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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November 10,
Topic: From RHP to PDE
Speaker: Guido Mazzuca - Tulane University
Abstract: In this lecture, we'll see how Riemann-Hilbert problems (RHP) are connected to PDEs.
I will show you, step-by-step, how to take a Riemann-Hilbert problem, build the associated Lax pair and show that the solution to the RHP can be related to a solution of a special PDE.
**Don't worry if you missed the last seminar.** This talk is self-contained.
Location: Gibson Hall 400D
Time: 3:00
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November 17,
Topic: From RHP to PDE: a new hope
Speaker: Guido Mazzuca - Tulane University
Abstract: In a crazy turn of events, the rebels, guided by the Jedi Joseph Liouville, were able to relate the solution to the emperor’s RHP to a system of ODEs. But the fight is far from over: a small group of rebels under the guidance of General Lax will move to the planet Pidies to find an ancient weapon that can finally defeat the emperor RHP!
If you are brave enough and you like PDEs and (sick) complex analysis tricks, please join, and may the force be with you!
Location: Gibson Hall 400D
Time: 3:00
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