HAROLD WIDOM Professor Emeritus M.S., Ph.D., University of Chicago My Page

Phone: 831-459-2652 Office: Baskin 353B Office Hours: By Appt. Email: widom_at_ucsc_dot_edu

RESEARCH INTERESTS

Harold Widom's early research was in the areas of integral equations and operator theory, in particular the determination of the spectra of semi-infinite Toeplitz matrices and Wiener-Hopf operators and the asymptotic behavior of the spectra of various classes of operators. The latter was looked at from the point of view of pseudodifferential operators (which generalize both integral and partial differential operators) on manifolds, and some new and unifying spectral asymptotic results were obtained using this approach.

A more recent interest is the theory of random matrices, which has become a very active area of research by mathematicians and physicists. For some questions integral operators play a central role. Widom has, with collaborators, used ideas from operator theory to obtain new results in random matrix theory. One such is the explicit representation in terms of Painlevé transcendents of the limiting distributions of the largest and smallest eigenvalues in many models of random matrices. These same distributions have since been shown to arise in numerous other physical models, in random growth models, and in asymptotic combinatorics.

SELECTED PUBLICATIONS

(with C. A. Tracy) Differential equations for Dyson processes, Comm. Math. Phys.252 (2004), 7-41.

(with J. Gravner and C. A. Tracy) Limit theorems for height fluctuations in a class of discrete space and time growth models,  J. Stat. Phys. 102 (2001) 1085-1132.

On the relation between orthogonal, symplectic and unitary matrix ensembles,  J. Stat. Phys. 94 (1999) 347-364.

Some classes of solutions to the Toda lattice hierarchy, Comm. Math. Phys.184 (1997) 653-667.

(with C. A. Tracy) On orthogonal and symplectic matrix ensembles, Comm. Math. Phys., Comm. Math. Phys. 177 (1996) 727-754.

(with C. A. Tracy) Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994) 38-72.

(with E. Basor and C. A. Tracy) Asymptotics of level-spacing distributions for random matrices, Phys. Rev. Letts 69 (1992) 5-8.

Asymptotic expansions for pseudodifferential operators on bounded domains. Lecture Notes in Math. 1152 (1985), Springer-Verlag.

A complete symbolic calculus for pseudodifferential operators, Bull. Sciences Math.104 (1980) 19-63.

Toeplitz determinants with singular generating functions, Amer. J. Math. 95 (1973) 333-383.

H_p sections of vector bundles over Riemann surfaces, Ann. Math. 94 (1971) 304-324.

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