William Goldman (mathematician)
| William Goldman | |
|---|---|
| Born |
November 17, 1955 Kansas City, United States |
| Nationality | American |
| Fields | Mathematics |
| Institutions | University of Maryland-College Park |
| Alma mater | Princeton, University of California, Berkeley |
| Doctoral advisor | Morris Hirsch and William Thurston |
William Mark Goldman (born 1955 in Kansas City, Missouri) is a professor of mathematics at the University of Maryland, College Park (since 1986). He received an A.B in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980.
Goldman has investigated geometric structures, in various incarnations, on manifolds since his undergraduate thesis, "Affine manifolds and projective geometry on manifolds" (supervised by William Thurston and Dennis Sullivan). This work led to work with Morris Hirsch and David Fried on affine structures on manifolds, and work in real projective structures on compact surfaces. In particular he proved that the space of convex real projective structures on a closed orientable surface of genus g > 1 is homeomorphic to an open cell of dimension 16g-16. With Suhyoung Choi, he proved that this space is a connected component (the "Hitchin component") of the space of equivalence classes of representations of the fundamental group in SL(3,R). Combining this with Suhyoung Choi's convex decomposition theorem, this led to a complete classification of convex real projective structures on compact surfaces.
His doctoral dissertation, "Discontinuous groups and the Euler class" (supervised by Morris W. Hirsch), characterizes discrete embeddings of surface groups in PSL(2,R) in terms of maximal Euler class, proving a converse to the Milnor-Wood inequality for flat bundles. Shortly thereafter he showed that the space of representations of the fundamental group of a closed orientable surface of genus g>1 in PSL(2,R) has 4g-3 connected components, distinguished by the Euler class.
With David Fried, he classified compact quotients of Euclidean 3-space by discrete groups of affine transformations, showing that all such manifolds are finite quotients of torus bundles over the circle. The noncompact case is much more interesting, as Margulis found complete affine manifolds with nonabelian free fundamental group. In his 1990 doctoral thesis, Todd Drumm found examples which are solid handlebodies using polyhedra which have since been called "crooked planes."
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