C.J.Mulvey

C.J.Mulvey MA (Cambridge) MSc (Sheffield) DPhil (Sussex)
Reader Emeritus in Mathematics

Recent Lectures and Seminar Talks

A fairly comprehensive description of current research interests may be found on my home page. Information on recent research articles and on forthcoming papers are also available from this page.

The following are abstracts of recently given research lectures and seminar talks, together with a link to the slides of the talk. It should be noted that these pdf files are intended for viewing rather than for printing, in the sense that some pages develop as you click through the file. There is inevitably considerable overlap between talks given to different audiences, so look before you link.

A list of research publications is also available from this page, together with a brief curriculum vitae and information on my contact details.

Quantal sets over an involutive quantale

In this joint work with Joel Zamora Ramos of the University of Sussex, we examine the ways in which the concepts of quantal set and of complete quantal set can be introduced in the context of involutive quantales. In particular, it is shown that obtaining a generalisation of the concept of completeness for quantal sets from that known from the work of Mulvey and Nawaz requires that the involutive quantale should be a locally Gelfand quantale. Applying these ideas, we arrive at the concepts of presheaf and of sheaf appropriate to this context.

Talk given at the Logic from Quantales workshop at the Computing Laboratory of the University of Oxford, United Kingdom in January, 2005, and in the Category Theory seminar in the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge, United Kingdom in February, 2005.

[PDF of Slides]

Quantal spaces

The concept of a quantale was motivated by ideas that at the outset were seen as rather distinct: on the one hand, the need to develop non-commutative versions of structures that had proved useful in the context of theoretical computer science, and, on the other hand, a wish to provide a more constructive foundation for quantum theory. In this talk I will be concerned with outlining work on quantales and quantal spaces which evolved out of these origins, giving along the way some idea of the motivation for considering non-commutative generalisations of topological spaces and of the way that spatiality can be conceived in that situation, and examining a particular case in which these aspects of the origins appear after all to be rather close.

Talk given in the Oxford Advanced Seminar on Informatic Structures at the Computing Laboratory of the University of Oxford, United Kingdom in November, 2004.

[PDF of Slides]

Quantum aspects of non-commutative logic

This talk will be concerned with outlining certain aspects of non-commutative logic which appear to have relevance to the quantum world. In particular, it will examine the way in which considering a theory within the context of such a logic yields in turn a space, itself non-commutative, of models of the theory. Indeed, it may be observed that any such space naturally emerges from a theory in this way. By way of illustration, we shall examine a situation within non-commutative geometry, namely that of the space of Penrose tilings, introduced formally by Connes as a particular C*-algebra, which can equivalently, and in a sense more intuitively, be described by introducing a particular non-commutative theory. More generally, we shall see that any C*-algebra can be classified by the spectrum obtained, again as a non-commutative space, by considering a particular theory describable in terms of the C*-algebra. Examining this theory, it may be shown that, in the case of C*-algebras of operators deriving from a quantum situation, it may be viewed as describing assertions about the nature of and relationships between quantum observables. One question that underlies these kinds of observation is whether indeed it is the space that yields the theory, or just possibly the theory that creates the space.

Talk given at the meeting on Iconoclastic Approaches to Quantum Gravity in Athens, Greece in June, 2004.

[PDF of Slides]

The Gelfand-Naimark theorem

This talk is concerned with two interconnected aspects of the Gelfand-Naimark theorem on the representation of C*-algebras. The first is the way in which the theorem can be proved constructively in the commutative case, even though the classical formulation of this result appears to be entirely dependent on ideas close to the Axiom of Choice. The second is the way in which the theorem can be proved in the non-commutative case, albeit without any claims to constructivity. Of course, there should be a third aspect, in which constructivity is added to non-commutativity to give the theorem as it ought to be proved. For the moment, one can at least see the way in which the approach to the non-commutative case derives naturally from that taken in proving the theorem constructively for commutative C*-algebras. The work currently under way attempts to provide the context in which this constructivity can be extended to the non-commutative case.

Talk given in the Algebra Colloquium of the Department of Mathematics of the University of Athens, Greece in June, 2004.

[PDF of Slides]

The space of Penrose tilings

The concept of quantale is an instance of that of residuated lattice that is adapted to the non-commutative logics that arise in both mathematics and physics. In this talk, we shall examine the way it may be applied to describe the non-commutative space of Penrose tilings, providing a particularly straightforward motivation of the approach taken by Connes in introducing a C*-algebra to represent this non-commutative geometric construct. In doing so, we shall more generally consider the way that quantales may be obtained by introducing propositional geometric theories within non-commutative logic, generalising the way that such theories may be applied to obtain spaces within constructive logic. In the present context, we shall see how the C*-algebra introduced by Connes is obtained by considering the theory of Penrose tilings within non-commutative logic.

Talk given at the meeting on Residuated Structures and Many-Valued Logics at the University of Patras, Greece in June, 2004.

[PDF of Slides]

Theories and space

The introduction of the theory of toposes by Lawvere and Tierney allowed, amongst many other things, the categorical techniques developed by Lawvere in his Columbia dissertation to be extended to theories other than those that are algebraic. The construction of the spectrum of a commutative ring, for example, may be viewed as that of creating the space needed to allow the existence of the commutative local ring freely generated by the commutative ring. In this case, the topos constructed over the topos in which the commutative ring lies classifies a propositional geometric theory, so is obtained by taking sheaves on a space, that is, a locale, constructed in the topos. To the extent that the classifying topos of any geometric theory may be obtained by constructing the classifying topos of an object, then considering within that topos a propositional geometric theory endowing that object with particular properties, any geometric theory may be examined by considering a space constructed in this way.

This talk considers the ways in which this approach has changed the way in which we think about mathematics constructively, examining the extent to which mathematics that had been considered to depend on the presence of the Axiom of Choice has been able to be rephrased in terms that involve no such dependence. The Hahn-Banach theorem, for example, becomes a statement about the spaces of functionals on a seminormed space and on a subspace of it (or, equivalently, about the theories of functionals on these spaces). The Axiom of Choice enters only if one wishes these spaces of functionals to be topological (or, equivalently, the theories to be complete in terms of models in the base topos). More generally, the approach to mathematics taken involves applying geometric theories to create the space in which the mathematics concerned can be allowed to unfold in a way that is constructive.

There are, however, situations in which mathematics has found it difficult to proceed, even by requiring the presence of the Axiom of Choice. While the construction of the spectrum of a commutative ring may be obtained classically (and, by the observations above, constructively) without difficulty, for noncommutative rings even the problem to be solved is not quite so evident. In the case of C*-algebras, the presence of a constructive Gelfand duality in the commutative case at least indicates the problem that one might wish to consider in the noncommutative case, and there are indications that one way forward in addressing that may be through recognition of the need to consider theories within noncommutative logic. The spaces considered in this context are once again those constructed by taking the Lindenbaum algebra of a propositional theory, in this case obtaining quantales, rather than locales.

While this consideration of noncommutative theories, and the corresponding concept of noncommutative space, is still at an early stage, there are indications that it provides a natural approach to the development of noncommutative mathematics. It is, of course, interesting to observe that the logic of the physical world, particularly that part of it that deals with quantum phenomena, is also noncommutative. The theories that might be developed to describe it constructively are therefore likely to need to be noncommutative, as may be the spaces to which they give rise. Perhaps it is this need not being taken into account that has required until now mathematics that is wholly dependent on the presence of the Axiom of Choice to be needed to describe the physics of a world that is evidently without this dependence..

Talk given at the Ramifications of Category Theory meeting held at the University of Firenze, Italy in November, 2003 to honour the fortieth anniversary of the doctoral dissertation of F.W. Lawvere.

[PDF of Slides]

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