Geometry and Topology
Workshop

All talks in Room G.207 - Alan Turing Building

15:30 Jonathan Pridham
University of Edinburgh
Shifted Poisson structures in derived geometry
Shifted Poisson structures have appeared in many guises over the past decades. I will try to give an overview and some examples from the perspective of derived geometry.

16:10 Kasia Rejzner
University of York
Quantization of gauge theories in (perturbative) algebraic quantum field theory
In this talk I will summarise some key results in quantization of gauge theories using the perturbative algebraic quantum field theory (pAQFT) framework. This framework allows for a rigorous construction of interacting QFT models using methods of deformation quantization and homological algebra. More recent result include treatment of boundaries and a non-perturbative formulation using C*-algebras.

16:50 Andrey Lazarev
University of Lancaster
Coderived categories and their applications
Derived categories of rings, algebraic varieties and, more generally, abelian categories, have been much studied in the past sixty years or so after their introduction by Grothendieck and Verdier. The construction of a derived category is based on the localization of the category of complexes (say, of abelian groups or R-modules for a ring R) by quasi-isomorphisms. The notion of a coderived category is more recent and the latter is constructed by localizing complexes by a class of morphisms more restrictive than quasi-isomorphisms. For some abelian categories (such as comodules over a coassociative coalgebra) the notion of a coderived category is every bit as canonical as that of a derived category for modules over a ring while for others there are several competing notions each of which has their advantages and downsides. The main application of coderived categories is a differential graded version of Koszul duality as developed by Leonid Positselski.

In this talk I will describe some examples of coderived categories related to comodules over the singular chain algebra of a topological space, to modules over the de Rham algebra of a smooth manifold or the Dolbeault algebra of a complex analytic manifold and relate them to more familiar categories arising in algebraic topology and differential geometry. The talk is based on joint work with J. Chuang and J. Holstein.

11:30 Mark Hagen
University of Bristol
Translation length spectra, injective spaces, and quasimorphisms
Many well-known finitely generated groups contain distorted infinite cyclic subgroups (Baumslag-Solitar groups, and, say, the integer Heisenberg group provide examples). On the other hand, many of the classes of groups defined in such a way as to exhibit features of nonpositive curvature have the property that their Z subgroups are quasi-isometrically embedded: CAT(0) groups, hyperbolic groups, etc. have this feature. By considering (stable) translation length, one can search for examples showing subtler behaviour, and Conner exhibited a group G whose Z subgroups are undistorted, but which has elements of arbitrarily small translation length on the Cayley graph. At the other extreme, we have translation-discrete groups, in which there is a uniform lower bound on the translation length of infinite order elements. In this talk, I will discuss the class of groups known to act properly and coboundedly on injective spaces, and how to use the barycentre construction in such spaces to prove translation-discreteness. In the special case of hierarchically hyperbolic groups, this result has an application to uniform exponential growth (via work of Abbott-Spriano-Ng), and there is a conjectural relationship to bounded cohomology. This is all joint work with Carolyn Abbott, Harry Petyt, and Abdul Zalloum.

13:45 Irene Pasquinelli
University of Bristol
How many times can two curves on a surface intersect?
Given a surface, one might want to understand how "complex" the surface is, in terms of curves. More specifically, we may ask how many times two curves on this surface can intersect. Of course, longer curves might intersect more times. KVol is a quantity measuring how many times curves can intersect, modulo their length. We will give an overview of some cases for which this quantity has been calculated, with particular focus on Veech surfaces, a class of flat surfaces with a rich group of symmetries.

This is joint work in progress with Julien Boulanger.

14:25 Mehdi Yazdi
King's College London
The algorithmic complexity of incompressible surfaces in 3-manifolds
A two-sided properly embedded surface in a compact 3-manifold is called incompressible if its fundamental group is injectively included in that of the ambient 3-manifold. Irreducible 3-Manifolds that contain an incompressible surface are called Haken. Many important results in 3-manifold topology and geometry were first obtained for Haken 3-manifolds, such as solution to the homeomorphism problem and Thurston's hyperbolisation theorem. We consider the algorithmic complexity of the following natural decision problem: given a triangulated closed orientable 3-manifold M and an integer g in binary, does M contain a closed incompressible surface of genus g? We show that the above decision problem is NP-complete. This is joint work with Marc Lackenby and Eric Sedgwick.

11:30 Timothy Logvinenko
Cardiff University
Categorification of generalised braids

11:50 Yongsheng Jia
University of Manchester
GGT meets symplectic geometry

12:10 Alex Levine
University of Manchester
Subsets of groups and context-free languages

14:00 Fraser Sanders
University of Manchester
Reduction by symmetries in symplectic and contact manifolds

14:20 Joe Parkin
University of Manchester
Tori in Groups of Lie Type

14:40 James Dolan
University of Liverpool
Lattice Angles in Lattice Polygons