Slides of some of my talks

This page contains slides of some of my talks.

2020

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2008

Before 2008

A topos presentation of C*-algebra based physical systems (jww Chris Heunen)
Lecture at Categorical Quantum Logic, Oxford.
Abstract: We show how a C*-algebra naturally induces a topos in which the family of its commutative subalgebras becomes a commutative C*-algebra. Its internal spectrum is a compact regular locale, and the Kochen-Specker theorem is equivalent to this spectrum having no points. (Quasi-)states become integrals, and self-adjoint elements become functions to the pertinent generalised real numbers (the interval domain). This provides a probabilistic interpretation of propositions in quantum theory. The topos-theoretic truth value of such a proposition is the collection of pure states of commutative subalgebras that make it true; in a physical interpretation these are the pure states for a classical observer making the proposition true. These results were motivated by a topos-theoretic approach of the Kochen-Specker theorem by Isham and co-workers. Our main tool is the use of the internal mathematics of a topos, such as the constructive Gelfand duality of Banaschewski and Mulvey, which simplifies the computations and provides very natural connections between internal and external reasoning.

Observational integration theory
TANCL07, Oxford.
Abstract: In this talk I will present a constructive theory of integration. The theory is constructive in the sense of Bishop or Brouwer, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. To be more precise we outline how to develop most of Bishop's theorems on integration theory that do not mention points explicitly. Coquand's constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop's spectral theorem. This talk illustrates the general theme of developing mathematics observationally, connecting ideas by Kolmogorov, von Neumann and Segal on the one hand and point-free (aka formal) topology on the other. This provides a nice illustration how ideas from logic (proof theory) can be used to obtain mathematical results. By generalizing Isham ideas on quantum theory we find that this integration theory is also applicable in the non-commutative context.

Observational integration theory
Invited lecture at the ASL Meeting in Montreal 2006.
Abstract:In this talk I will present a constructive theory of integration. The theory is constructive in the sense of Bishop or Brouwer, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. To be more precise we outline how to develop most of Bishop's theorems on integration theory that do not mention points explicitly. Coquand's constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop's spectral theorem. This talk illustrates the general theme of developing mathematics observationally, connection ideas by Kolmogorov, von Neumann and Segal on the one hand and point-free (aka formal) topology on the other. This provides a nice illustration how ideas from logic (proof theory) can be used to obtain mathematical results.
Finally, I will show (by constructing a model) that in this context the reals can not be proved to be uncountable and show how we live wi th this fact.

A constructive view on compact groups - constructive algebra applied to analysis
Talk given at the dagstuhl seminar Verification and Constructive Algebra. Dagstuhl January 2003.
Abstract: We claim that, contrary to Weyl's belief, constructive mathematics suffices for the applications of mathematics. To support our claim we prove the Peter-Weyl theorem in a constructive and natural way.
For this proof we need constructive integration theory, Gelfand theory and spectral theory. These theories will be outlined in the talk.
As proposed by Weyl we stress that mathematics should be build on basic observables or finite approximations.

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