Past Seminars

October 27  

Tristan Leger (Princeton University)

Title: L^p bounds for spectral projectors on hyperbolic surfaces

Abstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces. I will focus on the case where the spectral window has small width. Indeed the negative curvature assumption leads to improvements over the universal bounds of C.Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. The proof relies on new Strichartz and smoothing estimates for the Schrödinger semi-group, thus illustrating how dispersive PDE techniques can lead to new results in classical harmonic analysis. 

November 3

Joonhyun La (Princeton University/KIAS)

Title: Local well-posedness and smoothing of MMT kinetic wave equation

Abstract: In this talk, I will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).

November 10

Rajendra Beekie (Duke University)

Title: Uniform vorticity depletion and inviscid damping for periodic shear flows in the high Reynolds number regime

Abstract: We study the dynamics of the two-dimensional Navier Stokes equations linearized around a shear flow on a (non-square) torus which possesses exactly two non-degenerate critical points. The main task is to understand the associated Rayleigh and Orr-Sommerfeld equations, under the natural assumption that the linearized operator around the shear flow in the inviscid case has no discrete eigenvalues. We obtain linear inviscid damping and vorticity depletion estimates for the linearized flow that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to understand the behavior of the solution to Orr-Sommerfeld equations in three distinct regimes depending on the spectral parameter: the non-degenerate case when the spectral parameter is away from the critical values, the intermediate case when the spectral parameter is close but still separated from the critical values, and the most singular case when the spectral parameter is inside the viscous layer. This is based on joint work with Shan Chen (UMN) and Hao Jia (UMN)

November 17 (Zoom Only)

     Jacob Carruth (Princeton University)

Title: The Whitney Extension Problem for the Sobolev Space L^{2,p}(R^2) when p<2

Abstract: Let L^{2,p}(R^2) be the Sobolev space of real-valued functions on R^2 whose derivatives up to order 2 belong to L^p. For a subset E of R^2, let L^{2,p}(E) denote the space of restrictions to E of functions in L^{2,p}(R^2). Does there exist a bounded linear operator from L^{2,p}(E) to L^{2,p}(R^2) (an "extension operator")? When p > 2, a bounded linear extension operator exists (thanks to Fefferman, Israel, Luli). When p < 2, it is unknown whether a bounded linear extension operator exists in general. We will show how to construct such an operator for certain sets E. In a particularly interesting case, the problem reduces to an extension problem on a tree.

December 1 

    THIS TALK WILL TAKE PLACE AT 4PM IN ROOM 4213.03

    Joseph Miller (UT Austin)

    Title: On the effective dynamics of Bose-Fermi Mixtures

    Abstract:  In this talk, I will be discussing recent work with Esteban Cardenas and my advisor Natasa Pavlovic: https://arxiv.org/abs/2309.04638. In this work, we describe the dynamics of a Bose-Einstein condensate interacting with a degenerate Fermi gas, at zero temperature. First, we analyze the mean-field approximation of the many-body Schrödinger dynamics and prove emergence of a coupled Hartree-type system of equations. We obtain rigorous error control that yields a non-trivial scaling window in which the approximation is meaningful. Second, starting from this Hartree system, we identify a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical; this is one of the main contributions of the present article. In this regime, the bosons are much lighter and more numerous than the fermions. We then prove convergence to a coupled Vlasov-Hartee system of equations with an explicit convergence rate.

December 8

Title:

Abstract:

December 15 : There are two presentations scheduled for this day. 

     Quyuan Lin (Clemson University), 2-3 pm

     Title: Anisotropic Viscosities Estimation for the Stochastic Primitive Equations

Abstract: The primitive equations (PE) serve as a fundamental model for exploring large-scale oceanic and atmospheric dynamics. Introducing the element of randomness, we investigate the 3D PE perturbed by an additive noise. Our objective is to derive effective estimators for the viscosity within the model, grounded in observational data, and subsequently examine their asymptotic consistency and normality. In particular, our approach belongs to the category of the so-called spectral methods, where observations are conducted in Fourier space and continuously over a finite time interval. The novelty of our work lies in the simultaneous estimation of both horizontal and vertical viscosity, which are typically treated as distinct parameters.

Papageorgiou Effie (Tufts University and Paderborn University)  3-4 pm

Title: Asymptotic behavior of solutions to the heat equation on certain Riemannian manifolds

Abstract:  Here

February 3 Zoom (Recording)

Kevin D. Stubbs (UC Berkeley)

A Mathematical Invitation to Wannier Functions

Wannier functions, first proposed in the 1930s, have had a long history in computational chemistry as a practical means to speed up calculations. Stated in a mathematical language, Wannier functions are an orthonormal basis for certain types of spectral subspaces which are generated by the action of a translation group. In the 1980s however, it was realized that there is an intimate connection between Wannier functions and topology. In particular, Wannier functions with fast spatial decay exist if and only if a certain vector bundle is topologically trivial. Materials with non-trivial topology host a number of remarkable properties which are robust to physical imperfections. In this talk, I will give a brief introduction to topological materials and Wannier functions in periodic systems. I will then discuss my work on extending these results to systems without any underlying periodicity.

February 10 (In-Person)

Philip Greengard (Columbia University, Department of Statistics)

Efficient Fourier representations for Gaussian process regression

Over the last couple of decades a large number of numerical methods have been introduced for efficiently performing Gaussian process regression. Most of these methods focus on fast inversion of the covariance matrix that appears in the Gaussian density. In this talk I describe a slightly different approach to Gaussian process regression that relies on efficient weight-space representations of Gaussian processes. These representations -- complex exponential expansions with Gaussian coefficients -- have several advantages in Gaussian process regression tasks including theoretical guarantees, computational efficiency, and model-interpretability benefits.

February 17 Zoom (Recording)

Raghav Venkatraman (Courant Institute of Mathematical Sciences, Department of Mathematics)

Homogenization questions inspired by machine learning and the semi-supervised learning problem

This talk comprises two parts. In the first part, we revisit the problem of pointwise semi-supervised learning (SSL). Working on random geometric graphs (a.k.a point clouds) with few labeled points, our task is to propagate these labels to the rest of the point cloud. Algorithms that are based on the graph Laplacian often perform poorly in such pointwise learning tasks since minimizers develop localized spikes near labeled data. We introduce a class of graph-based higher order fractional Sobolev spaces, Hs, and establish their consistency in the large data limit, along with applications to the SSL problem. A crucial tool is recent convergence results for the spectrum of the graph Laplacian to that of the continuum. Obtaining optimal convergence rates for such spectra is an open question in stochastic homogenization. In the rest of the talk, we'll discuss how to get state-of-the-art and optimal rates of convergence for the spectrum, using tools from stochastic homogenization. The first half is joint work with Dejan Slepcev (CMU), and the second half is joint work with Scott Armstrong (Courant).

March 10 Zoom

Fushuai Jiang (Univeristy of Maryland, Department of Mathematics)

Quasi-optimal C2(Rn) Interpolation with Range Restriction

Experimental data often have range or shape constraints imposed by nature. For example, probability density or chemical concentration are non-negative quantities, and the trajectory design through an obstacle course may need to avoid two boundaries. In this talk, we investigate the theory of multivariate smooth interpolation with range restriction from the perspective of Whitney Extension Problems. Given a function defined on a finite set with no underlying geometric assumption, I will describe an O(N(log N)-n) procedure to compute a twice continuously differentiable interpolant that preserves a prescribed shape (e.g. nonnegativity) and whose second derivatives are as small as possible up to a constant factor (i.e., quasi-optimal). I will also provide explicit numerical results in one dimension. This is based on the joint works with Charles Fefferman (Princeton), Chen Liang (UC Davis), Yutong Liang (former UC Davis), and Kevin Luli (UC Davis).

March 17 Zoom (Recording)

Arthur Danielyan (University of South Florida, Department of Mathematics)

On a converse of Fatou's theorem

Fatou's theorem states that a bounded analytic function in the unit disc has radial limits a.e. on the unit circle T. This talk presents the following new theorem in the converse direction.
Theorem 1. Let E be a subset on T. There exists a bounded analytic function in the open unit disc which has no radial limits on E but has unrestricted limits at each point of T \ E if and only if E is an Fσ set of measure zero.
The sufficiency part of this theorem immediately implies a well-known theorem of Lohwater and Piranian the proof of which is complicated enough. However, the proof of Theorem 1 only uses the Fatou's interpolation theorem, for which too the author has recently suggested a new simple proof.
It turns out that for the Blaschke products, a well-known subclass of bounded analytic functions, Theorem 1 takes the following form. 
Theorem 2. Let E be a subset on the unit circle T. There exists a Blaschke product which has no radial limits on E but has unrestricted limits at each point of T \ E if and only if E is a closed set of measure zero.
The proof of the necessity part of Theorem 2 is completely elementary, but it still contains some methodological novelty. The proof of the sufficiency uses Theorem 1 as well as some known results on Blaschke products. (Theorem 2 is a joint result with Spyros Pasias.)

References.
1. A. A. Danielyan, On Fatou's theorem, Anal. Math. Phys. V. 10, Paper no. 28, 2020.
2. A. A. Danielyan, A proof of Fatou's interpolation theorem, J. Fourier Anal. Appl., V. 28, Paper no. 45, 2022.
3. A. A. Danielyan and S. Pasias, On a boundary property of Blaschke products, to appear in Anal. Mathematica

March 24 Zoom

Geet Varma (Royal Melbourne Institute of Technology, School of Science)

Weaving Frames Linked with Fractal Convolutions

Weaving frames have been introduced to deal with some problems in signal processing and wireless sensor networks. More recently, the notion of fractal operator and fractal convolutions have been linked with perturbation theory of Schauder bases and frames. However, the existing literature has established limited connections between the theory of fractals and frame expansions. In this paper we define Weaving frames generated via fractal operators combined with fractal convolutions. The aim is to demonstrate how partial fractal convolutions are associated to Riesz bases, frames and the concept of Weaving frames. This current view point deals with ones sided convolutions i.e both left and right partial fractal convolution operators on Lebesgue space Lp for 1 ≤ p < ∞ . Some applications via partial fractal convolutions with null function have been obtained for the perturbation theory of bases and weaving frames. 

March 31 Zoom (Recording)

Michael Perlmutter (UCLA, Department of Mathematics)

Geometric Scattering on Measure Spaces

Geometric Deep Learning is an emerging field of research that aims to extend the success of convolutional neural networks (CNNs) to data with non-Euclidean geometric structure. Despite being in its relative infancy, this field has already found great success in many applications such as recommender systems, computer graphics, and traffic navigation. In order to improve our understanding of the networks used in this new field, several works have proposed novel versions of the scattering transform, a wavelet-based model of CNNs for graphs, manifolds, and more general measure spaces. In a similar spirit to the original Euclidean scattering transform, these geometric scattering transforms provide a mathematically rigorous framework for understanding the stability and invariance of the networks used in geometric deep learning. Additionally, they also have many interesting applications such as drug discovery, solving combinatorial optimization problems, and predicting patient outcomes from single-cell data. In particular, motivated by these applications to single-cell data, I will also discuss recent work proposing a diffusion maps style algorithm with quantitative convergence guarantees for implementing the manifold scattering transform from finitely many samples of an unknown manifold.

April 14 Zoom

Nabil Fadai (University of Nottingham, Department of Mathematics)

Semi-infinite travelling waves arising in moving-boundary reaction-diffusion equations

Travelling waves arise in a wide variety of biological applications, from the healing of wounds to the migration of populations. Such biological phenomena are often modelled mathematically via reaction-diffusion equations; however, the resulting travelling wave fronts often lack the key feature of a sharp edge. In this talk, we will examine how the incorporation of a moving boundary condition in reaction-diffusion models gives rise to a variety of sharp-fronted travelling waves for a range of wave speeds. In particular, we will consider common reaction-diffusion models arising in biology and explore the key qualitative features of the resulting travelling wave fronts.

April 21 Zoom

Lu Zhang (Columbia University, Department of Applied Physics and Applied Mathematics)

Title

Abstract

April 28 Zoom

Chun-Kit Lai (San Francisco State University, Department of Mathematics)

Title

Abstract