Buck 9706865 The investigator collaborates with Prof. J. Simon, University of Iowa, to study knots and their applications. Knots are closed loops in space that may be tangled in complicated and essential ways. While knots usually have been studied as idealized one dimensional filaments, the focus here is on "physical knots," knots made of real physical stuff, from rope to DNA or other large flexible molecules. These are modeled as mathematical knots endowed with physical like properties such as self repelling energy or thickness. The twin goals of physical knot theory are to mathematically model and help understand the real physical systems, and to use the physically inspired measures of knot-complexity to develop novel methods for knot recognition and classification. The specific targets of this project include: determining precise relations between different measures of knot complexity, understanding critical points and the configuration space of polygonal knots, understanding the connections between parametrizations and topological properties of harmonic knots, extending knot energy ideas to other structures such as graphs, modeling gel electrophoresis of DNA loops, and using knot energies to help understand physics phenomena such as self-radiating tubes and knotted vortices. Knotting and tangling happen at every scale studied by science, from microscopic DNA loops to everyday rope to tangled magnetic field loops in the solar corona. The investigators and their students will study problems that are fundamental and arise in all the physical systems: How are knots and tangles created? What mathematical properties of the physical systems lead to knots becoming simplified or completely untangled? How do the mathematical properties of different kinds of knots influence their physical behavior? One focus task is to gain a better understanding of gel electrophoresis of DNA loops, hence of the gel process itself; this is one of the most important b iotechnology techniques, so a solid understanding is important. The project requires powerful computing and visualization, so it will train students, as well as stretch the technology, in these areas.