James W. Cannon

James Weldon Cannon (born January 30, 1943 in Bellefonte , Pennsylvania ) is an American mathematician who studies hyperbolic manifolds, geometric topology and geometric group theory.

Cannon received his PhD in 1969 from Cecil Edmund Burgess at the University of Utah ( Tame subsets of 2-spheres in euclidean 3-space ). From 1977 he was a professor at the University of Wisconsin – Madison and from 1986 at Brigham Young University . In 1971 he received a research grant from the Alfred P. Sloan Foundation ( Sloan Research Fellowship ).

In the 1970s he solved the double suspension problem of John Milnor by proving that the double suspension (suspension) of a homology sphere is a topological sphere . In 1979, together with Bryant and Larcher, he "almost" proved the characterization hypothesis for topological manifolds - manifolds in five and more dimensions that satisfy the disjoint disk property are topological manifolds (the proof was completed by Frank Quinn in 1983 ). Cannon lectured on this at the 1978 International Congress of Mathematicians. In the same year he introduced the term grope for objects that come close to a manifold, but are singular in a simple way (2-dimensional complexes with a peripheral circle ).

In the 1980s he turned to hyperbolic 3-manifolds, Klein groups, and geometric group theory. He investigated the combinatorial properties of Cayley graphs of Klein groups and their connection with the geometric properties of the operation of these groups in hyperbolic manifolds. In 1992 he was one of the co-authors of a book with William Thurston and others on automatic groups, that is, geometric group theory under algorithmic aspects. The Cannon-Thurston figure is named after him and Thurston .

In 1994 he proved what he called a combinatorial version of Riemann's mapping theorem of geometric group theory. He gave the necessary conditions for the operation of a group via homeomorphisms of a 2-sphere to be realized as Möbius transformations of the Riemann sphere . To do this, he carried out ever finer combinatorial subdivisions of the 2-sphere in order to introduce a conformal geometry in the borderline case. A related conjecture by Cannon (1998) asks about the characterization of hyperbolic groups with a 2-sphere as the edge, which Cannon worked on with William Floyd and Walter Parry, among others (introduction of Finite Subdivision Rules ), but which also has further effects on the Research by other mathematicians. Cannon, Parry and Floyd also applied their work on Finite Subdivision Rules to biology (pattern formation in organisms).

In 2012 he became a Fellow of the American Mathematical Society . He was invited speaker at the International Congress of Mathematicians in Helsinki in 1978 ( The characterization of topological manifolds of dimension ${\displaystyle n\geq 5}$ ).

Fonts

• The recognition problem. What is a topological manifold? , Bulletin AMS, Vol. 84, 1978, pp. 832-866, online
• Shrinking cell-like decompositions of manifolds. Codimension three , Annals of Mathematics, Volume 110, 1979, pp. 83-112.
• with JL Bryant, RC Lacher The structure of generalized manifolds having nonmanifold set of trivial dimension , in: Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) , Academic Press 1979, pp. 261-300
• The combinatorial structure of cocompact discrete hyperbolic groups , Geometriae Dedicata, Volume 16, 1984, pp. 123-148
• with David Epstein , Derek F. Holt, Silvio Levy, Michael S. Paterson , William Thurston Word processing in groups , Boston: Jones and Bartlett Publishers, 1992
• Almost convex groups , Geometriae Dedicata, Vol. 22, 1987, pp. 197-210
• The combinatorial Riemann mapping theorem , Acta Mathematica, Volume 173, 1994, pp. 155-234,
• with William Floyd, Walter Parry Finite subdivision rules , Conformal Geometry and Dynamics, Volume 5, 2001, pp. 153-196
• with Floyd, Parry Crystal growth, biological cell growth and geometry , in Pattern Formation in Biology, Vision and Dynamics , World Scientific, 2000, pp. 65–82
• with William Thurston Group invariant Peano curves , Geometry & Topology, Volume 11, 2007, pp. 1315–1355 (circulating as preprint since the mid-1980s, Cannon-Thurston illustration )

Homepage

• Homepage

Individual evidence

1. ^ Mathematics Genealogy Project
2. ↑ In his article in Bull. AMS, Volume 84, 1978, p. 832
3. ↑ Peter Teichner, What is a grope ..? , Notices AMS, September 2004