Marc Edward Culler (born November 22, 1953) is an American mathematician who works in

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 group (mathematics), groups and topology, topological and geometry, geometric pro ...

and

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 ...

. A native Californian, Culler did his undergraduate work at the

University of California at Santa Barbara The University of California, Santa Barbara (UC Santa Barbara or UCSB) is a public land-grant research university in Santa Barbara, California with 23,196 undergraduates and 2,983 graduate students enrolled in 2021–2022. It is part of the U ...

and his graduate work at

Berkeley Berkeley most often refers to: *Berkeley, California, a city in the United States **University of California, Berkeley, a public university in Berkeley, California * George Berkeley (1685–1753), Anglo-Irish philosopher Berkeley may also refer ...

where he graduated in 1978. He is now at the

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

. Culler is the son of Glen Jacob Culler who was an important early innovator in the development of the Internet.

Work

Culler specializes in

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, low dimensional topology, 3-manifolds, and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

. Culler frequently collaborates with

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. Life He graduated from Stuyvesant High School in 1962, and went on to earn a B.A. from Harvard Coll ...

and they have co-authored many papers. Culler and Shalen did joint work that related properties of representation varieties of

hyperbolic 3-manifold In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It ...

groups to decompositions of 3-manifolds. In particular, Culler and Shalen used the

Bass–Serre theory Bass–Serre theory is a part of the Mathematics, mathematical subject of group theory that deals with analyzing the algebraic structure of Group (math), groups Group action (mathematics), acting by automorphisms on simplicial Tree (graph theory), t ...

, applied to the function field of the SL(2,C)-

Character variety In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi, the G-''character variety of'' \pi is a space of equivalence classes of group homomorphisms from \pi to G: :\mathfrak(\pi,G)=\ ...

of a 3-manifold, to obtain information about

incompressible surface In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the surface of a suitcase is compressible, because ...

s in the manifold. Based on this work, Shalen, Cameron Gordon, John Luecke, and Culler proved the

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 A S ...

. Another important contribution by Culler came in a 1986 paper with

Karen Vogtmann Karen Vogtmann (born July 13, 1949 in Pittsburg, California''Biographies of Candidates 200 ...

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'' is a peer-reviewed journal focusing on algebra, analysis, geometry and topology. Its editorial board, , consists of 17 university-affiliated scholars in addition to the Editor-in-chief. Articles in the ''New ...

''. He was a

Sloan Foundation The Alfred P. Sloan Foundation is an American philanthropic nonprofit organization. It was established in 1934 by Alfred P. Sloan Jr., then-president and chief executive officer of General Motors. The Sloan Foundation makes grants to support o ...

Research Fellow (1986–1988) and a UIC University ScholarUIC College awards
/ref> (2008). In 2014, he became a Fellow 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, ...

.List of Fellows of the American Mathematical Society
/ref>

Selected publications

* 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.'' 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.

References

External links

Publications of Marc Culler

Marc Culler at the Mathematics Genealogy Project

Culler's home page at UIC
{{DEFAULTSORT:Culler, Marc 20th-century American mathematicians 21st-century American mathematicians Topologists University of Illinois Chicago faculty University of California, Berkeley alumni Sloan Research Fellows 1953 births Living people Fellows of the American Mathematical Society Institute for Advanced Study visiting scholars People from California