18.338
Eigenvalues of Random Matrices
Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of Julia helpful, but not required.
Fall 2004 · Infinite Random Matrix Theory
Topics
- Introduction
- The Hermite Ensemble: Wigner’s Semi-Circle Law
- The Laguerre Ensemble: Marcenko-Pastur Theorem
- The Jacobi Ensemble: McKay’s Random Graph Theorem
- The “Semi-Circular” Element: Central Limit Theorem for Infinite Random Matrices
- Free Cumulants in Free Probability
- Free Cumulants in Free Probability (cont.)
- Non-Crossing Partitions and Free Cumulants
- Free Cumulants: The Semi-Circular and “Free Poisson” Distributions
- Additive Free Convolution
- The R-Transform and the Marcenko-Pastur Theorem
- Multiplicative Free Convolution
- The S-Transform
- Dr. Anna Scaglione Talk
- Orthogonal Polynomials and the Classical Matrix Ensembles
- Project Progress Presentation
- Project Progress Presentation (cont.)
- Tracy Widom Distribution
- Eigenvalue Spectrum Fluctuations
- Dr. Roland Speicher Talk
- Free Probability and Fluctuations
- Zonal Polynomials and Random Matrices
- Symmetric Group Representations and Free Probability
- MOPs
- Connections and Open Problems
- Project Presentations
- Project Presentations (cont.)
Readings
Prerequisites & course information
Prereq: 18.701 or permission of instructor G (Spring) 3-0-9 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Fall 2004
- MIT OCW calendar ↗
- MIT OCW readings ↗
- MIT OCW syllabus ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.