
# Geometry and Topology Seminar

## Warwick Mathematics Institute, Term III, 2016-2017

 Thursday April 27, 15:00, room MS.01 Richard Schwartz (Brown, INI) Uncooperative PETs Abstract: This is a talk about work in progress. I'll start with a one-parameter family of PETs in the plane, which were inspired by my attempts to stop working on outer billiards. (PET stands for polytope exchange transformation.) These PETs produce beautiful recursive tilings of the plane which have been driving me crazy for some time. The PETs have four-dimensional compactifications which are also PETs, and these 4D PETs in turn have enhancements which are not so much classical dynamical systems as pseudo-group actions. I'll try to describe the whole odyssey, which goes from outer billiards on semi-regular octagons to the famous 24-cell, a 4D platonic solid, and I'll illustrate everything with computer demos. I'll also state at least one fully proved nontrivial theorem, so you can see that I am not just lost in a labyrinth of unproved experimental observations.

 Thursday May 4, 15:00, room MS.04 Marc Lackenby (Oxford) The complexity of unknot recognition Abstract: I will spend some time talking about the known algorithms to recognise the unknot. I will explain why there is little hope that they can be made to run in anything faster than exponential time. I'll then describe a new unknot recognition algorithm and will explain why we might be able to make it run faster.

 Thursday May 11, 15:00, room MS.04 Lars Louder (UCL) Generalized Freiheitssatz Abstract: Lyndon's theorem that a commutator in a free group is not a proper power, Magnus' Freiheitssatz, Wise's W-cycles conjecture, Stallings' theorem that injections of free groups inducing injections on abelianization are injections on conjugacy classes, and Baumslag's theorem on adjoining roots to subgroups of free groups are all special cases of a generalized Freiheitssatz. The simplest new case covered resembles the Borromean rings. This is joint work with Henry Wilton.

 Thursday May 18, 15:00, room MS.04 None (NA) NA Abstract: NA

 Thursday May 25, 15:00, room MS.04 None (NA) NA Abstract: NA

 Thursday June 1, 13:00, room PS1.28 Elisabetta Matsumoto (Georgia Tech) Non-euclidean virtual reality Abstract: Joint work with Vi Hart, Andrea Hawksley, and Henry Segerman. The properties of euclidean space seem natural and obvious to us, to the point that it took mathematicians over two thousand years to see an alternative to Euclid's parallel postulate. The eventual discovery of hyperbolic geometry in the 19th century shook our assumptions, revealing just how strongly our native experience of the world blinded us from consistent alternatives, even in a field that many see as purely theoretical. Non-euclidean spaces are still seen as unintuitive and exotic, but with direct immersive experiences we can get a better intuitive feel for them. The latest wave of virtual reality hardware, in particular the HTC Vive, tracks both the orientation and the position of the headset within a room-sized volume, allowing for such an experience. We use this nacent technology to explore the three-dimensional geometries of the Thurston/Perelman geometrisation theorem. This talk focusses on our simulations of $\HH^3$ and $\HH^2 \cross \EE$.

 Thursday June 1, 15:00, room MS.04 Alex Bartel and Aurel Page (Warwick) Can you hear torsion homology?... and group representations in the homology of 3-manifolds Abstract: Isospectral manifolds have many properties in common, and it is therefore natural to ask which invariants of Riemannian manifolds are isospectral invariants. Specialising to homological invariants: strongly isospectral manifolds have the same Betti numbers, and the Cheeger-Mueller formula implies that a certain alternating product of torsion homology and regulators must be identical for such manifolds. Do the individual torsion subgroups of homology groups have to be the same? In the first part of this talk, we will introduce representation-theoretic tools to study this question. This naturally leads to the question of which representations of a finite group $G$ can be realised as a homology group of a manifold with an action of $G$. In the second part of the talk, we will present a method based on surgery to solve this problem for 3-manifolds.

 Thursday June 8, 15:00, room MS.04 None (NA) NA Abstract: NA

 Thursday June 15, 15:00, room MS.04 Federica Fanoni (Max Planck) Curve graphs for infinite-type surfaces Abstract: There are various graphs associated to surfaces (of finite topological type), constructed using curves or arcs, which have been very useful in the study of Teichmueller space (the space of hyperbolic structures on a surface) and of the mapping class group. If the surface has infinite topological type (e.g. it has infinite genus), these graphs turn out to be not as interesting. I will discuss why and present an alternative construction which gives graphs with better properties. Joint work with Matt Durham and Nick Vlamis.

 Thursday June 22, 15:00, room MS.04 Derek Holt (Warwick) A new method for verifying the hyperbolicity of a finitely presented group Abstract: A finitely presented group is called hyperbolic if geodesic triangles in its Cayley graph are uniformly thin or, equivalently, if its Dehn function is linear. Although hyperbolicity of a group defined by a finite presentation is undecidable in general, the programs in the author's KBMAG package, which is implemented in GAP and in Magma, can verify hyperbolicity when the property holds. In this talk we describe new methods for proving hyperbolicity and for estimating the Dehn function that are based on small cancellation theory and the analysis of the curvature of van Kampen diagrams for the group. There is a GAP implementation by Marcus Pfeiffer and also a recent Magma implementation by the author. These methods are due to Richard Parker and many others. They have the disadvantage that they are not guaranteed to succeed on every hyperbolic group presentation, but when they do they are generally much faster than KBMAG. They can also sometimes be carried out by hand, and applied to infinite families of group presentations.

 Thursday June 22, 16:00, room MS.04 Mark Bell (UIUC) Trisections of 4-manifolds Abstract: We will discuss a decomposition of 4-manifolds by Gay and Kirby, known as a trisection, that is analogous to Heegaard splittings of 3-manifolds. In particular, we will look at a new upper bound on the genus of a trisection coming from the combinatorics of the manifold. This is joint work with Joel Hass and Hyam Rubinstein.

 Thursday June 29, 15:00, room MS.04 None (NA) NA Abstract: NA

Information on past talks. This line was last edited Wednesday, June 21, 2017 10:03:41 AM BST.