Seminars and Colloquia by Series

Existence of stationary solutions for some integro-differential equations with the double scale anomalous diffusion

Series
Analysis Seminar
Time
Wednesday, February 25, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Vitali Vougalter – University of Toronto –

The work is devoted to the investigation of the solvability of an integro-differential equation in the case of the double scale anomalous diffusion with a sum of two negative Laplacians in different fractional powers in $R^{3}$. The proof of the existence of solutions relies on a fixed point technique. Solvability conditions for the elliptic operators without the Fredholm property in unbounded domains are used.

The HRT Conjecture for a Symmetric (3,2) Configuration

Series
Analysis Seminar
Time
Wednesday, January 28, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shuang Guan – Tufts University –

The Heil-Ramanathan-Topiwala (HRT) conjecture is an open problem in time-frequency analysis. It asserts that any finite combination of time-frequency shifts of a non-zero function in $L^2(\mathbb{R})$ is linearly independent. Despite its simplicity, the conjecture remains unproven in full generality, with only specific cases resolved.
In this talk, I will discuss the HRT conjecture for a specific symmetric configuration of five points in the time-frequency plane, known as the $(3,2)$ configuration. Building upon restriction principles, we prove that for this specific setting, the Gabor system is linearly independent whenever the parameters satisfy certain rationality conditions (specifically, when one parameter is irrational and the other is rational). This result partially resolves the remaining open cases for such configurations. I will outline the proof methods, which involve an interplay of harmonic analysis and ergodic theory. This is joint work with Kasso A. Okoudjou.

Maximization of recurrent sequences, Schur positivity, and derivative bounds in Lagrange interpolation

Series
Analysis Seminar
Time
Wednesday, January 14, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitrii Ostrovskii – Georgia Institute of Technology –

Consider the following extremal problem: maximize the amplitude |X_T|, at time T, of a linear recurrent sequence X_1, X_2,... of order N < T, under natural constraints: (I) the initials are uniformly bounded; (II) the characteristic polynomial is R-stable, i.e., its roots are in the origin-centered disc of radius R. While the maximum at time T = N essentially follows from the classical Gautschi bound (1960), the general case T > N turns out to be way more challenging to handle. We find that for any triple (N,R,T), the amplitude is maximized when the roots coincide and have modulus R, and the initials are chosen to align the phases of fundamental solutions. This result is striking for two reasons. First, the same configuration of roots and initials is uniformly optimal for all T, i.e. the whole envelope is maximized at once. Second, we are not aware of any purely analytical proof: ours uses tools from algebraic combinatorics, namely Schur polynomials indexed by hook partitions. 

In the talk, I will sketch the proof of this result, making it as self-sufficient as possible under the circumstances. If time permits, we will discuss a related conjecture on the optimal error bounds in complex Lagrange interpolation.

The talk is based on the work https://arxiv.org/abs/2508.13554.

Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions

Series
Analysis Seminar
Time
Wednesday, November 19, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Donggeun Ryou – Indiana University Bloomington –

The Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem extends the classical restriction theorem for measures on smooth manifolds to fractal measures. We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions. The proof uses number fields to construct fractal measures in R^d. This work is joint with Robert Fraser and Kyle Hambrook.

Variable coefficient local smoothing and a projection problem in the Heisenberg group

Series
Analysis Seminar
Time
Wednesday, November 5, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Terence Harris – University of Wisconsin-Madison –

The Heisenberg projection problem asks whether there is an analogue of the Marstrand projection theorem in the first Heisenberg group, namely whether Hausdorff dimension of sets generically decreases under projection, for a natural family of projections arising from the group structure. This problem is still open, but I will discuss a recent improvement to the known bound obtained through a variable coefficient local smoothing inequality. 

 

Rather than going through the proof in detail, I will spend most of the talk introducing the problem and explaining the connection to averaging operators over curves, and explaining why these operators are Fourier integral operators satisfying Sogge's cinematic curvature condition. This condition was originally introduced by Sogge to generalise Bourgain's circular maximal theorem, but it turns out to have useful applications to projection theory. 

A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform

Series
Analysis Seminar
Time
Wednesday, October 29, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rahul Sethi – Georgia Institute of Technology –

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.   

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