Peter Shalen

From Wikipedia, the free encyclopedia

Peter B. Shalen is a American mathematician, working primarily in low dimensional topology. He is the "S" in JSJ decomposition.

His work with Marc Culler related properties of representation varieties of hyperbolic 3-manifold groups to decompositions of 3-manifolds. Based on this work, Culler, Cameron Gordon, John Luecke, and Shalen proved the cyclic surgery theorem. An important corollary of the theorem is that at most one nontrivial Dehn surgery (+1 or −1) on a knot can result in a simply-connected 3-manifold. This was an important piece of the Gordon-Luecke theorem that knots are determined by their complements. This paper is often referred to as "CGLS".

With John W. Morgan, he generalized his work with Culler, and reproved several foundational results of William Thurston.

Shalen graduated from Stuyvesant High School in 1962 and went on to earn an undergraduate degree in 1966 and his Ph.D. in 1972, both from Harvard University. After posts at Columbia University, Rice University, and the Courant Institute, Shalen joined the faculty of the University of Illinois at Chicago, where he currently has seven mathematical descendants.

[edit] Selected publications

Jaco, William H. and Shalen, Peter B. (1979). Seifert fibered spaces in 3-manifolds. Providence: American Mathematical Society. ISBN 0-821-82220-9.
Shalen, Peter B. Separating, incompressible surfaces in 3-manifolds. Invent. Math. 52 (1979), no. 2, 105–126.
Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds. Ann. of Math. (2) 117 (1983), no. 1, 109–146.
Culler, Marc; Gordon, C. McA.; Luecke, J.; Shalen, Peter B. Dehn surgery on knots. Ann. of Math. (2) 125 (1987), no. 2, 237–300.
Morgan, John W.; Shalen, Peter B. Valuations, trees, and degenerations of hyperbolic structures. I. Ann. of Math. (2) 120 (1984), no. 3, 401–476.
Morgan, John W.; Shalen, Peter B. Degenerations of hyperbolic structures. II. Measured laminations in 3-manifolds. Ann. of Math. (2) 127 (1988), no. 2, 403–456.
Morgan, John W.; Shalen, Peter B. Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston's compactness theorem. Ann. of Math. (2) 127 (1988), no. 3, 457–519.