and a Theorem on Geometric Rigidity

When: Thu, November 19, 2015 - 4:00pm
Where: MATH2300
Speaker: Xin Dong (UMD) -


Variation of Calabi-Yau metrics

When: Mon, November 23, 2015 - 12:00pm
Where: 1313.0
Speaker: Young-Jun Choi (KIAS) -


Introduction to the Black-Hole "No Hair Conjecture"

When: Thu, December 3, 2015 - 4:00pm
Where: MATH2300
Speaker: Yakov Shlapentokh-Rothman (Princeton) -
Abstract: 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.

A generalised Monge-Ampere equation

When: Tue, December 8, 2015 - 3:30pm
Where: MATH2300
Speaker: Vamsi Pingali (JHU) -
Abstract: 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.


Bifurcation of periodic solutions to the singular Yamabe problem on spheres

When: Thu, March 3, 2016 - 4:30pm
Where: 1313.0
Speaker: Bianca Santoro (CUNY) -
Abstract: In this talk, we describe how to obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are complete constant scalar curvature metrics on the complement of S^1 inside S^m, m ≥ 5, conformal to the round metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of S^m \ S^1. This is a joint work with R. Bettiol and P. Piccione.

Nodal sets in conformal geometry

When: Fri, March 4, 2016 - 4:00pm
Where: 1313.0
Speaker: Dmitry Jakobson (McGill) -