9705005 Calini This project concerns the connection between completely integrable partial differential equations and the topological properties of knotted closed curves which arise from models of vortex filament evolution. The principal investigator plans to show that questions regarding knot types, mechanisms for knotting and unknotting, stability of knot formations and classification of knots by means of standard representatives can be effectively addressed in this context. The two main tools that will be used are the periodic theory of relevant integrable equations (among which are the Continuous Heisenberg Model, the focusing Nonlinear Schroedinger and the sine- Gordon equations) with periodic boundary conditions, and the theory of Backlund transformations. The PI will use constructive methods for multiphase solutions to generate large classes of knot representatives and will study a precise relation between their knot type and the associated Floquet spectrum both theoretically and computationally. Backlund transformations will be used to investigate possible mechanisms for topological changes, to construct self-intersecting curves that represent transitions between different knot types and to produce examples of knots that are realized by curves with special properties (such as curves of constant torsion). Complex structures in the form of knotted and linked loops are present in many phenomena of the physical world: tornadoes, plasma loops and magnetic arches in stellar atmospheres display complicated vortex formations; mitochondrial DNA is a coiled, often knotted, molecule; bacteria strands are found to form tangled loops; links and knots are observed in certain stable mixtures of chemical media. This project concerns the connection between evolution equations whose structure is well-understood and the properties of knotted loops which arise in the study of vortex filament dynamics and DNA modeling. The principal goal of this inve stigation is the development of the mathematical tools necessary to effectively address questions regarding the mechanisms for knotting and unknotting, the stability of knot formations and the classification of knots and links. Such issues are gaining great importance in a number of applied fields: for example, an understanding of the complex vortex structures in the solar crown can provide information on how the sun's magnetic activity affects the earth's climate, while topological changes such as loop creation, knotting and unknotting appear to be at the heart of the replication mechanism of the DNA molecule, a fundamental question in cancer research.
