Marc Culler
Encyclopedia
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 University of Illinois at Chicago
. Culler is the son of Glen Jacob Culler
who was an important early innovator in the development of the Internet.
, low dimensional topology, 3-manifolds, and hyperbolic geometry
. Culler frequently collaborates with Peter Shalen
and they have co-authored many papers. Culler and Shalen did joint work that related properties of representation varieties of hyperbolic 3-manifold
groups to decompositions of 3-manifolds. In particular, Culler and Shalen used the Bass–Serre theory
, applied to the function field of the SL(2,C)-Character variety of a 3-manifold, to obtain information about
incompressible surface
s in the manifold. Based on this work, Shalen, Cameron Gordon
, John Luecke
, and Culler proved the cyclic surgery theorem
.
Another important contribution by Culler came in a 1986 paper with Karen Vogtmann
called "Moduli of graphs and automorphisms of free groups". This paper introduced an object that came to be known as Culler–Vogtmann Outer space.
Culler is one of the authors of a 1994 paper called "Plane curves associated to character varieties of 3-manifolds" which introduced the A-polynomial of a knot or, more generally, of a 3-manifold with one torus boundary component.
Culler is an editor of The New York Journal of Mathematics
. He was a Sloan Foundation Research Fellow (1986–1988) and a UIC University Scholar (2008).
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
and 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...
. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
where he graduated in 1978. He is now at 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...
. Culler is the son of Glen Jacob Culler
Glen Culler
Glen Jacob Culler was a professor of electrical engineering and an important early innovator in the development of the Internet. Culler joined the University of California, Santa Barbara mathematics faculty in 1959 and helped put the campus in the forefront of what would become the field of...
who was an important early innovator in the development of the Internet.
Work
Culler specializes in group theoryGroup theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, low dimensional topology, 3-manifolds, and hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
. Culler frequently collaborates with Peter Shalen
Peter Shalen
Peter B. Shalen is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.-Life: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...
and they have co-authored many papers. Culler and Shalen did joint work that related properties of representation varieties of hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
groups to decompositions of 3-manifolds. In particular, Culler and Shalen used the Bass–Serre theory
Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees...
, applied to the function field of the SL(2,C)-Character variety of a 3-manifold, to obtain information about
incompressible surface
Incompressible surface
In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
s in the manifold. Based on this work, Shalen, 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
John Edwin Luecke
John Edwin Luecke is an American mathematician who works in topology and knot theory. He got his Ph.D. in 1985 from the University of Texas at Austin and is now a professor in the department of mathematics at that institution.-Work:...
, and Culler 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...
.
Another important contribution by Culler came in a 1986 paper with Karen Vogtmann
Karen Vogtmann
Karen Vogtmann is a U.S. mathematician working primarily in the area of geometric group theory. She is known for having introduced, in a 1986 paper with Marc Culler, an object now known as the Culler–Vogtmann Outer space...
called "Moduli of graphs and automorphisms of free groups". This paper introduced an object that came to be known as Culler–Vogtmann Outer space.
Culler is one of the authors of a 1994 paper called "Plane curves associated to character varieties of 3-manifolds" which introduced the A-polynomial of a knot or, more generally, of a 3-manifold with one torus boundary component.
Culler is an editor of The New York Journal of Mathematics
The New York Journal of Mathematics
The New York Journal of Mathematics is a peer-reviewed, scholarly journal. It has an editorial board which , consists of 18 university-affiliated scholars in addition to the Editor-in-chief. It is, however, published entirely electronically...
. He was a Sloan Foundation Research Fellow (1986–1988) and a UIC University Scholar (2008).
Selected publications
- Culler, Marc; Shalen, Peter B.; Varieties of group representations and splittings of 3-manifolds. Annals of MathematicsAnnals 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. Annals of Mathematics (2) 125 (1987), no. 2, 237–300.
- Marc Culler and Karen Vogtmann; A group-theoretic criterion for property A. Proc. Amer. Math. Soc., 124(3):677—683, 1996. MR1307506