Description:

Title: Random Matrix Theory: A Journey from Eigenvalues to Aztec Diamonds, Part I

Abstract:

 For over three decades, I have taught graduate random matrix theory to students across mathematics, physics, computer science, biology, and beyond—emphasizing computational exploration and cross-disciplinary connections over technical proofs. These talks present a whirlwind tour through one of mathematics' most surprising success stories: how the statistical behavior of random eigenvalues unlocks deep mysteries in combinatorics, growth processes, and mathematical physics.  

We begin with the simple question: how do eigenvalues of large random matrices arrange themselves? This leads us through a remarkable mathematical landscape connecting orthogonal polynomials, determinantal point processes, and the Tracy-Widom distribution to seemingly unrelated problems: the longest increasing subsequences in random permutations, fluctuations in crystal growth, and the melting boundaries of random tilings like Aztec diamonds.

Through live computations and visual demonstrations, we'll witness the emergence of universal phenomena—the same mathematical objects appearing across vastly different contexts. We'll see how 19th-century orthogonal polynomial theory provides the key to 21st-century combinatorial problems, and how the Airy process governs everything from random matrix fluctuations to the shape of growing interfaces.

This interdisciplinary journey showcases why random matrix theory has become an essential tool across the sciences, revealing unexpected unity beneath mathematical diversity. No prior knowledge of random matrices assumed—only curiosity about beautiful mathematics.

Based on material from my ongoing course at MIT (github.com/mitmath/18338), condensing eleven lectures into a two-hour exploration of mathematical wonder.