Abstracts

**Dorothy Buck**
Three-Manifolds for DNA

The central axis of the famous DNA double helix is typicially topologically constrained or even circular, so that the DNA molecule can be modelled as a ribbon knot. The shape of this axis can influence which proteins interact with the underlying DNA. I will give an overview of some of the methods from 3-manifold topology that are used to model both these DNA molecules and a variety of DNA-protein reactions. We'll conclude with a few examples showing how the answers from these models aid biologists.

Knots through the lens of 3-manifold topology
The subject of 3-manifold topology in mathematics gives a rather beautiful solution to many problems in knot theory, one of them being the isotopy problem -- determining if one knot can be deformed to another. I'll describe the solution to this problem coming from the geometrization of 3-manifolds

Knotting nodes of light
Optical beams propagating in three-dimensional free space are complex scalar fields, and typically contain nodal lines (optical vortices) which may be thought of as generalized interference fringes and topological singularities of phase. Optical vortices were originally discovered by Hans Wolter in 1950 in his study of total internal reflection, and are now a major area of study of classical light fields, appearing ubiquitously in spatially varying wave functions. Random wave fields, representing optical speckle patterns scattered from rough surfaces, have a tangled skeleton of nodal lines, some of which are closed loops, and others are infinite, open lines. Computer simulations of random superpositions of plane waves indicate that these lines have the fractal properties of brownian random walks with characteristic scaling of the probability that pairs of loops are linked together. Holographically-controlled laser beams provide the opportunity to control the form of optical fields and the nodal lines within them. Using the theory of fibred knots, I will describe the design of superpositions of laser modes (effectively solutions of the 2+1 Schrödinger equation) which contain isolated knots and links. I will conclude by explaining how these mathematical fields have been experimentally realized by the experimental Optics group in Glasgow, UK.

Linked and Knotted fields: Light and Hydrodynamics
To tie a shoelace into a knot is a relatively simple affair. Tying a knot in a field is a different story, because the whole of space must be filled in a way that matches the knot being tied at the core. The possibility of such localized knottedness in a space-filling field has fascinated physicists and mathematicians ever since Kelvin’s 'vortex atom' hypothesis, in which the atoms of the periodic table were hypothesized to correspond to closed vortex loops of different knot types. An intriguing physical manifestation of the interplay between knots and fields is the possibility of having knotted dynamical excitations. I will discuss some remarkably intricate and stable topological structures that can exist in light fields whose evolution is governed entirely by the geometric structure of the field. A special solution based on a structure known as a Robinson Congruence that was re-discovered in different contexts will serve as a basis for the discussion. I will then turn to hydrodynamics and discuss topologically non-trivial vortex configurations in fluids.