# 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.

Peter Shalen | |
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Alma mater | Harvard College (BA, 1966); Harvard University (PhD, 1972) |

Known for | JSJ decomposition |

Scientific career | |

Fields | Mathematics, Topology |

Institutions | University of Illinois at Chicago |

## Life

He graduated from Stuyvesant High School in 1962,[1] and went on to earn a B.A. from Harvard College in 1966 and his Ph.D. from Harvard University in 1972.[2] 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).[3] 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".[4]

## 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.

## References

- "Stuyvesant Math Team". Archived from the original on 2007-10-01. Retrieved 2007-10-31.
- Curriculum Vitæ of Peter B. Shalen Retrieved 9 November 2010
- Sloan Research Fellowships
- 2017 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2016-11-06.

## External links

- Shalen's home page at UIC
- Art Rothstein's Stuyvesant Math Team page
- Peter Shalen at the Mathematics Genealogy Project