William Goldman (professor)
Encyclopedia
William Goldman 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at the University of Maryland, College Park
University of Maryland, College Park
The University of Maryland, College Park is a top-ranked public research university located in the city of College Park in Prince George's County, Maryland, just outside Washington, D.C...
(since 1986). He received his Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
in mathematics from the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
in 1980. He was on the Board of Governors for the The Geometry Center
The Geometry Center
The Geometry Center was a mathematics research and education center at the University of Minnesota. It was established by the National Science Foundation in the late 1980s and closed in 1998. The focus of the Center's work was the use of computer graphics and visualization for research and...
at the University of Minnesota
University of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
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
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, "Affine manifolds and projective geometry on manifolds" (supervised by William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
and Dennis Sullivan
Dennis Sullivan
Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...
).
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
Richard Schwartz
Richard Evan Schwartz is an American mathematician notable for his contributions to geometric group theory and to an area of mathematics known as billiards...
.
Professor Goldman also heads a research group at the University of Maryland called the Experimental Geometry Lab, a team developing software (primarily in Mathematica
Mathematica
Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
) to explore geometric structures and dynamics in low dimensions.
Papers
- William Goldman, "On the polynomial cohomology of affine manifolds", Invent. Math.Inventiones MathematicaeInventiones Mathematicae, often just referred to as Inventiones, is a mathematical journal published monthly by Springer Verlag. It was founded in 1966.-Editors:...
65 (1981/82), no. 3, 453–457. - William Goldman and Morris HirschMorris HirschMorris William Hirsch is an American mathematician, formerly at the University of California, Berkeley.A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of Edwin Spanier and Stephen Smale. His thesis was entitled Immersions of...
, "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.Crelle's JournalCrelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...
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.Annals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
(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