18.325: The Mathematics of Finite Random MatricesProfessor Alan EdelmanHandout #1, Tuesday, February 1, 2005 — Course OutlineSummary. This is a course on the mathematics and applications of finite random matrices. Our aim is totouch upon various branches of the study of finite random matrices—a consequence is that we will endup ling e ring on some areas longer than others. Our hope is that this course will confer:• some familiarity with several of the ma in thrusts of work in random matr ices—sufficient to giveyou some context for for mulating and see king known solutions to applications in engineering andphysics;• sufficient background and facility to let you read current research publications in the area of finiterandom matrix theory;• a s et of tools, both analytical and computational, for the analysis of new random matrices thatarise in new problems yo u may encounter.Course page. http://web.mit.edu/~18.325/www/Lecturer. Alan Edelman, [email protected]. URL: http://www-math.mit.edu/~edelman/.TA. Raj Rao, [email protected] List. You can add yourself to the class mailing list 18.325 us ing mailmaint on Athena.Content. The goal is for the course is, paradoxically, to be broad as well as deep. Our plan (unlikely tosurvive contact with the trials and tribulations of an actual semester her e at MIT) is to touch upon thefollowing broad areas while attempting to uncover deep insights into the underlying mechanisms thatunify these areas . This is a tentative list of topics that might be covered in the course; We will selectmaterial adaptively based on our background, interests, and rate of progress. If you are interested insome other topics, please let us know and we’d be happy to accomodate your interests.Matrix Calculus Matrix Jacobians, Computing Jacobians of 2 x 2 matrices. Jacobians of simplematrix factorizations.Wedge Products Notation to “simplify” computation of Matrix Jacobians for complicated matrixfactorizations.Classical Random Matrix Ensembles The Wisha rt, Gaussian and Jacobi ensembles, their jointeigenvalue densities, Haar distributed ortho gonal/unitary matrices, Stiefel/Grassman manifold.Matrix Integrals Classical orthogonal polynomials, the Cauchy-B inet theorem, correlation functions.Equilibrium Measure. The Hermite, Laguerre and Jacobi orthogonal polynomials. Interpretationof the limiting dis tribution as the equilibrium measure of (univaria te) orthogonal polynomials.Applications to physics.Fredholm Determinants. Tracy-Widom Distribution. Eigenvalue spacing s and the Riemann-ZetaHypothesis.Jack Polynomials. Multivariate orthogonal polynomials. Combinatorial aspects. Connections torandom matrices.Applications. Wireless Communications, Statistical Physics.Prerequisites. We assume that the reader has had an undergraduate course in Linear Algebr a (18.06) orits e quivalent and some exposure to probability (6.041 or 6.042 are more than sufficient).Requirements. Final projectMid-Term Project. You will be asked to read a paper on a topic of interest to you that involvesrandom matrix theory and present it via some mixture o f the following perspectives:• Write a description of greater clarity than the original publication, or• Devise an improved solution to the problem under consideration, and write up your improve-ment (with appropriate discussion of the original solution).• Implement the result in MATLAB in order to study its performance in practice. Considerationsinclude choice of random matrix result, design of good tests, interpretation of results, anddesign and analysis of heuristics for improving performance in practice.Semester project. The semester project can be an e xtension of the mid-term project if it sustainsyour interest. Otherwise, you will be asked to come up with some insights into a r andom matrixproblem that is of interest to you. Alternately you can use state-of-the-art tools developed recentlyto plot some eigenvalue statistics and compare it to pr e dictions. There will be more details onthis as we go through the semester.In this spirit, and since this is an advanced graduate class on a very active research area, thegrading will be like any other advance d graduate class on a cutting-edge topic. Please contactone of us for any clarifications.Textbooks. There are no textbooks covering a majo rity portion of the material we will be studying inthis course. We will be giving o ut cours e readers during the semester to help you study the materialbefo re the le c tur e s. There will be reserch papers handed out in class and pos ted o n the website.Guest Speakers.We will have guest speakers that will help give us their own perspective on their research involvingrandom matrices. While these spea kers will appear as pa rt of the Applied Math Colloquium series,attendance is strongly