ENYGMMa VI

Schedule 

12:30 pm - 1:30 pm Lunch in 13th Floor Lounge

1:30 pm  - 2:30 pm Talk by Sylvia Serfaty

2:30 pm  - 3:00 pm Tea break in Room 1302

3:00 pm  - 4:00 pm Talk by Azita Mayeli 

Titles and Abstracts

Sylvia Serfaty: Systems of points with Coulomb interactions 

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability.

We will first review these motivations, then present the ''mean-field'' derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

Azita Mayeli: On Time and Frequency concentration problem

The Heisenberg uncertainty principle states that a function and its Fourier transform cannot be compactly supported in both the time and frequency domains simultaneously. This leads to an intriguing question: Which functions with compact time support can be concentrated in the frequency domain, and conversely, which functions with limited Fourier support can be concentrated in the time domain? This question is referred to as the it concentration problem. 

In dimension d=1, this question was investigated in a series of Bell Labs papers by Landau, Slepian, Daubechies, and Polak in the 1960s to 1980s. They explored this through eigenfunctions as well as the asymptotic behavior of eigenvalues of special compact and self-adjoint localization operators. The eigenfunctions are known as Prolate spheroidal wave functions, and their eigenvalues exhibit surprising behavior, making them a useful  tool for addressing practical problems like interpolation, differentiation, and quadrature rules. 

However, tackling this problem in higher dimensions is indeed a challenge, and the outcomes heavily depend on the geometry of the time and frequency domains. 

In this talk, I will first review the solutions to the concentration problem in dimension d=1. Following that, I will introduce the localization operators in higher dimensions and discuss the non-asymptotic behavior of the eigenvalue distribution of these operators. This will include scenarios where the time domain is a hyper-cube, and the frequency domain can be either a hyper-cube or a unit ball.

To wrap up the talk, I will also introduce another representation of the localization operators, namely, the phase-space localizing operators, and  discuss some open problems in this field.