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A variety of models, especially Kitaev models, quantum Chern-Simons theory, and models from 3d quantum gravity, hint at a kind of lattice gauge theory in which the gauge group is generalized to a Hopf algebra. However, until recently, no general notion of Hopf algebra gauge theory was available. In this self-contained introduction, I will cover background on lattice gauge theory and Hopf algebras, and explain our recent construction of Hopf algebra gauge theory on a ribbon graph (arXiv:1512.03966). The resulting theory parallels ordinary lattice gauge theory, generalizing its structure only as necessary to accommodate more general Hopf algebras. All of the key features of gauge theory, including gauge transformations, connections, holonomy and curvature, and observables, have Hopf algebra analogues, but with a richer structure arising from non-cocommuntativity, the key property distinguishing Hopf algebras from groups. Main results include topological invariance of algebras of observables, and a gauge theoretic derivation of algebras previously obtained in the combinatorial quantization of Chern-Simons theory.

I will discuss the relation between topological field theories and gapped phases of matter. I will propose a general formalism to define a class of TFTs which can be realized by commuting projector Hamiltonians. This allows one to apply rigorous mathematical theorems about TFTs to gapped phases of matter. I will also discuss the role of generalized cohomology theories and spectra in the classification of SPT phases.

The (Freedman-Kitaev) topological model for quantum computation is an inherently fault-tolerant computation scheme, storing information in topological (rather than local) degrees of freedom with quantum gates typically realized by braiding quasi-particles in two dimensional media. I will give an overview of this model, emphasizing the mathematical aspects.

The aim of this talk is to give an introduction to modular categories, touching both basics and recent developments. I will begin with a quick reminder concerning tensor categories, in particular braided and symmetric ones, and notions like duality, fusion and spherical categories. We'll meet algebras in tensor categories, categories of modules, module categories and their connection. I will then focus on modular categories and some basic structure theory. We will consider two ways of obtaining modular categories: modularization and the Drinfeld center. (The important third one, quantum groups at root of unity, is too complicated to be discussed in any depth.) The Drinfeld center will be used to define Witt equivalence of modular categories and the Witt group. Several equivalent characterization of Witt equivalence, using module categories, will be discussed. The Witt group will be crucial for any (future) classification of modular categories, as well as for application to physics in condensed matter physics and conformal field theory.

I will give an introduction to the Kitaev quantum double models for Hopf C*-algebras. To this end I will introduce a graphical tensor-network notation to represent the algebraic objects and axioms. Using this notation I will then present the vertex- and plaquette symmetries of the model and discuss their interaction and the excitation structure they give rise to.

This talk will be a short introduction to the semisimple Hopf algebras over an algebraically closed field of characteristic 0 and their representation theories. It is intended to outline the main basic results about structure and known methods for the construction of semisimple Hopf algebras: extensions, twisting, Tannakian reconstruction. Basic notions concerning tensor categories will be introduced: braided structures, center construction, fiber functors. Special emphasis is given to the notion of fusion category. At the same time, the relations between these notions and those of the Hopf algebras are studied.

The purpose of this talk is twofold: one, to acquaint the wider community working mostly on Bell-Kochen-Specker contextuality with recent work on Spekkens’ contextuality that quantitatively demonstrates the sense in which Bell-Kochen-Specker contextuality is subsumed within Spekkens’ approach, and two, to argue that one can test for contextuality without appealing to a notion of sharpness which can needlessly restrict the scope of operational theories that could be considered as candidate explanations of experimental data. Testing contextuality in Spekkens’ approach therefore extends the range of experimental scenarios in which contextuality can be witnessed, and refines what it means to witness contextuality in the presence of inevitable noise in KS-type experiments. We will see this for both KS-uncolourability based logical contradiction type proofs of the KS theorem a la Kochen-Specker and statistical proofs on KS-colourable scenarios a la KCBS or Yu-Oh. While Bell-KS contextuality can be mathematically understood as an instance of the classical marginal problem, the same is not true of Spekkens' contextuality. The latter reduces to the classical marginal problem only under very specific conditions, being more general otherwise. All in all, we will argue that all you really need is Leibniz, i.e. identity of indiscernables, to make sense of contextuality in the most general context.

In order to perform foundational experiments testing the correctness of quantum mechanics, one requires data analysis tools that do not assume quantum theory. We introduce a quantum-free tomography technique that fits experimental data to a set of states and measurement effects in a generalised probabilistic theory (GPT). (This is in contrast to quantum tomography, which fits data to sets of density operators and POVM elements.) We perform an experiment on the polarization degree of freedom of single photons, and find GPT descriptions of the states and measurements in our experiment. We gather data for a large number of preparation and measurement procedures in order to map out the spaces of allowed GPT states and measurement effects, and we bound their possible deviation from quantum theory. Our GPT tomography method allows us to bound the extent to which nature might be more or less contextual than quantum theory, as measured by the maximum achievable violation of a particular noncontextuality inequality. We find that the maximal violation is confined to lie between 1.2±0.1% less than and 1.3±0.1% greater than the quantum prediction. Coauthors: Matthew Pusey, Robert Spekkens, Kevin Resch