Injectivity radius of representations of triangle groups and planar width of regular hypermaps

We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type with a given planar width

Fusion systems on bicyclic 2-groups

We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups (see [Craven-Glesser, 2012] and [Sambale, 2012]). As an application we prove Olsson's Conjecture for all blocks with bicyclic defect groups.Comment: 22 pages, shorted and some arguments replace

Buildings, Group Homology and Lattices

This is the author's PhD thesis, published at the Universit\"at M\"unster, Germany in 2010. It contains a detailed description of the results of arXiv:0903.1989, arXiv:0905.0071 and arXiv:0908.2713.Comment: 171 pages, PhD thesis, Universit\"at M\"unster, see http://nbn-resolving.de/urn:nbn:de:hbz:6-1748954938

Discrete Differential Geometry

Discrete Differential Geometry is a broad new area where differential geometry (studying smooth curves, surfaces and other manifolds) interacts with discrete geometry (studying polyhedral manifolds), using tools and ideas from all parts of mathematics. This report documents the 29 lectures at the first Oberwolfach workshop in this subject, with topics ranging from discrete integrable systems, polyhedra, circle packings and tilings to applications in computer graphics and geometry processing. It also includes a list of open problems posed at the problem session

Categorical post-quantum theories

In this thesis of two Parts, we investigate the application of categorical methods to modelling post-quantum theories. In Part I we study hyper-decoherence between quantum-like theories. Chapter 1 serves as an introduction to Categorical Probabilistic Theories which combine elements of Categorical Quantum Mechanics and Operational Probabilistic Theories, and to CPM categories which generalise the CPM construction of Selinger to allow for richer group symmetries. In Chapter 2 we study the theory of density hypercubes which exhibits a hyper-decoherence mechanism witnessing quantum theory as an effectful subtheory. We show that this hyper-decoherence process is probabilistic within the theory of density hypercubes and discuss some plausible operational interpretations of this. As a result, we side-step a no-go result regarding the existence of deterministic hyper-decoherence maps, showing that it is nevertheless possible for a post-quantum theory to possess probabilistic hyper-decoherence maps. In Chapter 3 we focus on a particular case of the CPM construction, where the symmetries are generated by the Galois group of a finite field extension. We discuss how to construct probabilistic theories which form towers of decoherence in bijection with the subfields of a Galois extension. These towers generalise the decoherence process of standard quantum theory. In Part II we study profunctorial methods and their application to spacetime and quantum supermaps. Chapter 4 serves as an introduction to profunctors, promonoidal categories and premonoidal categories, including the enriched version of the latter. Chapter 5 introduces some toy categories of causal curves in spacetime and discusses how we might upgrade the partial monoidal structure of such categories to a total tensor using both pre- and promonoidal categories. Chapter 6 makes this combination of pre- and promonoidal categories more formal, introducing the notion of a pro-effectful category. In the final Chapter 7 we describe how we can use the category of coend optics as a model of quantum combs. We describe the promonoidal structures on this category and their interpretation as horizontal and vertical composition of holes in monoidal categories. We also generalise coend optics to allow for a premonoidal base category, and point towards how the methods of this Chapter might be extended to include arbitrary quantum supermaps