Tim Cochran | |
---|---|

Born | April 7, 1955 |

Died | December 16, 2014 59) | (aged

Nationality | American |

Alma mater | University of California |

Known for | Cochran–Orr–Teichner (solvable) filtration |

Scientific career | |

Fields | Mathematics |

Institutions | Rice University |

Doctoral advisor | Robion Kirby |

Doctoral students | Shelly Harvey |

**Thomas** "**Tim**" **Daniel Cochran** (April 7, 1955 – December 16, 2014) was a professor of mathematics at Rice University specializing in topology, especially low-dimensional topology, the theory of knots and links and associated algebra.

Tim Cochran was a valedictorian for the Severna Park High School Class of 1973. Later, he was an undergraduate at the Massachusetts Institute of Technology, and received his Ph.D. from the University of California, Berkeley in 1982 (*Embedding 4-manifolds in S ^{5}*).

With his coauthors Kent Orr and Peter Teichner, Cochran defined the solvable filtration of the knot concordance group, whose lower levels encapsulate many classical knot concordance invariants.

Cochran was also responsible for naming the slam-dunk move for surgery diagrams in low-dimensional topology.

While at Rice, he was named an Outstanding Faculty Associate (1992–93), and received the Faculty Teaching and Mentoring Award from the Rice Graduate Student Association (2014)^{ [4] }

He was named a fellow of the American Mathematical Society ^{ [5] } in 2014, for *contributions to low-dimensional topology, specifically knot and link concordance, and for mentoring numerous junior mathematicians*.

- Cochran, T. (1984). "Four-manifolds which embed in but not in and Seifert manifolds for fibered knots".
*Inventiones mathematicae*.**77**: 173–184. doi:10.1007/BF01389141. - Cochran, Tim D. (1985). "Geometric invariants of link cobordism".
*Commentarii Mathematici Helvetici*.**60**: 291–311. doi:10.1007/BF02567416. - Cochran, Tim D. (1990). "Derivatives of links: Massey products and Milnor's concordance invariants".
*Memoirs of the American Mathematical Society*.**84**(427). - Cochran, Tim D.; Orr, Kent E. (1993). "Not all links are concordant to boundary links".
*Annals of Mathematics*.**138**(3): 519–554. doi:10.2307/2946555. - Cochran, Tim D.; Orr, Kent E.; Teichner, Peter (2003). "Knot Concordance, Whitney Towers and -signatures".
*Annals of Mathematics*.**157**(2): 433–519. doi: 10.4007/annals.2003.157.433 . - Cochran, Tim D.; Orr, Kent E.; Teichner, Peter (2004). "Structure in the Classical Knot Concordance Group".
*Commentarii Mathematici Helvetici*.**79**(1): 105–123. doi:10.1007/s00014-001-0793-6. - Cochran, Tim D. (2004). "Noncommutative Knot Theory".
*Algebraic and Geometric Topology*.**4**: 347–398. doi:10.2140/agt.2004.4.347. - Cochran, Tim D.; Teichner, Peter (2007). "Knot Concordance and von Neumann -invariants".
*Duke Mathematical Journal*.**137**(2): 337–379. doi:10.1215/S0012-7094-07-13723-2. - Cochran, Tim D.; Harvey, Shelly (2008). "Homology and Derived Series of Groups II: Dwyer's Theorem".
*Geometry and Topology*.**12**(1): 199–232. doi: 10.2140/gt.2008.12.199 . - Cochran, Tim D.; Harvey, Shelly; Leidy, Constance (2009). "Knot concordance and Higher-order Blanchfield duality".
*Geometry and Topology*.**13**(3): 1419–1482. doi: 10.2140/gt.2009.13.1419 . - Cochran, Tim D.; Harvey, Shelly; Leidy, Constance (2011). "Primary decomposition and the fractal nature of knot concordance".
*Mathematische Annalen*.**351**(2): 443–508. doi:10.1007/s00208-010-0604-5. - Cochran, Tim D.; Davis, Christopher William (2015). "Counterexamples to Kauffman's conjectures on slice knots".
*Advances in Mathematics*.**274**: 263–284. doi: 10.1016/j.aim.2014.12.006 .

In mathematics, **differential topology** is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the *geometric* properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology.

In an area of mathematics called differential topology, an **exotic sphere** is a differentiable manifold *M* that is homeomorphic but not diffeomorphic to the standard Euclidean *n*-sphere. That is, *M* is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one.

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 *filling*.

In knot theory, a **virtual knot** is a generalization of knots in 3-dimensional Euclidean space, **R**^{3}, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. Here is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).

In mathematics, a **Legendrian knot** often refers to a smooth embedding of the circle into , which is tangent to the standard contact structure on . It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane.

In the mathematical field of Riemannian geometry, M. Gromov's **systolic inequality** bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983; it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane.

**Michel André Kervaire** was a French mathematician who made significant contributions to topology and algebra.

In mathematics, the **Kervaire invariant** is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.

**André Haefliger** was a Swiss mathematician who worked primarily on topology.

In knot theory, an area of mathematics, the **link group** of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis,. Notably, the link group is not in general the fundamental group of the link complement.

**Wolfgang Lück** is a German mathematician who is an internationally recognized expert in algebraic topology.

In mathematics, and particularly homology theory, **Steenrod's Problem** is a problem concerning the realisation of homology classes by singular manifolds.

In mathematics, specifically in differential topology, a **Kervaire manifold** is a piecewise-linear manifold of dimension constructed by Michel Kervaire (1960) by plumbing together the tangent bundles of two -spheres, and then gluing a ball to the result. In 10 dimensions this gives a piecewise-linear manifold with no smooth structure.

**Matthias Kreck** is a German mathematician who works in the areas of Algebraic Topology and Differential topology. From 1994 to 2002 he was director of the Oberwolfach Research Institute for Mathematics and from October 2006 to September 2011 he was the director of the Hausdorff Center for Mathematics at the University of Bonn, where he is currently a professor.

**Shelly Lynn Harvey** is a professor of Mathematics at Rice University. Her research interests include knot theory, low-dimensional topology, and group theory.

**Martin George Scharlemann** is an American topologist who is a professor at the University of California, Santa Barbara. He obtained his Ph.D. from the University of California, Berkeley under the guidance of Robion Kirby in 1974.

**Paul Alexander Schweitzer**SJ is an American mathematician specializing in differential topology, geometric topology, and algebraic topology.

**Chern's conjecture for affinely flat manifolds** was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem.

**Rachel Roberts** is an American mathematician specializing in low-dimensional topology, including foliations and contact geometry. She is the Elinor Anheuser Professor of Mathematics at Washington University in St. Louis.

**Dan Burghelea** is a Romanian-American mathematician, academic, and researcher. He is an Emeritus Professor of Mathematics at Ohio State University.

- ↑ Tim Cochran at the Mathematics Genealogy Project
- ↑ "Rice mourns loss of mathematician Tim Cochran" . Retrieved December 20, 2014.
- ↑ "2 Rice mathematicians honored" . Retrieved December 20, 2014.
- ↑ "GSA honors those who support grad students at Rice" . Retrieved December 19, 2014.
- ↑ "List of Fellows of the American Mathematical Society" . Retrieved December 18, 2014.

- Tim Cochran's home page.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.