Fall 2025
Time & Location: All talks are on Wednesday in different places, at 3:00 PM .
Organizer: Katerina Gkogkou and Ken McLaughlin
September 8,
Topic: Power Spectrum Analysis for the Circular Unitary Ensemble
Speaker: Roman Riser - Tulane University
Abstract: The power spectrum has emerged as an effective tool for studying both system-specific and universal properties of quantum systems. In these 3 lectures we will study the power spectrum for the circular unitary ensemble (CUE). In the introduction, I will give an overview of results for the power spectrum. This will include a plot of the asymptotic limit for the CUE. We will compare it with numerical results for the zeros of the Riemann zeta function.
In the first lecture, I will derive a general representation for the power spectrum. Then we will review the definition and basic properties of the CUE and its joint probability distribution function of the eigenvalues. Next we deduce an exact representation of the power spectrum for the CUE with $N$ eigenvalues. In the second lecture, we will discuss the limit $N\rightarrow\infty$ and find a parameter free representation given in terms of the Painlev\'e V transcendent. In the last lecture, we will analyze the asymptotic formula and discuss its numerical evaluation.
Location: TBA
Time: 3:00
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October 20,
Topic: A Mini-Course on Riemann-Hilbert methods for integrable wave models.
Speaker: Deniz Bilman - University of Cincinnati
Abstract: This mini-course consists of a set of 5 90-minute long lectures that are aimed at graduate students from all areas.
Knowledge of complex analysis and differential equations should be enough to follow the lectures. There will be exercises.
Lecture 1: Integrable wave models: linearizing a flow
Introduces the scattering transform for the defocusing nonlinear Schrödinger equation and describes the analogy with Fourier transform methods for linear problems. Formulates a Riemann-Hilbert problem for the associated inverse-scattering transform.
Location: Gibson Hall, 400D
Time: 3:00
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October 21,
Topic: Riemann-Hilbert mini course, Lecture 2: Focusing nonlinear Schrödinger equation and solitons
Speaker: Deniz Bilman - University of Cincinnati
Abstract: Solitons from Lecture 0 come into the picture, and consequently Riemann-Hilbert problems with pole singularities. A nonlinear version of the superposition principle present for linear homogeneous problems is introduced.
Location: Gibson Hall, 400A
Time: 5:00
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October 22,
Topic: Mini course on Riemann-Hilbert Methods - Lecture 3: Lots of solitons and the dressing method
Speaker: Deniz Bilman - University of Cincinnati
Abstract: Describes the so-called “dressing” method: deriving differential equations satisfied by quantities extracted by the solution of a Riemann-Hilbert problem. Introduces a unified framework to capture arbitrary singularities in Riemann-Hilbert problems arising from direct scattering transform and its applications.
Location: Boggs 242
Time: 3:00
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