Contacts
Department of Mathematics & Applications
University of Milano - Bicocca
Via Cozzi, 55
20125 Milano, Italy

Building U5, III floor, Room 3065

Email: renzo.ricca@unimib.it
Direct line: (+39) 02 6448 5762
Department: (+39) 02 6448 5755
Fax: (+39) 02 6448 5705

Doctoral Courses (PhD, III level)

Physical Applications of Knot Theory (Torino, 2005-2008)

Course contents (20 hours):
  1. Introduction to space curves and knot theory
    • Fundamentals of space curves
    • Global geometric aspects of space curves
    • Knots, links and projections
    • Gauss linking number
    • Calugareanu-White invariant
    • Measures of structural complexity
    • Articles included:
      • By Ricca, R.L. (2005) on Knot Theory. Structural complexity.
      • By Hoste, J., Thistlethwaite M. & Weeks, J. (1998) on The first 1,701,936 knots
  2. Topological equivalence classes and change of topology
    • Topological equivalence classes for frozen fields
    • Change of topology
    • Measuring structural complexity: a test case
    • Articles included:
      • By Ricca, R.L. & Berger, M.A. (1996) on Topological ideas and fluid mechanics
      • By Ricca, R.L. (1998) on Applications of knot theory in fluid mechanics
  3. Vortex dynamics, knots and links
    • Elements of vortex dynamics
    • Vortex knots and links
    • Conservation laws and topology
    • Articles included:
      • By Moffatt, H.K. (1969) on The degree of knottedness of tangled vortex lines
      • By Arnold, V.I. (1974) The asymptotic Hopf invariant and its application
  4. Ideal magnetohydrodynamics of knotted and braided flux-tubes
    • Ideal magnetohydrodynamics and analogous Euler's flows
    • Lorentz force and inflexional disequilibrium of flux-tubes
    • Magnetic relaxation under topological constraints
    • Articles included:
      • By Moffatt, H.K. & Ricca, R.L. (1992) on Helicity and the Calugareanu invariant
      • By Freedman, M.H. (1988) A note on topology and magnetic energy in incompressible perfectly conducting fluids
  5. Magneto-elastic relaxation of braids and strings
    • Topological bounds on magnetic braids.
    • Elastic relaxation and supercoiled strings.
    • Elastic energy and knot type.
    • Articles included:
      • By Berger (1993) on Energy-crossing number relations for braided magnetic fields
      • By Katritch et al. (1996) on Geometry and physics of knots
      • By Sumners (1995) on Lifting the curtain: using topology to probe the hidden action of enzymes