Conference Agenda

The purpose of the Minisymposium "From Algebraic Geometry to Geometric Topology: Crossroads on Applications" is to bring together researchers who use algebraic, combinatorial and geometric topology in industrial and applied mathematics. These methods have already seen applications in: biology, physics, chemistry, fluid dynamics, distributed computing, robotics, neural networks and data analysis.

Reconnection is a fundamental event in many areas of science, including the interaction of vortices in classical and quantum fluids, magnetic flux tubes in magnetohydrodynamics and plasma physics, and site-specific recombination in DNA. The helicity of a collection of flux tubes can be calculated in terms of writhe, twist and linking among tubes. We discuss that the writhe helicity is conserved under anti-parallel reconnection [1]. We will discuss the mathematical similarities between reconnection events in biology and physics, and the relationship between iterated reconnection and curve topology. We will discuss helicity and reconnection in a tangle of confined vortex circles in a superfluid.

[1] Laing C.E., Ricca R.L. & Sumners D.W. (2015), Conservation of writhe helicity under anti-parallel reconnection, Nature Scientific Reports 5:9224/ DOI: 10.1038/srep09224.

The following is known as the geometric hypothesis of Banach: let V be an m-dimensional Banach space with unit ball B and suppose all n-dimensional subspaces of V are isometric (all the n-sections of B are affinely equivalent). In 1932, Banach conjectured that under this hypothesis V is isometric to a Hilbert space (the boundary of B is an ellipsoid). Gromov proved in 1967 that the conjecture is true for n=even and Dvoretzky derived the same conclusion under the hypothesis n=infinity. We prove this conjecture for n=5 and 9 and give partial results for an integer n of the form 4k+1.

The ingredients of the proof are classical homotopic theory, irreducible representations of the orthogonal group and geometric TOMOGRAPHY. Suppose B is an (n+1)-dimensional convex body with the property that all its n-sections through the origin are affinity equivalent to a fixed n-dimensional body K. Using the characteristic map of the tangent vector bundle to the n-sphere, it is possible to prove that if n=even, then K must be a ball and using homotopical properties of the irreducible subgroups of SO(5) and SO(9), we prove that if N=5,9, then K must be a body of revolution. Finally, we prove, using geometry tomography and topology that, if this is the case, then there must be a section of B which is an ellipsoid and consequently B must be also an ellipsoid.

We introduce a generalisation of the well-established Cucker-Smale model to complete Riemannian manifolds to study the influence of geometry and topology on the formation of flocks. The dynamics of the Cucker-Smale model facilitate the flocking of a group of particles in disordered motion into a coordinated one where all particle move parallelly with the center of mass. Despite their name, flocking models do not only illustrate the herding of animals but more generally the emergence of collective behaviour. The possible applications cover a broad spectrum of subjects such as linguistics, biology, opinion formation, sensor networks and robotics.

While the Cucker-Smale model already received much attention over the last decade, those efforts focused on particles moving in a Euclidean space. Chi, Choi and Ha raised the flocking realizability problem: Given a manifold and a group of particles, construct a dynamical system that leads to a collective movement as a flock at least asymptotically. We establish theorems about the convergence of the particles to a flocked state under the dynamics of our generalized model. Not only does this address the flocking realizability problem but it also lays the groundwork for further investigations of topological and geometric effects on the dynamics. As an example, we already established that the presence of curvature restricts the final flocked state into specific patterns and we are looking forward to further investigations in this direction. This is joint work done with S.-Y. Ha & D. Kim (Seoul National U., Seoul).

Local reconnection events are common in nature. One example is the action of recombination enzymes, and in particular of site-specific recombinases that recognize two short segments of DNA (the recombination sites), introduce two double-stranded breaks and recombine the ends. The local action of site-specific recombinases is a reconnection event which is modeled mathematically as a band surgery. The banding can be coherent or non-coherent, depending on the relative orientation of the recombination sites. Motivated by the unlinking of circular chromosomes after DNA replication, we have done extensive studies of coherent banding. In this talk I focus on more recent work that deals with non-coherent bandings. We use tools from low dimensional topology to investigate local reconnection between two sites in inverted repeats along a knot. We complement the analytical work with computer simulations. The numerical work provides a quantitative measure to distinguish among pathways of topology simplification by local reconnection, and also informs the search for bandings between specific pairs of knot or link types.

This is joint work with Allison Moore, Tye Lidman, Michelle Flanner and Koya Shimokaw.