Asymptotic Analysis and Probability in Riemann-Hilbert Problems: From Soliton Gases and Extreme Events to Random Matrices in a Random Environment
Ente: OFFICE OF MULTIDISCIPLINARY AC, ANALYSIS PROGRAM
Scadenza: 2029-07-31
Importo max: 219.999 EUR
Paese: US
Descrizione
This project uses and creates mathematics to understand how large and unexpected events arise in systems that follow precise mathematical laws. One part of the project studies unusually large waves, often referred to as rogue waves, that appear suddenly and are difficult to predict. These waves are modeled mathematically by nonlinear partial differential equations, and the investigator discovers properties of such waves through the analysis of these equations. Another part studies how large numbers of interacting waves can, taken together, produce orderly behavior on a much larger scale. The project also studies patterns that arise in large systems involving randomness, where individual outcomes may be unpredictable but overall behavior can often be described with mathematical precision. The investigator and collaborators develop mathematical theories that characterize this emergent behavior and can be used to predict it. Predicting such patterns has many applications ranging across the sciences. An overarching goal is to explain how reliable large-scale patterns emerge from many smaller interactions, and to explain the remarkably universal appearance of these patterns in very different settings. The project also contributes to education and workforce development. The investigator engaged graduate students and postdoctoral researchers as well as early-career mathematicians through collaborative research, workshops, and professional mentoring. In addition, the investigator participates in outreach activities that bring middle-school students to campus for mathematics workshops, with graduate students and postdoctoral researchers helping to design and lead activities. By strengthening the mathematical foundations needed to understand systems of remarkable complexity, while also supporting mentoring, training, and public engagement, the project advances the national interest through progress in fundamental mathematics and through better understanding of waves, randomness, and large interacting systems.
The investigator studies asymptotic and probabilistic problems arising in Riemann-Hilbert analysis, integrable nonlinear partial differential equations, random matrix theory, and discrete orthogonal polynomials. The project focuses on several connected directions. First, it develops Riemann-Hilbert methods for the analysis of extreme wave phenomena and for optimization problems involving large multi-soliton solutions. Second, it studies soliton gases for the Korteweg–de Vries equation, with the goal of deriving and justifying continuum kinetic descriptions from limits of many-soliton solutions. Third, it investigates nonclassical scaling regimes in random matrix theory, including questions related to the largest eigenvalue in basic statistical models. Fourth, it develops asymptotic methods for discrete orthogonal polynomials on quadratic q-lattices, with applications to random tiling models and the q-Racah random lozenge waterfall. The methods com
Istituzione: Tulane University
Sede: NEW ORLEANS, LA
PI: Kenneth T-R McLaughlin
Settori: Mathematical & Physical Sciences
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