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)
Title: Is speckle noise more challenging to mitigate than additive noise?
Abstract: In classical nonparametric regression problems, additive noise has been thoroughly explored. Speckle noise, prevalent in applications such as synthetic aperture radar, ultrasound imaging, and digital holography, has not received as much attention. In a recent joint work with Reihaneh Malekian and Arian Maleki, we studied the problem of estimating a function in the presence of both speckle and additive noises, commonly referred to as the de-speckling problem. Our focus is on investigating the minimax estimation error for estimating a $\beta$-H\"older continuous function and determining the rate of the minimax risk. This talk will be made accessible to a general audience without background in statistics.
April 24 (10:00 am, note special time)
Speaker: Mahir Hadžić (University College London)
Title: Accelerated shock formation for the energy-critical Euler-Poisson system
Abstract: The gravitational Euler-Poisson system provides a basic model of a self-gravitating isolated star. The simplest nontrivial solutions are the radially symmetric steady states known as Lane-Emden stars - some of their key features are the spatial inhomogeneity of the associated fluid density and the presence of vacuum boundaries, either at finite radii or at infinity. The rigorous description of the phase space around them is a challenging open question. I will provide an overview of known results and then explain a new instability mechanism of accelerated shock formation, which applies to perturbations of Lane-Emden stars for the energy-critical value of the polytropic pressure law. This is a joint work with Juhi Jang, Sung-Jin Oh, and Ely Sandine.
May 1
Speaker: Warren Li (Stanford University)
Title: On ODE Blowup for the focusing nonlinear wave equation
Abstract: We consider the focusing wave equation for all powers in all dimensions. It is well-known that the equation admits spatially homogeneous blow-up solutions, often dubbed ODE blow-up, terminating in a singular hypersurface at {t=T}. In this talk, we show both that we can construct solutions that (locally) blow-up on an arbitrary spacelike hypersurface, unique up to the choice of a function we call auxiliary scattering data, and that such blow-up hypersurfaces and auxiliary scattering data is stable to perturbations away from the singularity. For instance, we show smooth perturbations of the ODE blow-up solution yields a smooth spacelike blow-up hypersurface. This is based on joint work with Isti Kadar (ETH).
May 8
Speaker: Felix Ye (SUNY Albany)
Title: Geometric regularization of autoencoders via observed stochastic dynamics
Abstract: Many high-dimensional stochastic systems exhibit slow or metastable behavior that is effectively confined to an unknown low-dimensional manifold. I will discuss a geometric approach to learning reduced stochastic dynamics from short ambient data bursts. The method uses covariance information from the observed process to regularize an autoencoder so that its latent coordinates respect the tangent geometry of the underlying manifold. This leads to a learned latent SDE with controlled drift and diffusion errors, including an Itô-based encoder-pullback formula for the drift. I will describe the main consistency results, including propagation of chart errors to weak convergence of the learned ambient dynamics and to mean first-passage time statistics. The emphasis will be on the analytic and geometric ideas behind the construction.
May 15
Speaker: Amir Sagiv (NJIT)
Title: Sampling by Transport and the Approximation of Measures
Abstract: Transportation of measure underlies many contemporary methods in machine learning and statistics. Sampling, which is a fundamental building block in computational science, can be done efficiently given an appropriate measure-transport map. We ask: what is the effect of using approximate maps in such algorithms? We propose a new framework to analyze the approximation power of measure transport. This framework applies to existing algorithms, but also suggests new ones. At the core of our analysis is the theory of optimal transport regularity, approximationtheory, and an emerging class of inequalities, previously studied in the context of uncertainty quantification (UQ).
May 22
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