Past Seminars: 2019


  SPRING 2019

  • February 7, 2019
    Xin Zhou, UCSB and IAS
    Multiplicity One Conjecture in Min-max theory
    Abstract: I will present a recent proof of the Multiplicity One Conjecture in Min-max theory. This conjecture was raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for such manifolds with positive Ricci curvature, which says that there exist infinitely many pairwise non-isometric minimal hypersurfaces, and the Weighted Morse Index Bound Conjecture by Marques and Neves.

  • February 28, 2019
    Vincent R. Martinez (CUNY Hunter)
    Asymptotic enslavement in hydrodynamic equations and applications to data assimilation
    Abstract: In their 1967 seminal paper, Foias and Prodi captured a notion of finitely many degrees of freedom in the context of the two-dimensional (2D) incompressible Navier-Stokes equations (NSE). In particular, they proved that if a sufficiently large low-pass filter of the difference of two solutions converge to 0 asymptotically in time, then the corresponding high-pass filter of their difference must also converge to 0 in the long-time limit. In other words, the high modes are “eventually enslaved” by the low modes. One could thus define the number of degrees of freedom to be the smallest number of modes needed to guarantee this convergence for a given flow. This property has since led to several developments in the long-time behavior of solutions to the NSE, particularly to the mathematics of turbulence, but more recently to data assimilation. In this talk, we will give a survey of rigorous studies made for a certain approach to data assimilation that exploits this asymptotic coupling property as a feedback control. We will discuss these issues in the specific context of the 2D NSE, 2D surface quasi-geostrophic equation, as well recent joint work with N. Glatt-Holtz, A. Farhat, S. Macquarrie, and J.P. Whitehead on the 3D Boussinesq equation as it applies to mantle convection.

  • March 28, 2019
    Mohameden Ahmedou, Giessen University (Germany)
    Conical metrics of prescribed Gaussian and geodesic curvatures on compact surfaces
    Abstract: We consider the problem of finding conformal metrics with prescribed Gauss curvature and zero geodesic curvature. This amounts to solve a nonlinear Liouville equation under Neumann boundary condition and hence enjoys a variational structure. Moreover it turns out that, as far as the variational aspects are concerned, one has to distinguish between the "resonant" and "non resonant" case depending on whether or not the sum of the integrals of the Gauss curvature on the surfaces and the integral of the geodesic curvature on the boundary takes some explicit critical values. Indeed in the "non resonant" the associated variational problem is compact, while it is non compact in the resonant one. Using a Morse theoretical approach, we prove some existence results in the "non resonant " case and establish Morse Inequalities. A major role in our argument is played by some "boundary-weighted barycentric" set. In the resonant one we study the "critical points at Infinity" and derive some Euler-Poincaré type criterion of the existence of solutions.

  • April 3-April 4, 2019
    Two Days Conference: Workshop on Nonlinear Problems in Geometry
    Speakers:
    Pengfei Guan, Matt Gursky, Fengbo Hang, Lan-Hsuan Huang,
    Emmanuel Humbert, Ernst Kuwert, Yanyan Li, Gabriella Tarantello
    Abstracts

  • April 11, 2019
    Alexander Nabutovsky, University of Toronto
    Filling metric spaces
    Abstract: Uryson k-width of a metric space X measures how close X is to being k-dimensional. Several years ago Larry Guth proved that if M is a closed n-dimensional manifold, and the volume of each ball of radius 1 in M does not exceed a certain small constant e(n), then the Uryson (n-1)-width of M is less than 1. This result is a significant generalization of the famous Gromov's inequality relating the volume and the filling radius that plays a central role in systolic geometry. Guth asked if a much stronger and more general result holds true: Is there a constant e(m)>o such that each compact metric space with m-dimensional Hausdorff content less than e(m) always has (m-1)-dimensional Uryson width less than 1? Note that here the dimension of the metric space is not assumed to be m, and is allowed to be arbitrary. Such a result immediately leads to interesting new inequalities even for closed Riemannian manifolds. In my talk I am are going to discuss a joint project with Yevgeny Liokumovich, Boris Lishak and Regina Rotman towards the positive resolution of Guth's problem.

  • May 2, 2019
    Alberto Setti, University of Insubria
    Gradient estimates for Green's kernel via fake distances and applications to the inverse mean curvature flow
    Abstract: We prove new gradient and decay estimates for the kernel of the p-Laplacian on manifolds with Ricci curvature bounded from below which behave well as p tends to one and are obtained via the study of the fake distance associated to the kernel. Using the approximation by p-Laplacian kernels strategy pioneered by J. Moser we will apply our estimates to study existence and properties of weak solutions of the inverse mean curvature flow starting from a relatively compact set on a large class of manifolds satisfying Ricci lower bounds. This is joint work with L. Mari and M. Rigoli.

  • May 9, 2019
    Dennis Kriventsov, Rutgers University
    A spiral interface with positive Alt-Caffarelli-Friedman limit at the origin
    Abstract: I will discuss an example of a pair of continuous nonnegative subharmonic functions, each vanishing where the other is positive, which have a strictly positive limit for the Alt-Caffarelli-Friedman monotonicity formula at the origin, but for which the origin is not a point of differentiability for the boundary of their supports. Time permitting, I will also discuss some further progress on related problems.This is based on joint work with Mark Allen.

  • May 16, 2019
    Kazuo Yamazaki, University of Rochester
    Three-dimensional magnetohydrodynamics system forced by space-time white noise
    Abstract: The magnetohydrodynamics system consists of the Navier-Stokes equations forced by Lorentz force, coupled with the Maxwell's equations from electromagnetism. This talk will be relatively expository about the direction of research on stochastic PDE forced by space-time white noise, with a new result on the three-dimensional magnetohydrodynamics system forced by space-time white noise. In short, the fact that the noise is white in not only time but also space forces the solution to become extremely rough in spatial variable, its regularity akin to those of distributions, so that it becomes difficult for the non-linear term to become well-defined in any classical sense because there is no universal agreement on a product of a distribution with another distribution. Our discussion should also include following systems of equations: Kardar-Parisi-Zhang equation, Boussinesq system. The following notions and techniques may also be included in our discussions: Feynman diagrams, local subcriticality, paracontrolled distributions, renormalizations, regularity structures, rough path theory, Wick products, Young's integral.



  FALL 2019

  • September 12, 2019: No seminar

  • September 19, 2019
    Mohameden Ahmedou, Giessen University (Germany)
    Morse theory and the resonant $Q$-curvature problem
    Abstract: The Q-curvature is a scalar quantity which plays a central role in conformal geometry, in particular in the search of high order conformal invariants. In this talk we address the problem of finding conformal metrics of prescribed Q-curvature on four riemannian manifolds in the so called resonant case, that is when the total integral of the Q-curvature is a multiple of the one of the four-dimensional round sphere. This geometric problem has a variational structure with a lack of compactness. Using some topological tools of the theory of critical points at infinity of Abbas Bahri, combined with a refined blow-up analysis, we extend the full Morse theory, including Morse inequalities, to this non-compact geometric variational problem and derive existence and multiplicity results. This is a joint work with C.B. Ndiaye (Howard University)

  • October 3, 2019: TBA

  • October 24, 2019
    Niclas Linne, Giessen University (Germany)
    The prescribed mean curvature problem on Riemannian manifolds with boundary
    Abstract: I will introduce the prescribed mean curvature problem on Riemannian manifolds with boundary. This problem is closely related to the constant mean curvature problem and the Yamabe problem, which have become very famous during the last decades. I will explain the technical problems and present some results. Finally I will focus on four dimensional manifolds. I will present our results and give some ideas of the proof. We mainly used variational methods, Morse theory and the theory of "critical points at infinity", which goes back to Abbas Bahri.

  • October 31, 2019
    One-day Symposium
    Monica Clapp (Universidad Nacional Autonoma de Mexico), Luca Martinazzi (University of Padua, Italy), Tristan Riviere (ETH Zurich, Swizterland), Yi Wang (Johns Hopkins)
    Schedule

  • November 7, 2019
    Vincent Martinez, CUNY-Hunter College
    Learning Seminar on Incompressible Euler and Navier-Stokes: Issues in Regularity and Well-posedness, Part I
    Abstract: We will introduce the incompressible Euler and Navier-Stokes equations over the whole space or periodic domain. We will discuss weak solutions and in particular focus on distinguishing features between two- and three-dimensions, as it regards the issue of global regularity. The necessary background on functional spaces will be developed as needed.

  • November 14, 2019
    Jinggang Xiong, Beijing Normal University
    Optimal boundary regularity for fast diffusion equations in bounded domains
    Abstract: We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity. This is a joint work with Tianling Jin.

  • November 21, 2019
    Vincent Martinez, CUNY-Hunter College
    Learning Seminar on Incompressible Euler and Navier-Stokes: Issues in Regularity and Well-posedness, Part II
    Abstract: We develop the well-posedness for the 2D and discuss regularity criterion in 3D that ensures continuation of the solution beyond a given time. Time-permitting, we discuss the celebrated Beale-Kato-Majda criterion, and some cautionary examples in the scale Holder spaces.

  • December 5, 2019
    Aidin Murtha, CUNY-Hunter College
    Learning Seminar on Incompressible Euler and Navier-Stokes: Issues in Regularity and Well-posedness, Part III
    Abstract: We discuss a recent paper of J. Bourgain and D. Li, where ill-posedness of the 3D Euler Equations is established in the scale of the critical Sobolev spaces. Here, ill-posedness derives from the instantaneous loss of continuity of the solution operator. We distill the main ideas by considering the 2D case, which was developed in a set of lecture notes by T. Yoneda.

  • December 12, 2019
    Yizhong Zheng (Graduate Center, CUNY)
    Uniform Lipschitz continuity of isoperimetric profile on compact surfaces under normalized Ricci flow
    Abstract: We study the isoperimetric profile function h(\xi,g(t)) on a compact Riemannian manifold M under varying of metrics g(t), where \xi is the volume ratio. We show that h(\xi,g(t)) is jointly continuous when metrics g(t) vary continuously. We also show that h^2(\xi,g(t)) is uniform Lipschitz when M is a compact surface and g(t) is evolving under the normalized Ricci flow.