Alex Clark

 


Professor


 

Office College House 108

Phone (0116) 252 5670

Alex.Clark_AT_le.ac.uk


Alex

Research

Link to Research Related International Network

Research Interests

  • Minimal Sets of Foliations and Flows Recently I have been investigating the role of matchbox manifolds as minimal sets of foliations.  A matchbox manifold is a space that is locally the product of an open set of some Euclidean space of a fixed dimension and a totally disconnected space. A particularly important class of matchbox manifolds consists of solenoids. A solenoid is the total space of a fiber bundle projection onto a closed manifold with a profinite structure group. These are natural generalizations of covering spaces and arise naturally in foliations.  In  Embedding solenoids in foliations we show how these solenoids can be embedded as minimal sets of smooth foliations and that they occur precisely in those settings to which Thurston stability does not apply. One of the intriguing suggestions of this research is that solenoids are quite common minimal sets of foliations.  In Homogeneous matchbox manifolds we establish that all homogeneous matchbox manifolds with a smooth structure are solenoids. This brings to a natural conclusion Bing's conjectured characterization of one-dimensional solenoids and at the same time makes an important connection between the dynamic property of equicontinuity of foliated spaces and the purely topological property of homogeneity.  It follows from this work that any equicontinuous matchbox manifold is a solenoid. In Classification of matchbox manifolds we determine the extent to which we can classify  solenoids by the dynamics of the associated pseudogroup, which draws an interesting connection with the pro-homotopy groups of the solenoids and generalized versions of the Borel Conjecture. With Robbert Fokkink we explored the embeddings of solenoids in Euclidean space and continuous foliations and the bihomogeneity of solenoids, and together with Olga Lukina we  investigated the structure of the ends of the leaves of matchbox manifolds in The Schreier continuum and ends.

  • Tiling Spaces Tiling spaces are another natural class of matchbox manifolds, but are fundamentally different from solenoids in that they cannot be equicontinuous . These spaces are constructed by taking the closure of the orbit of an individual tiling under the action of some group of isometries. All Euclidean translational tiling spaces fiber over a torus with a Cantor fiber, but the fiber bundle is not principal, unlike the case of homogeneous solenoids.  With Lorenzo Sadun we have drawn ties between the cohomology and dynamics of tiling spaces. In particular in the paper When shape matters: deformations of tiling spaces we use cohomology to determine when the dynamics of a tiling space are sensitive to changes in the size and shape of tiles in the tiling underlying the tiling space. With John Hunton we have characterized tiling spaces that occur as codimension one attractors of smooth diffeomorphisms of manifolds.

  • Linking of Minimal Sets of Flows With Michael Sullivan we developed a purely topological way of measuring the linking of one-dimensional minimal sets of flows and applied these techniques to aperiodic minimal sets of the Lorenz  template in The linking homomorphism of one-dimensional minimal sets.

  • Affine Maps of Compact Abelian Groups Rigidity and Ergodic Properties with Robbert Fokkink



 

Selected Publications and Preprints

 

Links

Recent Grants

Ph D Students

  • Ahmed Al-Hindawe
  • Petra Staynova
  • Dina Abuzaid; graduated 2016, Lecturer at King Abdulaziz University
  • Dan Rust (Contains a link to download Grout), graduated 2016, Currently at University of Bielefeld
  • James Walton ; Blog explaining thesis (first 3 years supervised by John Hunton) Currently at York University
  • Sheila McCann, graduated August 2013, currently an RA in Leicester

Teaching and Seminars

Autumn 2015

Investigations in Mathematics, Second Year Module

  • Ph D Seminar for Current Students

  • Pure Maths Seminar


    Complete List of Publications and Preprints

     

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    Contact details

    Department of Mathematics
    University of Leicester
    University Road
    Leicester LE1 7RH
    United Kingdom

    Tel.: +44 (0)116 252 3917
    Fax: +44 (0)116 252 3915

    Campus Based Courses

    Undergraduate: mathsug@le.ac.uk
    Postgraduate Taught: mathspg@le.ac.uk

    Postgraduate Research: pgrmeccluster@le.ac.uk

    Distance Learning Course  

    Actuarial Science: actsciadmin@le.ac.uk