# Aspects of hyperbolicity in geometry, topology and dynamics

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

We will hold a three day workshop at the University of Warwickentitled ``Aspects of hyperbolicity in geometry, topology and dynamics''.We have received commitments from 11 internationally respected researchers, andintend to invite an additional 4. Workshop time will be divided between focuseddiscussion groups and research talks on cutting edge topics.We expect this interaction to generate new directions of research and inspirefuture research proposals. The workshop will include a poster session, an exhibition ofmathematical images and a celebration (funded by the Warwick Mathematics Instituteand participants) in honour of Professor Caroline Series' 60th birthday.Geometry has a history going back several millennia. Traditionally geometers studiedeuclidean or ``flat'' geometry where the angles of any triangle sum to 180 degrees.In modern times other geometries have become of great interest to scientistsand mathematicians. Notably, in spherical and hyperbolic geometry the angles ofany triangle sum to more than or less than 180 degrees respectively.Of these modern geometries, hyperbolic is the richest, and arises naturally inmany different contexts.Topology, one of the great triumphs of 20th century mathematics,is the study of shapes without regard to distance or angle.From the topological point of view, a sphere and cube are same;from the geometric point of view they are distinct: the spherehas much greater symmetry. One of the great insights of thelast 30 years has been that the geometry of greatest symmetryfor most 3-dimensional shapes is hyperbolic geometry.Dynamics studies how systems evolve with time, a classicalexample being the motion of the planets. Celestial mechanicsin particular gives rise to intractable chaotic systems.However there are much more tractable systems arising out ofhyperbolic geometry, due to the rapid divergence of straight lines.

### Planned Impact

Our workshop will be theoretical in nature. Beneficiaries will include: - Attendees due to increased collaboration and publication - Mathematicians working in the fields of low-dimensional topology, geometric group theory, dynamics and Teichmuller theory. - The UK mathematical community due to exposure to the invited experts - Graduate students who will be helped to advance their own research programmes The subject also has broader implications which we outline below. Many practical problems can be interpreted in terms of configuration spaces of various objects. One much studied example of this is in robotics, where one is interested in analysing the topology and geometry of possible configurations of complex mechanical structures. In many cases, these are determined by polynomial constraints, and give rise to spaces of a similar type to the representation varieties we describe here. As discussed more fully in the case for support the dynamics of Markoff triples arise in several contexts; among these are applications of interest to physicists. For example, it is closely linked to the spectra of quasi-periodic Schrodinger operator and to the magnetic Laplacian. Of course, all the topics listed fit into a much broader picture. Aspects of topology are playing an ever increasing role in biology, dynamic systems in climate modelling and so on. While we will not be dealing specifically with these applied topics, a deeper understanding of foundational issues is always of value. The case for support additionally refers to visualisations of various structures associated to representation varieties and dynamical systems. Many of our expected participants have particular expertise in this. The use of computer graphics is now ubiquitous, and it is hoped that the interaction at out meeting will help to stimulate the development of new techniques. In particular the workshop includes an exhibition of mathematical images open to the public.