Seminars and Colloquia by Series

Matrix generalization of the cubic Szegő equation

Series
Analysis Seminar
Time
Wednesday, March 27, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruoci Sun – Georgia Tech

This presentation is devoted to studying matrix solutions of the cubic Szegő equation, leading to the matrix Szegő equation on the 1-d torus and on the real line. The matrix Szegő equation enjoys a Lax pair structure, which is slightly different from the Lax pair structure of the cubic scalar Szegő equation introduced in Gérard-Grellier [arXiv:0906.4540]. We can establish an explicit formula for general solutions both on the torus and on the real line of the matrix Szegő equation. This presentation is based on the works Sun [arXiv:2309.12136, arXiv:2310.13693].

Stein's mathod and stability for sharp constants in functional inequalities

Series
Analysis Seminar
Time
Wednesday, February 21, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Max Fathi – Etablissement public experimental, Paris, France –

In this talk, I will present some joint works with Tom Courtade on characterizing probability measures that optimize the constant in a given functional inequalitiy via integration by parts formulas, and how Stein's method can be used to prove quantitative bounds on how close almost-optimal measures are to true optimizers. I will mostly discuss Poincaré inequalities and Gaussian optimizers, but also some other examples if time allows it.

Positive curvature implies existence of isoperimetric sets?

Series
Analysis Seminar
Time
Wednesday, January 24, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Federico Glaudo – Princeton University –

Over the past decade, a rich theory of existence for the isoperimetric problem in spaces of nonnegative curvature has been established by multiple authors.
We will briefly review this theory, with a special focus on the reasons why one may expect the isoperimetric problem to have a solution in any nonnegatively curved space: it is true for large enough volumes, it is true if the ambient is 2-dimensional, and it is true under appropriate assumptions on the ambient space at infinity.

The main topic of the talk will be the presentation of a counterexample to this "intuition": a 3-dimensional manifold of positive sectional curvature without isoperimetric sets for small volumes.
This is a joint work with G. Antonelli.

Spectral monotonicity under Gaussian convolution

Series
Analysis Seminar
Time
Wednesday, December 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eli Putterman – Tel Aviv University

The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.

Spectral monotonicity under Gaussian convolution

Series
Analysis Seminar
Time
Wednesday, December 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eli Putterman – Tel Aviv University –

The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.  

On the curved trilinear Hilbert transform

Series
Analysis Seminar
Time
Wednesday, November 15, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bingyang Hu – Auburn University –

The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator

$$

H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R

$$

is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1

 

The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:

 

1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;

 

2). a structural analysis of suitable maximal "joint Fourier coefficients";

 

3). a level set analysis with respect to the time-frequency correlation set. 

 

This is a joint work with my postdoc advisor Victor Lie from Purdue.

 

Higher dimensional fractal uncertainty

Series
Analysis Seminar
Time
Wednesday, November 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex Cohen – MIT –

The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n. 

On the spectral synthesis for the unit circle in ${\mathcal F} L_s^q ({\mathbf R}^2)$

Series
Analysis Seminar
Time
Wednesday, October 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
TBA
Speaker
Masaharu Kobayashi – Hokkaido University –

Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $ \| f \|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2}\, ( |{\mathcal F}[f](\xi)| \,( 1+ |\xi| )^s )^q\, d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$. We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ with $1\frac{2}{q^\prime}$, where $q^\prime$ denotes the conjugate exponent of $q$. This is joint work with Prof. Sato (Yamagata University).

Convergence of Frame Series: from Hilbert Space to Modulation Space

Series
Analysis Seminar
Time
Wednesday, October 4, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pu-Ting Yu – Georgia Tech

It is known that if $\{x_n\}$ is a frame for a separable Hilbert space, then there exist some sequences $\{y_n\}$ such that $x= \sum x_n$, and this sum converges in the norm of H. This equation is called the reconstruction formula of x. In this talk, we will talk about the existence of frames that admit absolutely convergent and unconditionally convergent reconstruction formula. Some characterizations of such frames will also be presented. Finally, we will present an extension of this problem about the unconditional convergence of Gabor expansion in Modulation spaces.

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