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FieldsMathematics, Topology Known forJSJ decomposition

NamePeter Shalen RoleMathematician

InstitutionsUniversity of Illinois at Chicago Alma materHarvard College (BA, 1966);
Harvard University (PhD, 1972) EducationHarvard University, Stuyvesant High School

Volume and homology for hyperbolic 3 orbifolds arithmetic groups peter shalen

Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.

He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard College in 1966 and his Ph.D. from Harvard University in 1972. After posts at Columbia University, Rice University, and the Courant Institute, he joined the faculty of the University of Illinois at Chicago.

Shalen was a Sloan Foundation Research Fellow in mathematics (1977—1979). He was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to three-dimensional topology and for exposition".

Work

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.

Selected publications

Jaco, William H. & Shalen, Peter B. (1979). Seifert fibered spaces in 3-manifolds. Providence: American Mathematical Society. ISBN 0-8218-2220-9.

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.