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 PDEs or in geometry, but not necessarily in both. The length of the talks is usually between 50 and 80 minutes, as chosen by the speaker.
- September 8, 2 PM, MATH1311 (note special time and place, joint with Dynamics Seminar)
Dan Romik (UC Davis)
Title: The moving sofa problem
Abstract: As everyone knows from real-life experience, moving sofas around corners is often tricky: not every sofa shape will fit. The moving sofa problem is a mathematical question that elegantly captures the subtleties involved even in a simplified two-dimensional setting. It asks for the planar shape of maximal area that can be moved around a right-angled corner in a corridor of width 1. The problem has been open for 50 years, and has a complicated conjectured solution known as Gerver's sofa, proposed in 1992 - a shape whose boundary has 18 distinct pieces. In this talk I will explain the mathematics of this fascinating problem, and tell about a new approach to the study of the problem that I developed recently, and some additional related results. The talk will be self-contained, will require no prerequisite knowledge beyond standard calculus, and will include many entertaining animations of moving sofas.
- October 4
Wolfgang Ziller (University of Pennsylvania)
Title: Graph manifolds, Nullity and the Nomizu conjecture
Abstract: 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.
- October 25
Yanir Rubinstein (UMD)
Title: Kahler-Einstein metrics, canonical random point processes and birational geometry (following Berman)
- November 1
Dan Cristofaro Gardiner (Harvard)
Title: Beyond the Weinstein conjecture
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.
- November 8
Yi Wang (JHU)
Title: A fully nonlinear Sobolev trace inequality
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.
- November 15
- February 7
Matt Dellatorre (UMD)
Title: Grauert tubes and the homogoenous MA equation (after Guillemin-Stenzel and Lempert-Szoke).
- February 14
Alex Waldron (Stony Brook)
Title: Long-time existence for Yang-Mills flow
Abstract: After briefly describing the problem, I'll give a simplified description of the proof that YM flow over a four-manifold doesn't blow up in finite time.
- February 21
- February 28
Paolo Piccione (Sao Paolo)
Title: Teichmuller theory and collapse of flat manifolds.
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).
- March 7 (MATH 1308 Note special place!)
Herman Gluck (University of Pennsylvania)
Title: Germs of fibrations of spheres by great circles always extend to the whole sphere
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.
- March 14
Jeffrey Case (Penn State) CANCELLED due to inclement weather
Title: An invariant operator on CR pluriharmonic functions
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.
- March 21
- March 28
- March 30
Alpar Meszaros (UCLA)
Title: BV estimates in optimal transport and applications
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).
- April 4
Paul Feehan (Rutgers)
Title: The Lojasiewicz-Simon gradient inequality and applications to energy discreteness and gradient flows in gauge theory
Abstract: The Lojasiewicz-Simon gradient inequality is a generalization, due to Leon Simon (1983), to analytic or Morse-Bott functionals on Banach manifolds of the finite-dimensional gradient inequality, due to Stanislaw Lojasiewicz (1963), for analytic functions on Euclidean space. We shall discuss several recent generalizations of the Lojasiewicz-Simon gradient inequality and a selection of their applications, such as global existence and convergence of Yang-Mills gradient flow over four-dimensional manifolds and discreteness of the energy spectrum for harmonic maps from Riemann surfaces into analytic Riemannian manifolds.
- April 18
- April 25, MATH1308 (Note special room!)
Artem Pulemotov (University of Queensland)
Title: The prescribed Ricci curvature problem on homogeneous spaces
Abstract: We will discuss the problem of recovering an invariant Riemannian metric on a compact homogeneous space from its Ricci curvature
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