Peter B. Shalen (born c. 1946) is an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, working primarily in

low-dimensional topology In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the theory of 3-manifolds and 4-manifolds, knot theory, ...

. He is the "S" in JSJ decomposition.

Life

He graduated from

Stuyvesant High School Stuyvesant High School ( ) is a co-ed, State school, public, college-preparatory, Specialized high schools in New York City, specialized high school in Manhattan, New York City. The school, commonly called "Stuy" ( ) by its students, faculty, a ...

in 1962, and went on to earn a B.A. from

Harvard College Harvard College is the undergraduate education, undergraduate college of Harvard University, a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Part of the Harvard Faculty of Arts and Scienc ...

in 1966 and his Ph.D. from

Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...

in 1972. After posts at

Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...

Rice University William Marsh Rice University, commonly referred to as Rice University, is a Private university, private research university in Houston, Houston, Texas, United States. Established in 1912, the university spans 300 acres. Rice University comp ...

, and the Courant Institute, he joined the faculty of the

University of Illinois at Chicago The University of Illinois Chicago (UIC) is a public research university in Chicago, Illinois, United States. Its campus is in the Near West Side community area, adjacent to the Chicago Loop. The second campus established under the Universi ...

. Shalen was a Sloan Foundation Research Fellow in mathematics (1977—1979). In 1986 he was an invited speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

Berkeley, California Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Anglo-Irish bishop and philosopher George Berkeley. It borders the cities of Oakland, Cali ...

. He was elected as a member of the 2017 class of

Fellow A fellow is a title and form of address for distinguished, learned, or skilled individuals in academia, medicine, research, and industry. The exact meaning of the term differs in each field. In learned society, learned or professional society, p ...

s of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

"for contributions to three-dimensional topology and for exposition".2017 Class of the Fellows of the AMS

, retrieved 2016-11-06.

Work

His work with Marc Culler related properties of representation varieties of

hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

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 In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...

(+1 or −1) on a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

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 William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...

Selected publications

* *Shalen, Peter B. ''Separating, incompressible surfaces in 3-manifolds.''

Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing ...

52 (1979), no. 2, 105–126. * Culler, Marc; Shalen, Peter B. ''Varieties of group representations and splittings of 3-manifolds.''

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

(2) 117 (1983), no. 1, 109–146. * Culler, Marc; Gordon, C. McA.; Luecke, J.; Shalen, Peter B. ''Dehn surgery on knots.''

(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.''

(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.''

(2) 127 (1988), no. 3, 457–519.

References

External links

Shalen's home page at UIC
* 1940s births 20th-century American mathematicians 21st-century American mathematicians American topologists Stuyvesant High School alumni Harvard College alumni Columbia University faculty Rice University faculty University of Illinois Chicago faculty Living people Courant Institute of Mathematical Sciences faculty Fellows of the American Mathematical Society Mathematicians from New York (state) Year of birth missing (living people) {{US-mathematician-stub