Speaker: Samson Abramsky Title: Axiomatics of No-Cloning and No-Deleting Abstract: We review the notions of strongly- or dagger-compact category, which form the core ingredient of Categorical Quantum Mechanics as developed by Abramsky and Coecke. We also describe free constructions of such categories, and how they factor through the construction of free traced monoidal categories. Then we look at the axiomatics of No-Cloning and No-Deleting, key foundational principles of Quantum Information, from this perspective. Joyal's Lemma (`Boolean categories are preorders') precludes classical behaviour in a closed category when the monoidal structure is cartesian. On the other hand, there are many non-trivial examples of *-autonomous categories, which form the models for Multiplicative Linear Logic. We give limitative results showing the impossibility of having natural diagonals, or natural deleting, non-trivially in a dagger-compact setting. ----------------------------------------- Speaker: Robin Cockett Title: Itegories and partial combinatory algebras Abstract: Flow diagrams are used extensively in Computer Science especially in the construction of optimizing compilers. But what exactly is the categorical semantics of flow diagrams? Itegories are extensive restriction categories whose coproduct is uniformly traced. These categories, I claim, provide a basic semantics of flow diagrams. Our aim is to show that, when such a category has a stack object, it also has a partial combinatory algebra and so can simulate all computable functions. Notably very little logic and arithmetic is needed in this development. Much of the development involves getting good descriptions of itegories. In particular, we explore formulating the (particle style) trace in terms of a "Kleene wand" and reformulating extensive restriction categories in terms of local disjunction. ------------------------- Speaker: Nicola Gambino Title: Trace of analytic functors Abstract: Analytic functors, introduced by Joyal within the theory of combinatorial species of structures, are functorial counterparts of exponential power series. Following work of Ryu Hasegawa, I will discuss how analytic functors support a notion of trace. --------------------------------------- Speaker: Lou Kauffman Title: Anyonic Topological Computation and Quantum Algorithms for Knot Polynomnials Abstract: This talk will review the q-deformed spin network approach to constructing unitary representations of the Artin Braid Group and applications of these representations to quantum information theory, quantum computation and the quantum computation of the Jones polynomial, colored Jones polynomials and the Witten-Reshetikhin invariant. ------------------------------------ Speaker: Paul-Andre Mellies Title: Functorial boxes in string diagrams Abstract: String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in proof-theory (like Jean-Yves Girard's proof-nets) and in concurrency theory (like Robin Milner's bigraphs). This is not the case however, at least because string diagrams do not accomodate boxes --- a key ingredient in these pictorial systems. In this short tutorial, based on our accidental rediscovery of an idea by Robin Cockett and Robert Seely, we explain how string diagrams may be extended with a notion of functorial box to depict a functor separating an inside world (its source category) from an outside world (its target category). We expose two elementary applications of the notation: first, we characterize graphically when a faithful balanced monoidal functor F:C -> D transports a trace operator from the category D to the category C, and we then exploit this to construct well-behaved fixpoint operators in cartesian closed categories generated by models of linear logic; second, we explain how the categorical semantics of linear logic induces that the exponential box of proof-nets decomposes as two enshrined functorial boxes. --------------------------------------- Speaker: Greg Meredith Title: Knots as processes We present an encoding of knots as processes in Milner's \pi-calculus enjoying the property that knots are ambient isotopic if and only if their encodings as processes are weakly bisimilar-up-to a commutation property. --------------------------------------------- Speaker: Prakash Panangaden Title: Internal Traces Abstract: I describe a new notion of trace due to Yannick Delbeceque that allows one to obtain a trace in situations where there is no trace according to the existing definition. This trace is defined as a map from an object A to I the tensor unit. For an element of A, i.e. an arrow from I to A the trace is obtained by composition. I will describe the axioms and the motivating example of stochastic relations. It is possible that this structure is interesting from the point of view of categorical quantum mechanics. --------------------------------------------- Speaker: Eric Paquette Title: Quantum decoherence Abstract: The main topic of this talk is one of the most interesting application of traces in quantum mechanics. I'll introduce the basics of quantum mechanics via the usual postulates. Then, I'll speak of the so-called measurement problem and then, I'll move to quantum decoherence in order to give a partial answer to one of the issues raised by the measurement problem. ---------------------------------------------- Speaker: Dusko Pavlovic Title: A monadic view of traces and intruders Abstract: I will describe a monad over the 2-category of symmetric monoidal categories such that traced categories are just its algebras. This view turns out to be useful for reasoning about the Intruder-in-the-Middle attacks on authentication protocols: security analysis amounts to reconstructing the intruder from his trace. ----------------------------------------------- Speaker: Tim Porter Title: Formal homotopy quantum field theory and traces. Abstract: Topological QFTs led to Homotopy QFTs in an attempt to handle spaces and cobordisms "with structure". If the background space of the HQFT is a homotopy 2-type it can be modelled by a 2-group or crossed module. The talk will outline joint work with Turaev on a formal algebraic/combinatorial model for such an HQFT and the corresponding algebraic gadget. These structures naturally obey a trace formula, BUT the theory at present does not give a significant contribution of the second level structure to that formula and this raises questions of what sort of higher trace might detect such structure. The talk will give an introduction to HQFTs and will discuss the clssification results, before going on to the formal version and its interpretation. ---------------------------------------------- Speaker: Tarmo Uustalu (joint with Varmo Vene) Title: Comonadic notions of computation We argue that symmetric monoidal comonads provide a means to structure context-dependent notions of computation such as dataflow computation (computation on streams) and attribute grammar evaluation, the coKleisli category of the comonad serving as the category of computations. We pay particular attention to the partial trace structures arising from solutions to guarded recursive definitions in this setting. This continues the work in the 90s by Brookes, Geva and Stone on the use of computational comonads in intensional semantics. ------------------------------------------ Speaker: Benoit Valiron Title: A categorical model for a quantum lambda-calculus. Abstract: In this talk I will present a higher-order language for quantum computation, and discuss a categorical model. The language is a computational lambda-calculus in Moggi-style call-by-value, with a type system based on intuitionistic linear logic to capture the non-duplicability of quantum bits. ------------------------------------------ Speaker: Derek Wise Title: Loops, Bubbles, and Trace Divergences in Quantum Field Theory Abstract: The notorious "infinities" in quantum field theory are intimately related to the general notion of trace. For example, the problematic Feynman diagrams in field theory are the ones involving "feedback loops" like those in trace diagrams. Other examples include 2d gauge theories with noncompact gauge group, which have divergences whenever we try taking the "trace" of a cobordism. Indeed, the existence of certain kinds of traces is a major distinction between well-behaved topological field theories and their real real-world counterparts in physics. In this talk I will explore the relationships between traces and topology in quantum field theories. In particular, I will consider examples from 2d Yang-Mills theory, and from p-form electromagnetism in p+1 dimensions, which has divergences related to certain higher dimensional analogues of "trace".