{"id":222,"date":"2023-06-11T19:13:57","date_gmt":"2023-06-12T02:13:57","guid":{"rendered":"https:\/\/live-usc-dornsife.pantheonsite.io\/sergey-lototsky\/?page_id=222"},"modified":"2023-06-28T17:55:01","modified_gmt":"2023-06-29T00:55:01","slug":"math-606-in-summer-2023-introduction-to-random-matrices","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-606-in-summer-2023-introduction-to-random-matrices\/","title":{"rendered":"Math 606 in Summer 2023: Introduction to Random Matrices"},"content":{"rendered":"\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <div class=\"html-content\">\n<h1 align=\"center\">Class number 054&#8211;39482R<br \/>\n<span style=\"color: red;\"> Class meetings: MW, 9:30am-12:30pm, KAP 166. <\/span><\/h1>\n<h2 align=\"center\">Information on this and related pages changes frequently.<\/h2>\n<h2><strong> Instructor: <\/strong> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/\" target=\"_blank\" rel=\"noopener\">Sergey Lototsky<\/a>.<\/h2>\n<h2><strong>Office:<\/strong> KAP 248D.<br \/>\n<strong> Phone:<\/strong> 213 740 2389<br \/>\n<strong> E-mail:<\/strong> lototsky usc edu.<\/h2>\n<p><strong>Office Hours:<\/strong> MW before and after the class. Appointments at other time are welcome.<\/p>\n<p><strong> Course objective: <\/strong>To learn main terminology and results related to random matrices, such as <strong><br \/>\n<\/strong><\/p>\n<ul>\n<li>Gaussian ensembles (GOE, GUE, GSE) and some of the fundamental math behind them (Lie groups, Lie Algebras, Quaternions, etc.);<\/li>\n<li>Circle, semi-circle, and other laws, and the names behind them;<\/li>\n<li>Saddle point method, Method of moment, Stieltjes trasnform, Dyson Brownian motion, and other technical tools.<\/li>\n<\/ul>\n<p><strong> Course work: <\/strong>Class participation, homework assignments, final presentation.<\/p>\n<p><strong> Official grading scheme: <\/strong>20% class participation, 40% homework assignments, 40% final presentation.<\/p>\n<p><strong>Main reference:\u00a0 <\/strong>G. Livan, M. Novaes, P. Vivo.\u00a0 Introduction to random matrices. Theory and practice. SpringerBriefs in Mathematical Physics, 26. <em>Springer, Cham,<\/em> 2018,<strong> ix+124 pp.\u00a0<\/strong><\/p>\n<p>The book is available in electronic form from USC libraries.<\/p>\n<p>A longer list of <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/RandomMatrices-OrganizedReferences.pdf\" target=\"_blank\" rel=\"noopener\">references<\/a>.<\/p>\n<p><strong>An example of a book review from <\/strong><em>Math reviews<\/em> [<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/FrizHarier-BookRev-MathRev-1.pdf\" target=\"_blank\" rel=\"noopener\">Edition 1<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/FrizHarier-BookRev-MathRev-2.pdf\" target=\"_blank\" rel=\"noopener\">Edition 2<\/a>] <strong>and<\/strong> <strong>from<\/strong> the <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/FrizHarier-BookRev-BulletAMS.pdf\" target=\"_blank\" rel=\"noopener\">Bulletin of the AMS<\/a><\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/RandomMatrices.pdf\" target=\"_blank\" rel=\"noopener\">Homework problems and more<\/a>.<\/p>\n<p><strong>Our progress<\/strong>.<\/p>\n<p><strong>May 17. <\/strong>An overview of the class, the book, and some foundational material from <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/LinAlgebra-advanced.pdf\" target=\"_blank\" rel=\"noopener\">Linear Algebra<\/a> and <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Convergence-Pics.pdf\" target=\"_blank\" rel=\"noopener\">Probability<\/a>.<\/p>\n<p><strong>May 22. <\/strong>Random eigenvalues vs. iid random variables; circle\/semi-circle\/quarter-circle laws; an overview of <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Quaternions.pdf\" target=\"_blank\" rel=\"noopener\">quaternions<\/a>.<\/p>\n<p><strong>May 24.<\/strong> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/AsymptotIntegration.pdf\" target=\"_blank\" rel=\"noopener\">Asymptotic Integration<\/a>; more about the circle laws.<\/p>\n<p><strong>May 29.<\/strong> No class: Memorial Day.<\/p>\n<p><strong>May 31.<\/strong> From the joint pdf of matrix entries to the joint pdf of eigenvalues and eigenvectors; <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Vandermonde.pdf\" target=\"_blank\" rel=\"noopener\">Vandermonde and his determinant<\/a>.<\/p>\n<p><strong>June 5.<\/strong>\u00a0 Proving the circle laws: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/WeakConvergence.pdf\" target=\"_blank\" rel=\"noopener\">weak convergence of probability measures<\/a>, method of moments, Stieltjes transform, and more.<\/p>\n<p><strong>June 7.<\/strong> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/OrthPoly.png\" target=\"_blank\" rel=\"noopener\">Orthogonal polynomials<\/a>; determinant processes; examples of bulk vs edge asymptotics for eigenvalues.<\/p>\n<p><strong>June 12.<\/strong> Wishart-Laguerre ensembles; Marchenko-Pastur theorem.<\/p>\n<p><strong>June 14.<\/strong> Free probability and its connections to random matrices.<\/p>\n<p><strong>June 19.<\/strong> No class: Federal holiday\/USC non-instructional day.<\/p>\n<p><strong>June 21.<\/strong> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/PainleveEqns.pdf\" target=\"_blank\" rel=\"noopener\">Painleve equations<\/a> and related topics; Dyson&#8217;s Brownian motion.<\/p>\n<p><strong>June 26.\u00a0<\/strong> More on Dyson&#8217;s Brownian motion, matrix norms, and kernels\/determinant processes.<\/p>\n<p><strong>June 28.<\/strong> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/FinalDiscussion.pdf\" target=\"_blank\" rel=\"noopener\">A concluding discussion<\/a>.<\/p>\n<\/div>\n<div class=\"contact-panel\"><\/div>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":385,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-222","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - 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