Linear Systems and Differential Equations in Random Matrix Theory

Explore a 50-minute semi-plenary talk by Gordon Blower from Lancaster University, UK, on linear systems and differential equations in random matrix theory. Delve into the solution of nonlinear differential equations using linear systems, focusing on Hankel integral operators and Fredholm determinants. Examine the connections between Schrödinger differential operators and Hankel integral operators, and discover solutions to the sinh-Gordon PDE and Painlevé III' transcendental ordinary differential equation. Learn about applications in random matrix theory and MIMO wireless communications. Investigate topics such as controllability and observability operators, Howland operators, theta and tau functions, scattering functions, matrix potentials, and the Gelfand-Levitan equation. Gain insights into Hankel determinants, Painlevé equations, random matrix models, and the free logarithmic Sobolev inequality. This collaborative work with Yang Chen and Ian Doust offers a comprehensive exploration of advanced mathematical concepts and their practical applications.

Intro
Plan
Controllability and observability operators
Evolution of the linear system
Howland operators via linear systems
Theta and tau functions
Classical tau functions and PDE
Linear system for solving the sinh-Gordon equation
Scattering functions
Solving the coupled ODE
Matrix potentials
The bracket operation
Potentials and derivatives
Solution of the coupled ODE
Matrix potential in Gelfand-Levitan equation
Hankel determinant for deformed Laguerre weight
Painleve III' equations
Random matrix model
Equilibrium potential
Free logarithmic Sobolev inequality