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Topological Dynamics in the Physical and Biological Sciences

16th July 2012 to 21st December 2012

Organisers: Konrad Bajer (Warsaw), Tom Kephart (Vanderbilt), Yoshi Kimura (Nagoya), Keith Moffatt (Cambridge) and Andrzej Stasiak (Lausanne)

Scientific Advisory Committee: Jason Cantarella (Georgia), Andrew Gilbert (Exeter), Raymond Goldstein (Cambridge), Boris Khesin (Toronto), Shigeo Kida (Kyoto), Mikhail Monastyrski (Moscow), Sergey Nazarenko (Warwick), Wilma Olsen (Rutgers), Renzo Ricca (Milan), De Witt Sumners (Florida State), Lynn Zechiedrich (Houston, USA)

Programme Theme

The programme is intended to stimulate interaction between applied mathematicians, biologists and physicists who frequently encounter dynamical problems that have some explicit or implicit topological content. We use the term 'topological' to convey the idea of structures, e.g. knots, links or braids in 3D, that exhibit some measure of invariance under continuous deformation. Dynamical evolution is then subject to the topological constraints that express this invariance. A basic common problem is to determine minimum energy structures (and routes towards these structures) permitted by such constraints; and to explore mechanisms, e.g.diffusive, by which such constraints may be broken.

When formulated in terms of the mathematical objects and issues, the current view of the common topological denominator is summarised below. We expect to add to this list during the Programme.

Tubes in R3

Surfaces in R3

Lines in R3

Lines on manifolds

Singularities

  • Knotted flux tubes in MHD (helicity, minimisation problems, topological aspects of dynamo theory)
  • Knots on vortex tubes (analogies and differences with MHD, dynamical implications of knottedness)
  • Knotted molecules (DNA, proteins, minimisation of drag on a knotted tubular structure)
  • Dynamics of protein folding, coiling and supercoiling (folding of knotted proteins)
  • Spectrum of knots and links under different minimisation principles
  • Biological membranes (minimisation problems, implications of topology for phase transitions)
  • Fermi surfaces (electron trajectories on Fermi surfaces, topological phase transitions)
  • Knots and links and braids in streamlines and magnetic field lines (decay of topological invariants in the ideal medium limit)
  • Quantum vortices in super bundles, topological implications on statistical mechanics)
  • Quantum implications of classical topological constrains
  • Application of knot theory to statistical mechanics
  • Electron trajectories on Fermi surfaces of complex topology
  • Formation of singularities in the Navier-Stokes and Euler equations
  • Topological changes in parametric evolution of magnetic fields (singularity formation under footpoint shuffling)
Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons