Knots and Applications
In collaboration with INdAM and the Dipartimento di Matematica e Applicazioni, Milano Bicocca
1 May 2011 - 31 July 2011Aims
Program description:
In recent years there has been a growing interest for "applicable" aspects of classical knot theory in 3D, motivated by the discovery of new concepts and techniques with a strong geometric flavor, both in pure and applied mathematics. This, in turn, has given new impetus to such disparate areas as geometric topology, singularity theory, geometric dynamics, topological fluid mechanics, topological quantum field theory, and others. Meanwhile, continuous progress in computational techniques and numerical simulation on the one hand, and applied sciences such as polymer physics, DNA biology, neural networks, etc., on the other, have benefited from these developments.
New results on the groundstate energy of physical knots and links, for example, show that local and global geometric aspects, topological classification and complexity issues, including framing, chirality, etc., are all important issues in the development of a topological field theory. There is therefore a call for more work towards a "geometric" knot theory, accompanied by advanced numerical modeling of tight/ideal knots and numerical simulations of structural complexity issues.
This intensive research period is devoted to explore analytical, numerical and applied aspects of geometric knot theory, with particular emphasis on the following topics: Theory: aspects of hyperbolic geometry, classification issues, differential geometric aspects of knots and links, analytical and parametric representation of knots, links and braids, tubular knots, invariants for applications, energy functionals, chirality issues, Seifert surfaces. Numerics: implementation of energy functionals, local/global minimum, tightening and ideal shapes, role of framing and chirality, twist and satellites. Applications: magnetic fields, elastic systems, energy quantization, quantum entanglement, DNA biology.
Advisory Board:
L.H. Kauffman (U. Illinois at Chicago, USA)
E-mail: kauffman@uic.edu
K.C. Millett (U. California at Santa Barbara, USA)
E-mail: millett@math.ucsb.edu
C. Petronio (U. Pisa, Italy)
E-mail: petronio@dm.unipi.it
R.L. Ricca (U. Milano-Bicocca, Italy)
E-mail: renzo.ricca@unimib.it
D. Rolfsen (U. British Columbia, Canada)
E-mail: rolfsen@math.ubc.ca
D.W. Sumners (Florida State U., USA)
E-mail: sumners@math.fsu.edu
Foundations:
Kauffman, L.H., On Knots. Annals of Mathematics Studies 115, Princeton University Press (1987), Princeton NJ.
Rolfsen, D., Knots and Links. AMS Chelsea Publishing (2003), Providence RI.
Applications:
Calvo, J.A., Millett, K.C., Rawdon, E.J. & Stasiak, A. (Eds.) Physical and Numerical Models in Knot Theory. Series on Knots and Everything 36, World Scientific (2005), Singapore.
Millett, K.C. & Sumners, D.W. (Eds.) Random Knotting and Linking. Series on Knots and Everything 7, World Scientific (1994), Singapore. Ricca, R.L. (Ed.) Lectures on Topological Fluid Mechanics. Lecture Notes in Mathematics 1973, Springer-Verlag (2009), Heidelberg.