A native of New York City, Wolff was a specialist in analysis, particularly harmonic analysis. Professor Wolff made numerous highly original contributions to the mathematical fields of Fourier Analysis, Partial Differential Equations, and Complex Analysis. A recurrent theme of his work was the application of finite combinatorial ideas to infinite, continuous problems.
Wolff grew up in a mathematical environment. His uncle, Clifford Gardiner was a professor at NYU's Courant Institute of Mathematics for many years and Wolff's mother Lucile was a technical editor of volume 1 of the English translation of the celebrated book Methods of Mathematical Physics by Courant and Hilbert.
While a Berkeley graduate student Wolff surprised mathematicians worldwide with a new proof of Lennart Carleson's corona theorem about bounded analytic functions in the plane. His papers with Peter Jones of Yale University on harmonic measure showed that a random moving particle in the plane will, with large probability, first hit a given frontier set along a given set of finite length.
Wolff also worked in analytic problems arising in the fundamental equations of quantum mechanics. He proved new results on uniqueness of solutions of partial differential equations such as the Schrodinger equation by recasting all previously used inequalities in terms of oscillatory integrals. With Barry Simon, also at Caltech, he provided fundamental criteria for the localization of electrons in random media that have been used in virtually all related work in the past fifteen years. In a 1995 paper, he disproved three different outstanding conjectures about steady state of heat flows in three-dimensional space.
His two papers on the Kakeya problem (in 1995, 1997) gave new bounds on the size of subsets that include a line segment in every direction. This result gave sharper bounds on several important operators in Fourier analysis and differential equations and it established a link between discrete combinatorial mathematics and continuous harmonic analysis. The implications of this work are under intense study by analysts worldwide.
Wolff's most recent work has been similarly groundbreaking. In a paper to appear in the Annals of Mathematics, he obtained a striking new understanding of the wave equation that promises to have significant effect on the study of nonlinear physics.
Wolff was mild-mannered and unassuming but broke through his shyness to be a mentor and teacher with enormous impact on his graduate students, postdocs and coauthors.
Wolff earned his bachelor's degree in 1975 from Harvard, where he often played poker with his fellow student Bill Gates. He received his doctorate in mathematics at UC Berkeley, and afterward was acting assistant professor at the University of Washington and an NSF Postdoctoral Fellow at the University of Chicago.
He came to Caltech in 1982 as an assistant professor and was named full professor in 1986. From 1986 to 1989 he was a professor of mathematics at New York University, was at Caltech from 1988 to 1992, and from 1992 to 1996 was at Berkeley. He returned to Caltech in 1995, where he was a professor at the time of his death.
Among his major awards were the 1999 Bocher Prize and the 1985 Salem Prize as well as a Sloan Fellowship and invited named lecture series at the University of Chicago and Stanford. He was a member of the editorial boards of three publications: Communications in Analysis and Geometry, AMS Electronic Journal of Research Announcements, and the Journal of Functional Analysis.
He is survived by his wife, Carol Shubin, a mathematics professor at Cal State Northridge; two sons, James Herbert Wolff, age 3, and Richard Thomas Wolff, age 5; his parents Frank and Lucile Wolff and his sisters Virginia and Caroline.
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