**University of Oxford, August 23-24, 2008**

The workshop began with some tutorials on quantum theory, given by Bob Coecke and Andreas Doering.

**Bob Coecke, Oxford, Computer Science**

*Tutorial 1: Quantum formalism*

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**Andreas Doering, Imperial College London, Physics**

*Tutorial 2: Conceptual issues*

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We also had talks by 8 invited speakers.

**John Barrett, Nottingham, Mathematics**

*Knots and links in braided quantum field theory*

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I will explain some aspects of braided quantum field theory: how it is a generating function of knots and links, and also a little about the relation to 3d quantum gravity.

**Louis Crane, Kansas State, Mathematics**

*Model categories in quantum gravity*

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We examine a plausible physical hypothesis which would allow us to model quantum regions as differential graded hopf algebras or as differential graded categories.

**Chris Fewster, York, Mathematics**

*The locally covariant approach to quantum field theory in curved spacetimes*

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A long-established guiding principle for quantum field theory in curved spacetimes is that it should be formulated in a local and geometrically covariant way. Recently, Brunetti, Fredenhagen and Verch have given an elegant formulation of this principle in categorical terms: a theory should be understood as a functor from a category of spacetimes to a category of star-algebras. This conceptual clarification is at the heart of a number of important results for QFT in general spacetimes, such as a proof of the spin-statistics theorem and the perturbative construction of interacting QFT. This talk reviews the locally covariant approach and some of its applications, and also describes joint work with Verch that refines and sharpens the framework.

**Keith Hannabuss, Oxford, Mathematics**

*Categories and non-associative C*-algebras in quantum field theory*

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**Simone Perdrix, University of Oxford, Computer Science**

*Bases in diagrammatic quantum protocols*

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**Mehrnoosh Sadrzadeh, Paris VII, PPS**

*What is the vector space content of what we say? … a categorical approach to distributed meaning*

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**Steve Vickers, University of Birmingham, Computer Science**

*Locales via bundles*

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Locale theory, a "point-free" approach to topology, can be understood as viewing topological spaces as the spaces of models for logical theories of a particular kind. It has been found effective in a variety of fields, including pure mathematics (deriving from algebraic geometry) and computer science (notably in Abramsky's thesis). One of its most compelling virtues is that it interacts well with constructive mathematics, including the internal mathematics of toposes. In fact, it has a more satisfactory body of constructively valid results than does ordinary point-set topology. Locales have also appeared in the topos approaches to quantum mechanics of Isham and Doering at Imperial and (more explictly) of Heunen, Landsman and Spitters at Nijmegen. Locales have been motivated in a variety of ways, but the path to the constructive virtues can be long and stony. I shall outline a conceptual development in terms of bundles, and going back to a technical result described by Joyal and Tierney in 1984 and known even earlier. Essentially it says that bundles over a space X are equivalent to topological spaces in the internal mathematics of "bundles of sets" (local homeomorphisms) over X. This fits an intuition that "bundle" means a space (the fibre) parametrized by points of the base space. However, there are simple examples to show that this cannot work with a point-set approach to topology - essentially because arbitrary bundles cannot be approximated closely enough by local homeomorphisms. Instead, spaces must be replaced by locales. A key notion is that of "geometric" reasoning, preserved under pullback of bundles, that is more restricted than topos-valid reasoning. It is hoped that the topos-internal reasoning of the Imperial and Nijmegen groups, insofar as it is geometric, can be expressed more intuitively as fibrewise reasoning for bundles.

**Simon Willerton, Sheffield, Mathematics**

*Two 2-traces*

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In a monoidal category there is a notion of a trace for certain types of endomorphisms — for example the trace of an endomorphism of a finite dimensional vector space. In a monoidal bicategory there are two different notions of trace for endomorphisms which in various cases are 'dual'; I will describe these traces diagrammatically. Several examples from assorted areas will be given and I hope that the audience will come up with some more.

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