William Goldman (mathematician)

William Goldman (born 1955) is a professor of mathematics at the University of Maryland, College Park (since 1986). He received his Ph.D. in mathematics from the University of California, Berkeley in 1980. He was on the Board of Governors for the The Geometry Center at the University of Minnesota from 1994 to 1996. He is married and has three children.

Goldman has investigated geometric structures, in various incarnations, on manifolds since his undergraduate thesis at Princeton University, "Affine manifolds and projective geometry on manifolds" (supervised by William Thurston and Dennis Sullivan).

With John Parker, he examined the complex hyperbolic ideal triangle group representations. These are representations of hyperbolic ideal triangle groups to the group of holomorphic isometries of the complex hyperbolic plane such that each standard generator of the triangle group maps to a C-reflection and the products of pairs of generators to parabolics. The space of representations for a given triangle group (modulo conjugacy) is parametrized by a half-open interval. They showed that the representations in a particular range were discrete and conjectured that a representation would be discrete if and only if it was in a specified larger range. This has become known as the Goldman–Parker conjecture and was eventually proven by Richard Schwartz.

Professor Goldman also heads a research group at the University of Maryland called the Experimental Geometry Lab, a team developing software (primarily in Mathematica) to explore geometric structures and dynamics in low dimensions.

Selected publications

Papers

• William Goldman, "On the polynomial cohomology of affine manifolds", Invent. Math. 65 (1981/82), no. 3, 453–457.
• William Goldman and Morris Hirsch, "A generalization of Bieberbach's theorem", Invent. Math. 65 (1981/82), no. 1, 1–11.
• William Goldman, "Invariant functions on Lie groups and Hamiltonian flows of surface group representations", Invent. Math. 85 (1986), no. 2, 263–302.
• William Goldman and John J. Millson, "Local rigidity of discrete groups acting on complex hyperbolic space", Invent. Math. 88 (1987), no. 3, 495–520.
• William Goldman, "Geometric structures on manifolds and varieties of representations", Geometry of group representations (Boulder CO, 1987), 169–198, Contemp. Math., 74, Amer. Math. Soc., Providence, RI, 1988.
• William Goldman, "Topological components of spaces of representations", Invent. Math. 93 (1988), no. 3, 557–607.
• William Goldman and John Parker, "Complex hyperbolic ideal triangle groups", J. Reine Angew. Math. 425 (1992), 71–86.
• William Goldman, Michael Kapovich, and Bernhard Leeb; "Complex hyperbolic manifolds homotopy equivalent to a Riemann surface", Comm. Anal. Geom. 9 (2001), no. 1, 61–95.
• William Goldman, "Ergodic theory on moduli spaces", Ann. of Math. (2) 146 (1997), no. 3, 475–507.

Books

• William Goldman, Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1999. xx+316 pp. ISBN 0-19-853793-X

External links

• Faculty page at the University of Maryland, College Park