Informal Geometric Analysis Archives for Academic Year 2016
Graph manifolds, Nullity and the Nomizu conjecture
When: Tue, October 4, 2016 - 3:30pm
Speaker: Wolfgang Ziller (U Penn) -
In many geometric problems the curvature tensor has a large nullity space. We show that under certain regularity assumptions a Riemannian manifold with almost maximal nullity is isometric to a graph manifold. As an application we show that the Nomizu conjecture holds for finite volume manifolds.
Kahler-Einstein metrics, canonical random point processes and birational geometry (following Berman)
When: Tue, October 25, 2016 - 3:30pm
Speaker: Yanir Rubinstein (UMD) -
Beyond the Weinstein conjecture
When: Tue, November 1, 2016 - 4:00pm
Where: MATH 2300
Speaker: Dan Cristofaro Gardiner (Harvard) -
Abstract: The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the ``standard" contact structure on S^3 has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.
A fully nonlinear Sobolev trace inequality
When: Tue, November 8, 2016 - 4:00pm
Speaker: Yi Wang (Johns Hopkins) -
Abstract: The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2 u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $\int -u \sigma_k(D^2 u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$. This is joint work with Jeffrey Case.
Grauert tubes and the homogoenous MA equation (after Guillemin-Stenzel and Lempert-Szoke)
When: Tue, February 7, 2017 - 4:00pm
Speaker: Matt Dellatorre (UMD) -
Long-time existence for Yang-Mills flow
When: Tue, February 14, 2017 - 4:00pm
Speaker: Alex Waldron (Stony Brook) -
Teichmuller theory and collapse of flat manifolds
When: Tue, February 28, 2017 - 4:00pm
Speaker: Paolo Piccione (Sao Paolo) -
Abstract: I will describe the Teichmuller space of flat metrics on a compact manifold, and the boundary of this space, which consists of (isometry classes) of flat orbifolds obtained by collapse. The Teichmuller space is described in terms of the isotypic components of the holonomy representation. I will prove that every compact flat orbifold can be obtained by collapsing flat metrics on some compact Bieberbach manifold. An application to the Yamabe problem on noncompact manifold will also be discussed. This is a joint work with R. Bettiol (UPenn) and A. Derdzinski (OSU).
Germs of fibrations of spheres by great circles always extend to the whole sphere
When: Tue, March 7, 2017 - 4:00pm
Where: Kirwan Hall 1308
Speaker: Herman Gluck (U Penn) -
Abstract: we prove that every germ of a smooth fibration of an odd-dimensional round sphere by great circles extends to such a fibration of the entire sphere, a result previous known only in dimension three. This is joint work with Patricia Cahn and Haggai Nuchi.
An invariant operator on CR pluriharmonic functions
When: Tue, March 14, 2017 - 4:00pm
Where: Math 2300
Speaker: Jeffrey Case (Penn State) -
Abstract: The P-prime operator is a CR invariant operator on CR pluriharmonic functions and is closely related to a sharp Moser--Trudinger-type inequality in CR manifolds. I will describe some analytic and geometric properties of this operator, and in particular use it to solve a nonlinear PDE of critical order which is the CR analogue of the Q-curvature prescription problem. This talk is based on joint works with Paul Yang and Chin-Yu Hsiao.
BV estimates in optimal transport and applications (Note special time and place)
When: Thu, March 30, 2017 - 3:30pm
Where: Kirwan Hall 3206
Speaker: Alpar Meszaros (UCLA) -
Abstract: In this talk the main question that I will consider is the regularity of solutions of certain variational problems in optimal transport. In particular I will be interested in the Wasserstein projection of a measure with BV density on the set of measures with densities bounded by a given BV function f. I will show that the projected measure is of bounded variation as well with a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an $L^\infty$ bound, where one can prove that the total variation decreases by the projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, one can obtain BV estimates for solutions of some non-linear parabolic PDEs by means of optimal transport techniques. The talk is based on a joint work with G. De Philippis (SISSA, Italy), F. Santambrogio (Orsay, France) and B. Velichkov (Grenoble, France).
The Lojasiewicz-Simon gradient inequality and applications to energy discreteness and gradient flows in gauge theory
When: Tue, April 4, 2017 - 4:00pm
Speaker: Paul Feehan (Rutgers) -
The prescribed Ricci curvature problem on homogeneous spaces
When: Tue, April 25, 2017 - 4:00pm
Where: Kirwan Hall 1308
Speaker: Artem Pulemotov (University of Queensland) -
Abstract: We will discuss the problem of recovering an invariant Riemannian metric on a compact homogeneous space from its Ricci curvature