CUNY GC Harmonic Analysis and PDE Seminar

Organizers: Han Li, Weilin Li, Vincent Martinez, Azita Mayeli

Friday, 1:00-3:00pm EST, (GC 6421, hybrid format; please email vrmartinez-at-hunter-dot-cuny-dot-edu or amayeli-at-gc-dot-cuny-dot-edu for passcode)

Spring 2023 Schedule

January 27

Hiatus

February 3 Zoom (Recording)

Kevin D. Stubbs (Institute of Pure and Applied Mathematics, UCLA)

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, H^s, 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).

February 24

Hiatus

March 3 Zoom

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

(Rescheduled to March 24)

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 F_σ 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 L^p 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 7

Hiatus

April 14 Zoom (Recording)

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 (Recording)

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

Coupling physics-deep learning inversion

In recent years, there is an increasing interest in applying deep learning to geophysical/medical data inversion. However, direct application of end-to-end data-driven approaches to inversion have quickly shown limitations in the practical implementation. Indeed, due to the lack of prior knowledge on the objects of interest, the trained deep learning neural networks very often have limited generalization. In this talk, we introduce a new methodology of coupling model-based inverse algorithms with deep learning for two typical types of inversion problems. In the first part, we present an offline-online computational strategy of coupling classical least-squares based computational inversion with modern deep learning based approaches for full waveform inversion to achieve advantages that can not be achieved with only one of the components. In the second part, we present an integrated data-driven and model-based iterative reconstruction framework for joint inversion problems. The proposed method couples the supplementary data with the partial differential equation model to make the data-driven modeling process consistent with the model-based reconstruction procedure. We also characterize the impact of learning uncertainty on the joint inversion results for one typical inverse problem.

April 28 Zoom (Recording)

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

On measure and topological Erdos similarity problems

In this talk, we explore an analogous problem in the topological setting. Instead of sets with positive measure, we investigate the collection of dense sets and in the collection of generic sets (dense G_δ and complement has Lebesgue measure zero). We refer to such pattern as topologically universal and generically universal respectively. We will show that Cantor sets on R^d are never topologically universal and Cantor sets with positive Newhouse thickness on R¹ are not generically universal. This gives a positive partial answer to a question by Svetic concerning the Erdos similarity problem on Cantor sets. Moreover, we also obtain a higher dimensional generalization of the generic universality problem. This is a joint work with John Gallagher, who was a Master student in SFSU, and Eric Weber from Iowa State University.

May 5

Hiatus

May 12 Zoom

Laura Cladek (Institute for Advanced Study, School of Mathematics)

Additive Energy for the Sphere and Decoupling

We describe some new results about improved bounds on additive energy estimates for finite subsets of the sphere, a problem which is closely connected to the $ell_2$ decoupling theorem of Bourgain and Demeter. Our techniques are motivated by versions of the crossing lemma for simple graphs, an important graph-theoretic tool used in incidence geometry to prove Szemer'{e}di-Trotter type theorems. We highlight some interesting key geometric differences between the discrete and discretized (fractal) version of the problem, the latter which considers discretized additive energy estimates for fractional dimensional Ahlfors-David regular subsets of the sphere. This is joint work with Terence Tao from UCLA.

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