Peter Shalen
Encyclopedia
Peter B. Shalen is an American
mathematician
, working primarily in low-dimensional topology
. He is the "S" in JSJ decomposition
.
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).
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
.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, working primarily in low-dimensional topology
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
. He is the "S" in JSJ decomposition
JSJ decomposition
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson...
.
Life
He graduated from Stuyvesant High SchoolStuyvesant High School
Stuyvesant High School , commonly referred to as Stuy , is a New York City public high school that specializes in mathematics and science. The school opened in 1904 on Manhattan's East Side and moved to a new building in Battery Park City in 1992. Stuyvesant is noted for its strong academic...
in 1962, and went on to earn a B.A. from Harvard College
Harvard College
Harvard College, in Cambridge, Massachusetts, is one of two schools within Harvard University granting undergraduate degrees...
in 1966 and his Ph.D. from Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
in 1972. After posts at Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
, Rice University
Rice University
William Marsh Rice University, commonly referred to as Rice University or Rice, is a private research university located on a heavily wooded campus in Houston, Texas, United States...
, and the Courant Institute, he joined the faculty of the University of Illinois at Chicago
University of Illinois at Chicago
The University of Illinois at Chicago, or UIC, is a state-funded public research university located in Chicago, Illinois, United States. Its campus is in the Near West Side community area, near the Chicago Loop...
.
Shalen was a Sloan Foundation Research Fellow in mathematics (1977—1979).
Work
His work with Marc CullerMarc Culler
Marc Edward Culler is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the...
related properties of representation varieties of hyperbolic 3-manifold groups to decompositions of 3-manifolds. Based on this work, Culler, Cameron Gordon
Cameron Gordon (mathematician)
Cameron Gordon is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem...
, John Luecke, and Shalen proved the cyclic surgery theorem
Cyclic surgery theorem
In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is not a Seifert-fibered space and r,s are slopes on T such that their Dehn fillings have cyclic...
. An important corollary of the theorem is that at most one nontrivial Dehn surgery
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
(+1 or −1) on a knot can result in a simply-connected 3-manifold. This was an important piece of the Gordon-Luecke theorem
Gordon-Luecke theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that every homeomorphism between two complements of knots in the 3-sphere extends to give a self-homeomorphism of the 3-sphere...
that knots are determined by their complements. This paper is often referred to as "CGLS".
With John W. Morgan
John Morgan (mathematician)
John Willard Morgan is an American mathematician, well known for his contributions to topology and geometry. He is currently the director of the Simons Center for Geometry and Physics at Stony Brook University.-Life:...
, he generalized his work with Culler, and reproved several foundational results of 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...
.
Selected publications
- Shalen, Peter B. Separating, incompressible surfaces in 3-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:...
52 (1979), no. 2, 105–126. - Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds. Ann. of MathAnnals 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) 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.