Marc Culler | |
---|---|

Photograph by Roberta Devlin | |

Born | November 22, 1953 67) | (age

Nationality | American |

Alma mater | University of California, Berkeley |

Known for | cyclic surgery theorem Culler–Vogtmann Outer space |

Scientific career | |

Fields | geometric group theory low-dimensional topology |

Doctoral advisor | John Robert Stallings |

**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 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.

Culler specializes in group theory,^{ [1] } low dimensional topology, 3-manifolds, and hyperbolic geometry. Culler frequently collaborates with Peter Shalen and they have co-authored many papers.^{ [2] }^{ [3] } 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 surfaces 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".^{ [4] } 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"^{ [5] } which introduced the A-polynomial of a knot^{ [6] } or, more generally, of a 3-manifold with one torus boundary component.

Culler is an editor of * The New York Journal of Mathematics *.^{ [7] } He was a Sloan Foundation Research Fellow^{ [8] } (1986–1988) and a UIC University Scholar^{ [9] }^{ [10] } (2008). In 2014, he became a Fellow of the American Mathematical Society.^{ [11] }

- Culler, Marc; Shalen, Peter B.;
*Varieties of group representations and splittings of 3-manifolds.*Annals of Mathematics. (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. MR 1307506

**William Paul Thurston** was an American mathematician. He was 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. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

In knot theory, a **figure-eight knot** is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.

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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 look like our universe does to a small enough observer. This is made more precise in the definition below.

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 is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.

**Algorithmic topology**, or **computational topology**, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory.

**Peter B. Shalen** is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.

In mathematics, **hyperbolic Dehn surgery** is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.

**John Willard Morgan** is an American mathematician known for his contributions to topology and geometry. He is a Professor Emeritus at Columbia University and a member of the Simons Center for Geometry and Physics at Stony Brook University.

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Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early, significant stimulus in knot theory would arrive later with Sir William Thomson and his vortex theory of the atom.

In the mathematical subject of geometric group theory, a **Dehn function**, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the *area* of a *relation* in that group in terms of the length of that relation. The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive. The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface.

**Karen Vogtmann** is an American 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. The Outer space is a free group analog of the Teichmüller space of a Riemann surface and is particularly useful in the study of the group of outer automorphisms of the free group on *n* generators, Out(*F*_{n}). Vogtmann is a professor of mathematics at Cornell University and The University of Warwick.

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 fundamental group, then the distance between *r* and *s* is at most 1. Consequently, there are at most three Dehn fillings of *M* with cyclic fundamental group. The theorem appeared in a 1987 paper written by Marc Culler, Cameron Gordon, John Luecke and Peter Shalen.

**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.

In geometric group theory, the **Rips machine** is a method of studying the action of groups on **R**-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991.

**Marc Lackenby** is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory.

**Nathan Michael Dunfield** is an American mathematician, specializing in Topology.

- ↑ People in Geometric Group Theory
- ↑ Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds. Ann. of Math. (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
- ↑ Culler, Marc; Vogtmann, Karen (1986), "Moduli of graphs and automorphisms of free groups" (PDF),
*Inventiones Mathematicae*,**84**(1): 91–119, Bibcode:1986InMat..84...91C, doi:10.1007/BF01388734, S2CID 122869546. - ↑ D. Cooper, M. Culler, H. Gillet, D. D. Long and P. B. Shalen. Plane curves associated to character varieties of 3-manifolds. Invent. Math., 118(1):47--84, 1994.
- ↑ The A-polynomial and the FFT, in PDF format
- ↑ New York Journal of Mathematics
- ↑ Sloan Research Fellowships
^{[ permanent dead link ]} - ↑ UIC News: 2008 University Scholar Marc Culler
- ↑ UIC College awards
- ↑ List of Fellows of the American Mathematical Society

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