James W. Cannon

Last updated
James W. Cannon
Born (1943-01-30) January 30, 1943 (age 80)
NationalityAmerican
Citizenship United States
Alma materPh.D. (1969), University of Utah
Known forwork in low-dimensional topology, geometric group theory
AwardsFellow of the American Mathematical Society
Sloan Fellowship
Scientific career
Fields Mathematics
Institutions University of Wisconsin-Madison
Brigham Young University
Doctoral advisor Cecil Burgess
Doctoral students Colin Adams

James W. Cannon (born January 30, 1943) is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.

Contents

Biographical data

James W. Cannon was born on January 30, 1943, in Bellefonte, Pennsylvania. [1] Cannon received a Ph.D. in Mathematics from the University of Utah in 1969, under the direction of C. Edmund Burgess.

He was a professor at the University of Wisconsin, Madison from 1977 to 1985. [1] In 1986 Cannon was appointed an Orson Pratt Professor of Mathematics at Brigham Young University. [2] He held this position until his retirement in September 2012. [3]

Cannon gave an AMS Invited address at the meeting of the American Mathematical Society in Seattle in August 1977, an invited address at the International Congress of Mathematicians in Helsinki 1978, and delivered the 1982 Mathematical Association of America Hedrick Lectures in Toronto, Canada. [1] [4]

Cannon was elected to the American Mathematical Society Council in 2003 with the term of service February 1, 2004, to January 31, 2007. [2] [5] In 2012 he became a fellow of the American Mathematical Society. [6]

In 1993 Cannon delivered the 30-th annual Karl G. Maeser Distinguished Faculty Lecture at Brigham Young University. [7]

James Cannon is a devout member of the Church of Jesus Christ of Latter-day Saints. [8]

Mathematical contributions

Early work

Cannon's early work concerned topological aspects of embedded surfaces in R3 and understanding the difference between "tame" and "wild" surfaces.

His first famous result came in late 1970s when Cannon gave a complete solution to a long-standing "double suspension" problem posed by John Milnor. Cannon proved that the double suspension of a homology sphere is a topological sphere. [9] [10] R. D. Edwards had previously proven this in many cases.

The results of Cannon's paper [10] were used by Cannon, Bryant and Lacher to prove (1979) [11] an important case of the so-called characterization conjecture for topological manifolds. The conjecture says that a generalized n-manifold , where , which satisfies the "disjoint disk property" is a topological manifold. Cannon, Bryant and Lacher established [11] that the conjecture holds under the assumption that be a manifold except possibly at a set of dimension . Later Frank Quinn [12] completed the proof that the characterization conjecture holds if there is even a single manifold point. In general, the conjecture is false as was proved by John Bryant, Steven Ferry, Washington Mio and Shmuel Weinberger. [13]

1980s: Hyperbolic geometry, 3-manifolds and geometric group theory

In 1980s the focus of Cannon's work shifted to the study of 3-manifolds, hyperbolic geometry and Kleinian groups and he is considered one of the key figures in the birth of geometric group theory as a distinct subject in late 1980s and early 1990s. Cannon's 1984 paper "The combinatorial structure of cocompact discrete hyperbolic groups" [14] was one of the forerunners in the development of the theory of word-hyperbolic groups, a notion that was introduced and developed three years later in a seminal 1987 monograph of Mikhail Gromov. [15] Cannon's paper explored combinatorial and algorithmic aspects of the Cayley graphs of Kleinian groups and related them to the geometric features of the actions of these groups on the hyperbolic space. In particular, Cannon proved that convex-cocompact Kleinian groups admit finite presentations where the Dehn algorithm solves the word problem. The latter condition later turned out to give one of equivalent characterization of being word-hyperbolic and, moreover, Cannon's original proof essentially went through without change to show that the word problem in word-hyperbolic groups is solvable by Dehn's algorithm. [16] Cannon's 1984 paper [14] also introduced an important notion a cone type of an element of a finitely generated group (roughly, the set of all geodesic extensions of an element). Cannon proved that a convex-cocompact Kleinian group has only finitely many cone types (with respect to a fixed finite generating set of that group) and showed how to use this fact to conclude that the growth series of the group is a rational function. These arguments also turned out to generalize to the word-hyperbolic group context. [15] Now standard proofs [17] of the fact that the set of geodesic words in a word-hyperbolic group is a regular language also use finiteness of the number of cone types.

Cannon's work also introduced an important notion of almost convexity for Cayley graphs of finitely generated groups, [18] a notion that led to substantial further study and generalizations. [19] [20] [21]

An influential paper of Cannon and William Thurston "Group invariant Peano curves", [22] that first circulated in a preprint form in the mid-1980s, [23] introduced the notion of what is now called the Cannon–Thurston map. They considered the case of a closed hyperbolic 3-manifold M that fibers over the circle with the fiber being a closed hyperbolic surface S. In this case the universal cover of S, which is identified with the hyperbolic plane, admits an embedding into the universal cover of M, which is the hyperbolic 3-space. Cannon and Thurston proved that this embedding extends to a continuous π1(S)-equivariant surjective map (now called the Cannon–Thurston map) from the ideal boundary of the hyperbolic plane (the circle) to the ideal boundary of the hyperbolic 3-space (the 2-sphere). Although the paper of Cannon and Thurston was finally published only in 2007, in the meantime it has generated considerable further research and a number of significant generalizations (both in the contexts of Kleinian groups and of word-hyperbolic groups), including the work of Mahan Mitra, [24] [25] Erica Klarreich, [26] Brian Bowditch [27] and others.

1990s and 2000s: Automatic groups, discrete conformal geometry and Cannon's conjecture

Cannon was one of the co-authors of the 1992 book Word Processing in Groups [17] which introduced, formalized and developed the theory of automatic groups. The theory of automatic groups brought new computational ideas from computer science to geometric group theory and played an important role in the development of the subject in 1990s.

A 1994 paper of Cannon gave a proof of the "combinatorial Riemann mapping theorem" [28] that was motivated by the classic Riemann mapping theorem in complex analysis. The goal was to understand when an action of a group by homeomorphisms on a 2-sphere is (up to a topological conjugation) an action on the standard Riemann sphere by Möbius transformations. The "combinatorial Riemann mapping theorem" of Cannon gave a set of sufficient conditions when a sequence of finer and finer combinatorial subdivisions of a topological surface determine, in the appropriate sense and after passing to the limit, an actual conformal structure on that surface. This paper of Cannon led to an important conjecture, first explicitly formulated by Cannon and Swenson in 1998 [29] (but also suggested in implicit form in Section 8 of Cannon's 1994 paper) and now known as Cannon's conjecture, regarding characterizing word-hyperbolic groups with the 2-sphere as the boundary. The conjecture (Conjecture 5.1 in [29] ) states that if the ideal boundary of a word-hyperbolic group G is homeomorphic to the 2-sphere, then G admits a properly discontinuous cocompact isometric action on the hyperbolic 3-space (so that G is essentially a 3-dimensional Kleinian group). In analytic terms Cannon's conjecture is equivalent to saying that if the ideal boundary of a word-hyperbolic group G is homeomorphic to the 2-sphere then this boundary, with the visual metric coming from the Cayley graph of G, is quasisymmetric to the standard 2-sphere.

The 1998 paper of Cannon and Swenson [29] gave an initial approach to this conjecture by proving that the conjecture holds under an extra assumption that the family of standard "disks" in the boundary of the group satisfies a combinatorial "conformal" property. The main result of Cannon's 1994 paper [28] played a key role in the proof. This approach to Cannon's conjecture and related problems was pushed further later in the joint work of Cannon, Floyd and Parry. [30] [31] [32]

Cannon's conjecture motivated much of subsequent work by other mathematicians and to a substantial degree informed subsequent interaction between geometric group theory and the theory of analysis on metric spaces. [33] [34] [35] [36] [37] [38] Cannon's conjecture was motivated (see [29] ) by Thurston's Geometrization Conjecture and by trying to understand why in dimension three variable negative curvature can be promoted to constant negative curvature. Although the Geometrization conjecture was recently settled by Perelman, Cannon's conjecture remains wide open and is considered one of the key outstanding open problems in geometric group theory and geometric topology.

Applications to biology

The ideas of combinatorial conformal geometry that underlie Cannon's proof of the "combinatorial Riemann mapping theorem", [28] were applied by Cannon, Floyd and Parry (2000) to the study of large-scale growth patterns of biological organisms. [39] Cannon, Floyd and Parry produced a mathematical growth model which demonstrated that some systems determined by simple finite subdivision rules can results in objects (in their example, a tree trunk) whose large-scale form oscillates wildly over time even though the local subdivision laws remain the same. [39] Cannon, Floyd and Parry also applied their model to the analysis of the growth patterns of rat tissue. [39] They suggested that the "negatively curved" (or non-euclidean) nature of microscopic growth patterns of biological organisms is one of the key reasons why large-scale organisms do not look like crystals or polyhedral shapes but in fact in many cases resemble self-similar fractals. [39] In particular they suggested (see section 3.4 of [39] ) that such "negatively curved" local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue.

Selected publications

See also

Related Research Articles

<span class="mw-page-title-main">Differential topology</span> Branch of mathematics

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.

<span class="mw-page-title-main">William Thurston</span> American mathematician

William Paul Thurston was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.

In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

<span class="mw-page-title-main">Discrete geometry</span> Branch of geometry that studies combinatorial properties and constructive methods

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

<span class="mw-page-title-main">Geometric topology</span> Branch of mathematics studying (smooth) functions of manifolds

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

<span class="mw-page-title-main">Low-dimensional topology</span> Branch of 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 theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

<span class="mw-page-title-main">Geometric group theory</span> Area in mathematics devoted to the study of finitely generated groups

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.

<span class="mw-page-title-main">3-manifold</span> Mathematical space

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean 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 and close 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.

<span class="mw-page-title-main">Kleinian group</span> Discrete group of Möbius transformations

In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B3 in R3. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H3, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

<span class="mw-page-title-main">Hyperbolic group</span> Mathematical concept

In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.

<span class="mw-page-title-main">Dennis Sullivan</span> American mathematician (born 1941)

Dennis Parnell Sullivan is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a distinguished professor at Stony Brook University.

Brian Hayward Bowditch is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.

In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space by an arithmetic Kleinian group.

<span class="mw-page-title-main">William Floyd (mathematician)</span> American mathematician

William J. Floyd is an American mathematician specializing in topology. He is currently a professor at Virginia Polytechnic Institute and State University. Floyd received a PhD in mathematics from Princeton University 1978 under the direction of William Thurston.

<span class="mw-page-title-main">Mladen Bestvina</span> Croatian-American mathematician

Mladen Bestvina is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.

In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston (1982), states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.

<span class="mw-page-title-main">Finite subdivision rule</span> Way to divide polygon into smaller parts

In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.

In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those spaces.

References

  1. 1 2 3 Biographies of Candidates 2003. Notices of the American Mathematical Society, vol. 50 (2003), no. 8, pp. 973–986.
  2. 1 2 "Newsletter of the College of Physical and Mathematical Sciences" (PDF). Brigham Young University. February 2004. Archived from the original (PDF) on February 15, 2009. Retrieved September 20, 2008.
  3. 44 Years of Mathematics. Brigham Young University. Archived 2016-10-22 at the Wayback Machine Accessed July 25, 2013.
  4. The Mathematical Association of America's Earle Raymond Hedrick Lecturers. Mathematical Association of America. Accessed September 20, 2008.
  5. 2003 Election Results. Notices of the American Mathematical Society vol 51 (2004), no. 2, p. 269.
  6. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
  7. Math Professor to Give Lecture Wednesday at Y. Deseret News. February 18, 1993.
  8. Susan Easton Black.Expressions of Faith: Testimonies of Latter-Day Saint Scholars. Foundation for Ancient Research and Mormon Studies, 1996. ISBN   978-1-57345-091-1.
  9. J. W. Cannon, The recognition problem: what is a topological manifold? Bulletin of the American Mathematical Society, vol. 84 (1978), no. 5, pp. 832–866.
  10. 1 2 J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three. Annals of Mathematics (2), 110 (1979), no. 1, 83–112.
  11. 1 2 J. W. Cannon, J. L. Bryant and R. C. Lacher, The structure of generalized manifolds having nonmanifold set of trivial dimension. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 261–300, Academic Press, New York-London, 1979. ISBN   0-12-158860-2.
  12. Frank Quinn. Resolutions of homology manifolds, and the topological characterization of manifolds. Inventiones Mathematicae, vol. 72 (1983), no. 2, pp. 267–284.
  13. John Bryant, Steven Ferry, Washington Mio and Shmuel Weinberger, Topology of homology manifolds, Annals of Mathematics 143 (1996), pp. 435-467; MR 1394965
  14. 1 2 J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups. Geometriae Dedicata, vol. 16 (1984), no. 2, pp. 123–148.
  15. 1 2 M. Gromov, Hyperbolic Groups, in: "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.
  16. R. B. Sher, R. J. Daverman. Handbook of Geometric Topology. Elsevier, 2001. ISBN   978-0-444-82432-5; p. 299.
  17. 1 2 David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. Levy, Michael S. Paterson, William P. Thurston. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. ISBN   0-86720-244-0. Reviews: B. N. Apanasov, Zbl   0764.20017; Gilbert Baumslag, Bull. AMS, doi:10.1090/S0273-0979-1994-00481-1; D. E. Cohen, Bull LMS, doi:10.1112/blms/25.6.614; Richard M. Thomas, MR 1161694
  18. James W. Cannon. Almost convex groups. Geometriae Dedicata, vol. 22 (1987), no. 2, pp. 197–210.
  19. S. Hermiller and J. Meier, Measuring the tameness of almost convex groups. Transactions of the American Mathematical Society vol. 353 (2001), no. 3, pp. 943–962.
  20. S. Cleary and J. Taback, Thompson's group F is not almost convex. Journal of Algebra, vol. 270 (2003), no. 1, pp. 133–149.
  21. M. Elder and S. Hermiller, Minimal almost convexity. Journal of Group Theory, vol. 8 (2005), no. 2, pp. 239–266.
  22. J. W. Cannon and W. P. Thurston. Group invariant Peano curves. Archived 2008-04-05 at the Wayback Machine Geometry & Topology, vol. 11 (2007), pp. 1315–1355.
  23. Darryl McCullough, MR 2326947 (a review of: Cannon, James W.; Thurston, William P. 'Group invariant Peano curves'. Geom. Topol. 11 (2007), 1315–1355), MathSciNet; Quote::This influential paper dates from the mid-1980s. Indeed, preprint versions are referenced in more than 30 published articles, going back as early as 1990"
  24. Mahan Mitra. Cannon–Thurston maps for hyperbolic group extensions. Topology, vol. 37 (1998), no. 3, pp. 527–538.
  25. Mahan Mitra. Cannon–Thurston maps for trees of hyperbolic metric spaces. Journal of Differential Geometry, vol. 48 (1998), no. 1, pp. 135–164.
  26. Erica Klarreich, Semiconjugacies between Kleinian group actions on the Riemann sphere. American Journal of Mathematics, vol. 121 (1999), no. 5, 1031–1078.
  27. Brian Bowditch. The Cannon–Thurston map for punctured-surface groups. Mathematische Zeitschrift, vol. 255 (2007), no. 1, pp. 35–76.
  28. 1 2 3 James W. Cannon. The combinatorial Riemann mapping theorem. Acta Mathematica 173 (1994), no. 2, pp. 155–234.
  29. 1 2 3 4 J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension 3. Transactions of the American Mathematical Society 350 (1998), no. 2, pp. 809–849.
  30. J. W. Cannon, W. J. Floyd, W. R. Parry. Sufficiently rich families of planar rings. Annales Academiæ Scientiarium Fennicæ. Mathematica. vol. 24 (1999), no. 2, pp. 265–304.
  31. J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196.
  32. J. W. Cannon, W. J. Floyd, W. R. Parry. Expansion complexes for finite subdivision rules. I. Conformal Geometry and Dynamics, vol. 10 (2006), pp. 63–99.
  33. M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry. In: Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1–17, Springer, Berlin, 2002; ISBN   3-540-43243-4.
  34. Mario Bonk and Bruce Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geometry & Topology, vol. 9 (2005), pp. 219–246.
  35. Mario Bonk, Quasiconformal geometry of fractals. International Congress of Mathematicians. Vol. II, pp. 1349–1373, Eur. Math. Soc., Zürich, 2006; ISBN   978-3-03719-022-7.
  36. S. Keith, T. Laakso, Conformal Assouad dimension and modulus. Geometric and Functional Analysis, vol 14 (2004), no. 6, pp. 1278–1321.
  37. I. Mineyev, Metric conformal structures and hyperbolic dimension. Conformal Geometry and Dynamics, vol. 11 (2007), pp. 137–163.
  38. Bruce Kleiner, The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity. International Congress of Mathematicians. Vol. II, pp. 743–768, Eur. Math. Soc., Zürich, 2006. ISBN   978-3-03719-022-7.
  39. 1 2 3 4 5 J. W. Cannon, W. Floyd and W. Parry. Crystal growth, biological cell growth and geometry. Pattern Formation in Biology, Vision and Dynamics, pp. 65–82. World Scientific, 2000. ISBN   981-02-3792-8, ISBN   978-981-02-3792-9.