Seminars and Colloquia Schedule

Approximation of differential operators on unknown manifolds and applications

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, October 16, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and https://gatech.zoom.us/j/98355006347
Speaker
John HarlimPennsylvania State University

I will discuss the numerical approximation of differential operators on unknown manifolds where the manifolds are identified by a finite sample of point cloud data. While our formulation is general, we will focus on Laplacian operators whose spectral properties are relevant to manifold learning. I will report the spectral convergence results of these formulations with Radial Basis Functions approximation and their strengths/weaknesses in practice. Supporting numerical examples, involving the spectral estimation of various vector Laplacians will be demonstrated. Applications to solve elliptic PDEs will be discussed. To address the practical issue with the RBF approximation, I will discuss a weak approximation with a higher-order local mesh method that not only promotes sparsity but also allows for an estimation of differential operators with nontrivial Cristoffel symbols such as Bochner and Hodge Laplacians.

Non-linear mean-field systems in flocking and sampling

Series
Stochastics Seminar
Time
Thursday, October 17, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sayan BanerjeeUNC

Mean-field particle systems are well-understood by now. Typical results involve obtaining a McKean-Vlasov equation for the fluid limit that provides a good approximation for the particle system over compact time intervals. However, when the driving vector field lacks a gradient structure or in the absence of convexity or functional inequalities, the long-time behavior of such systems is far from clear. In this talk, I will discuss two such systems, one arising in the context of flocking and the other in the context of sampling (Stein Variational Gradient Descent), where there is no uniform-in-time control on the discrepancy between the limit and prelimit dynamics. We will explore methods involving Lyapunov functions and weak convergence which shed light on their long-time behavior in the absence of such uniform control.

 

Based on joint works with Amarjit Budhiraja, Dilshad Imon (UNC, Chapel Hill), Krishnakumar Balasubramanian (UC Davis) and Promit Ghosal (UChicago).

An interesting variational problem related to the Cwikel-Lieb-Rozenblum inequality and its solution

Series
Math Physics Seminar
Time
Friday, October 18, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Clough 280
Speaker
Tobias ReidGeorgia Tech

The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical estimate on the number of bound states for Schrödinger operators. In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our new proof highlights a natural but overlooked connection of the CLR inequality with bounds for maximal Fourier multipliers from harmonic analysis and leads to a variational problem that can be reformulated in terms of a variant of Hadamard’s three-lines lemma. The solution of this variational problem relies on some interesting complex analysis techniques. (Based on joint work with T. Carvalho-Corso, D. Hundertmark, P. Kunstmann, S. Vugalter)

Regularity for Semialgebraic Hypergraphs and Applications

Series
Combinatorics Seminar
Time
Friday, October 18, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hans Hung-Hsun YuPrinceton University

A semialgebraic hypergraph is a hypergraph whose edges can be described by a system of polynomial inequalities. Semialgebraic hypergraphs appear in many problems in discrete geometry. There has been growing interest in semialgebraic hypergraphs since the discovery that they satisfy strong regularity lemmas, where between most parts, the hypergraph is either complete or empty. In this talk, I will talk about an optimal regularity lemma along these lines and several applications. Based on joint work with Jonathan Tidor.

Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

Series
CDSNS Colloquium
Time
Friday, October 18, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Mark van den BoschLeiden University

Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns?

 

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations; based on applications. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. In the deterministic setting, multidimensional stability of planar waves on the whole space has been obtained, and we show to what extend we can do this in the stochastic case.

 

The metastability result above is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long-time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

Series
CDSNS Colloquium
Time
Friday, October 18, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Mark van den BoschLeiden University

Talk is in-person; zoom link if needed: <br />
<br />
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT0... />
<br />

 

Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns?

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations; based on applications. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. In the deterministic setting, multidimensional stability of planar waves on the whole space has been obtained, and we show to what extend we can do this in the stochastic case.

The metastability result above is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long-time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.