ACTIVITIES AT THE GRADUATE CENTER

Here is the list of seminars organized at the Graduate Center weekly seminars bulletin (updated every Tuesday).
Here, you will find the conferences in nonlinear analysis we are organizing at the Graduate Center.

Nonlinear Group Study

This meeting aims to investigate nonlinear problems arising in Differential geometry and mathematical physics. It gives graduates students the opportunity to be familiar with a wide range of open problems, and learn tools from the calculus of variations to tackle some of these questions. Our meeting takes place

Goals of these seminars is to discuss techniques that are used nonlinear problems arising in applied mathematics,physics or differential geometry. These events are sponsored by the Initiative for the Theoretical Sciences.

Those participating in the Nonlinear Analysis and PDE seminar may also be interested in the Geometric Analysis Seminar which meets Tuesdays in the same room 6496 starting at 3pm.

SCHEDULE 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
  TBA
  
  
• May 9, 2019
  Dennis Kriventsov, Rutgers University
  TBA
  
  
• 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.
  
  

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