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
Nonlinear Analysis and PDEs
CUNY Graduate Center, 365 Fifth Avenue, NYC
Room 6496, 4:15pm--5:15pm
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
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
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
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
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
In the resonant one we study the "critical points at Infinity" and
derive some Euler-Poincaré type criterion of the existence of
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
April 11, 2019
Alexander Nabutovsky, University of Toronto
Filling metric spaces
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
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
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
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
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.