Organizers: Ioakeim Ampatzoglou, Dan Ginsberg, Weilin Li, Vincent Martinez, and Azita Mayeli
Time: Friday, 2:00-4:00pm EST
Location (in-person and hybrid): GC 9116
Zoom: link Passcode: HAPDE2024
Current Schedule (Spring 2026)
January 30
Speaker: Kevin Dembski (Duke)
Title: Singularity Formation in the Incompressible Porous Medium Equation without Boundary Mass
Abstract: In this talk, I will discuss recent work on the problem of singularity formation in the incompressible porous medium (IPM) equation. We construct Lipschitz continuous solutions of the IPM equation which vanish on the boundary of the domain and blow-up in finite time. At the blow-up point, the flow is hyperbolic with points approaching the boundary from the interior and escaping tangent to the boundary.
February 6
Speaker: Genevieve Romanelli (CUNY)
Title: Form Uniqueness for Weakly Spherically Symmetric Graphs
Abstract: Dirichlet forms are generalizations of the Laplacian which are especially useful for studying infinite graphs. Importantly, unlike with traditional graph Laplacians on finite graphs, there may not be a unique operator and Dirichlet form associated to a graph. I will provide two characterizations for uniqueness of the Dirichlet form on graphs satisfying a certain spherical symmetry constraint, one via graph structure and the other via boundary capacity. Time permitting, I will give some stability results. This work was joint with Luis Hernandez, Sean Ku, Jun Masamune, and Radoslaw Wojciechowski.
February 13
Speaker: Han Li (CUNY)
Title: Whitney extension problem for the fractional Sobolev spaces
Abstract: Given a function space X defined on R^n, a subset E of R^n and a natural number m, if every function in X is m-th differentiable, for each function F in X, one obtains a family of m-th order Taylor polynomials, one at each point of E. The jet space of X on E of order m is defined as the collection of all such families. The Whitney extension problem asks: How can one construct an operator to recover functions in X from the information in its jet space?
In this talk, we present our result for the homogeneous fractional Sobolev space, which extends the previous results. Specifically, we present the existence of a bounded linear extension operator from the jet space of $L^{s,p}(R^n)$ on E of order equal to the integer part of s to $L^{s,p}(R^n)$ for any subset E of R^n, with the condition that n/p is less than the fractional part of s. Our approach builds upon the classical method of Whitney extension and uses the exponentially decreasing path.
February 20
Speaker: Maxime Van De Moortel (Rutgers University)
Title: Late-time asymptotics for the Klein-Gordon equation on a Schwarzschild black hole
Abstract: It has long been conjectured that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping and the involvement of long-range scattering. We will present our recent resolution of this problem, establishing that, contrary to previous expectations, solutions with sufficiently localized initial data decay polynomially in time. Time permitting, we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums. The talk is based on joint work(s) with Federico Pasqualotto and Yakov Shlapentokh-Rothman.
February 27
(No seminar)
March 6
Speaker: Elias Hess-Childs (Carnegie Mellon University)
Title: Turbulent phenomena in a universal total anomalous dissipator
Abstract: Anomalous dissipation describes the tendency for a turbulent fluid to dissipate energy at a constant rate independent of the molecular viscosity, despite viscosity being the ultimate mechanism of dissipation. It is a cornerstone of phenomenological turbulence theory and is taken as a basic axiom in Kolmogorov’s highly successful K41 theory. Despite this, a rigorous mathematical demonstration of these effects in fluid models remains elusive.
To study anomalous dissipation in a more tractable setting, recent work has focused on constructing incompressible vector fields that induce persistent energy loss in scalar advection-diffusion equations in the vanishing noise limit.
In this talk, I will provide an overview of anomalous dissipation and discuss my recent work with Keefer Rowan, where we construct a universal total anomalous dissipator—a vector field that completely dissipates any initial data in unit time in the vanishing noise limit. Building on this construction, we then construct a vector field exhibiting several further hallmarks of turbulence, including Richardson dispersion, anomalous regularization, and intermittency.
March 13
Speaker: Luke Peilen (Temple University)
Title: Local Laws and Fluctuations for Super-Coulombic Riesz Gases
Abstract: Coulomb and Riesz gases are interacting particle systems with a wide range of applications in random matrix theory, approximation theory, convex geometry, and diverse areas of physics. We study the statistical mechanics of general Riesz gases at mesoscopic and microscopic length scales, proving controls on fluctuations of linear statistics down to microscopic length scales and establishing for the first time a CLT for fluctuations of linear statistics for general two-dimensional Riesz gases.
A novel technical difficulty involves the development of a transport method for general Riesz gases, building on work of Leblé and Serfaty for Coulomb gases, to understand the behavior of the partition function under small perturbations of the external potential. Our study involves several questions concerning degenerate, singular elliptic PDE and fractional operators.This is based on joint work with S. Serfaty.
March 20
(No seminar) CUNY closed
March 27
April 3
(No seminar) CUNY closed for spring break
April 10
(No seminar)
April 17
Speaker: Hao Xing (The Graduate Center, CUNY)
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April 24
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May 1
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May 8
Speaker: Felix Ye (SUNY Albany)
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May 15
Speaker: Amir Sagiv (NJIT)
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May 22
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