# Harold Widom

Title | Professor Emeritus of Mathematics, Distinguished Professor |

Division | Physical & Biological Sciences |

Department | PBSci-Mathematics Department |

Phone | 831-459-2652 |

FAX | 831-459-4511 |

Web Site | http://widom.math.ucsc.edu |

Office | McHenry Building Room #4188 |

Campus Mail Stop | Mathematics Department |

1156 High Street Santa Cruz, CA 95060 |

### 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 PainlevĂ© 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.

### Biography, Education and Training

M.S., Ph.D., University of Chicago### Honors, Awards and Grants

American Mathematical Society Fellow### 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.