I co-organize the ISU analysis seminar with Walton Green.
Spring Semester 2026 Schedule
January 12, 2026
Presenter: Kiril Datchev (Purdue)
Title: Low frequency scattering and decay of waves
Abstract: Low frequency waves are sensitive to the large-scale geometry of the environment through which they travel and of scatterers with which they interact. Their analysis has implications for wave evolution, and for the scattering matrix and phase. We study these using resolvent asymptotics, and present a robust method for deriving such asymptotics, based in part on an identity of Vodev and on boundary pairing. We focus on two-dimensional Euclidean scattering because of the rich phenomena observed in this setting, but other dimensions work just as well, and the method also applies to more general geometric situations.
This project is joint work with Tanya Christiansen, and parts are also joint work with Colton Griffin, Pedro Morales, and Mengxuan Yang.
March 2, 2026
Presenter: Antoine Prouff (Purdue)
Title:
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March 16, 2026
Presenter: Jared Wunsch (Purdue)
Title:
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March 30, 2026
Presenter: Rahul Sethi (Georgia Tech)
Title: A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform
Abstract: Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $\mu_\alpha$-relatively dense (where $d\mu_\alpha(x) \approx x^{2\alpha+1}\, dx$) for $\alpha > -1/2$, and $\operatorname{supp} \mathcal{F}_\alpha(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_\alpha(\mathbb{R}^+)} \lesssim \|f\|_{L^2_\alpha(E)},$$ for all $f\in L^2_\alpha(\mathbb{R}^+)$, where the constants in $\lesssim$ do not depend on $R > 0$. Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on $R$. In contrast, our techniques yield bounds that are independent of $R$, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation. This is joint work with Ben Jaye.
April 27, 2026
Presenter: Ben Foster (Washington University in St. Louis)
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