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, structure, space, models, and change. History On ...

, 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 structure theory of 3-manifolds and 4-manifolds, knot th ...

. He is the "S" in

JSJ decomposition In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: : Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isot ...

Life

He graduated from

Stuyvesant High School Stuyvesant High School (pronounced ), commonly referred to among its students as Stuy (pronounced ), is a State school, public university-preparatory school, college-preparatory, Specialized high schools in New York City, specialized high school ...

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

Harvard College Harvard College is the undergraduate college of Harvard University, an Ivy League research university in Cambridge, Massachusetts. Founded in 1636, Harvard College is the original school of Harvard University, the oldest institution of higher lea ...

in 1966 and his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

from

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

in 1972. After posts at

Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

Rice University William Marsh Rice University (Rice University) is a Private university, private research university in Houston, Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranke ...

, and the

Courant Institute The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU), and is among the most prestigious mathematics schools and mathematical sciences research cente ...

, he joined the faculty of the

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

. 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 Nevanlinna Prize (to be rename ...

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 Irish bishop and philosopher George Berkeley. It borders the cities of Oakland and Emer ...

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

Fellow A fellow is a concept whose exact meaning depends on context. In learned or professional societies, it refers to a privileged member who is specially elected in recognition of their work and achievements. Within the context of higher education ...

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 Marc Edward Culler (born November 22, 1953) 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 ...

related properties of representation varieties of

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

to decompositions of 3-manifolds. Based on this work, Culler, Cameron Gordon, John Luecke, and Shalen proved the

cyclic surgery theorem In three-manifold, three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a Compact space, compact, Connected space, connected, Orientability, orientable, Irreducibility (mathematics), irreducible three-ma ...

. 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 cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

can result in a

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...

3-manifold. This was an important piece of the

Gordon–Luecke theorem In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a me ...

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. Thurston ...

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 managing editors ...

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 the ...

(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 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) {{US-mathematician-stub