RandomMatrixTheory
The Unreasonable Utility of Random Matrix TheoryIn 1959 Eugene Wigner delivered the Richard Courant Lecture entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences at New York University. Perhaps the best evidence for his argument concerning the interplay of mathematics and the applied sciences actually comes from random matrix theory (RMT), to which he himself made pioneering contributions. Historically RMT emerged from diverse sources  Adolf Hurwitz's 1897 study on invariant measures of the classical groups; Wishart's 1928 investigations of the singular values of random, real rectangular matrices; and Wigner's famous "Semicircle Law" for the global density of eigenvalues of a random symmetric matrix. And since his observations the remarkable synergy between abstract areas of pure mathematics and a bewildering array of diverse applications in contemporary studies of RMT has grown. Some of the modern examples include:
Yet another example is the role of integrable systems in RMT as an essential description of the averages, correlations and fluctuations arising in the theory. This was the theme of the keynote lecture that Professor Percy Deift of the Courant Institute NYU gave as the MSRIClay Distinguished Professor for the program ``Random matrices, interacting particle systems, and integrability'' at the MSRI in 2010. Some introductory articles on these aspects of RMT are Dei_2000, Dia_2005 and Cip_1999, whereas an encyclopediac resource is the monograph For_2010. Deift's observation is even more true now. We now have additional problems arising in RMT, such as the recent work on the singular values of products of rectangular matrices with random, complex i.i.d normal variables WF_2016, Str_2014 (the socalled Ginibre ensembles), where higher order nonlinear differential equations (fourth order in this case) are key to an understanding of the problem. This generalises a famous result of Tracy and Widom in 1994 TW_1994a treating the smallest squaredsingular value of a single random matrix with unitary symmetry, the socalled hard edge problem. Here they identified the third Painlevé equation as the relevant integrable system. This identification is not "accidental" or "gratuitous" in any sense but is in fact an essential step which allows the employment of powerful tools from analysis for studying the averages, correlations, large deviations etc of the models. The projects envisaged here will exploit this connection by employing the tools of RMT and integrable systems in related but novel applications and so will suit students with a background in probability theory and classical analysis. Cip_1999 B. Cipra. A prime case of chaos, 1999. Dei_2000 P. Deift. Integrable systems and combinatorial theory. Notices Amer. Math. Soc., 47(6):631640, 2000. Dia_2005 P. Diaconis. What is a random matrix? Notices Amer. Math. Soc., 52(11):13481349, 2005. ESW_2014 A. Edelman, B.D. Sutton, and Y. Wang. Random matrix theory, numerical computation and applications. In Modern aspects of random matrix theory, volume~72 of Proc. Sympos. Appl. Math., pages 5382. Amer. Math. Soc., Providence, RI, 2014. For_2010 P.J. Forrester. Log Gases and Random Matrices, volume~34 of London Mathematical Society Monograph. Princeton University Press, Princeton NJ, first edition, 2010. Joh_2006 I.M. Johnstone. High dimensional statistical inference and random matrices. In International Congress of Mathematicians. Vol. I, pg 307333. Eur. Math. Soc., Zürich, 2006. May_1972 R.M. May. Will a large complex system be stable? Nature, 238(5364):413414, August 1972. MS_2005 F. Mezzadri and N.C. Snaith, editors. Recent perspectives in random matrix theory and number theory, volume 322 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2005. SS_2010 T. Sasamoto and H. Spohn. Onedimensional KardarParisiZhang equation: An exact solution and its universality. Phys. Rev. Lett., 104:230602, Jun 2010. Str_2014 E. Strahov. Differential equations for singular values of products of Ginibre random matrices. J. Phys. A, 47(32):325203, 27, 2014. TW_1994a C.A. Tracy and H. Widom. Level spacing distributions and the Bessel kernel. Comm. Math. Phys., 161(2):289309, 1994. TV_2004 A.M. Tulino and S. Verdú. Random matrix theory and wireless communications. Found. Trends Commun. Inf. Theory, 1(1):1182, 2004. WF_2016 N.S. Witte and P.J. Forrester. Singular values of products of Ginibre random matrices. 2016. RMT occupies an interface between many subdisciplines of mathematics and is increasingly relevant in a diversity of applications. As of 2012012 the ISI database has 5744 published articles with RM in the title or abstract and the yearly amounts are given in the Figure below. CombinatoricsRandom permutations, longest increasing subsequences, nonintersecting random walkers, polynuclear growth models last passage percolation models, queuing models. Mathematical StatisticsPrincipal components analysis, sample covariance matrices, Wishart distribution, null and nonnull SCM, Zonal and Jack polynomials. AnalysisFredholm determinants, Painlevé equations, orthogonal and biorthogonal polynomials, RiemannHilbert problems. Algebraic GeometryMatrix Integrals and genus expansions. Number TheoryMoments of the Riemann Zeta function, distribution of the nontrivial zeros. Applications
