A Fields Institute Sponsored Workshop

Smooth Structures in Logic, Category Theory and Physics

Dept. of Mathematics & Statistics

University of Ottawa

May 1-3, 2009

Note: Some Slides of Talks Available Below

Smooth Structures in Logic, Category Theory and Physics

Dept. of Mathematics & Statistics

University of Ottawa

May 1-3, 2009

Note: Some Slides of Talks Available Below

Since the 1970's, studies of categories of "smooth spaces", i.e. spaces with sufficient structure to define differentiation, have been developed by numerous researchers. Such settings make possible, for example, a more general, abstract version of differential geometry. One of the issues that the more categorically minded researchers have investigated is the fact that the category of manifolds and smooth maps is not cartesian closed, i.e. the collection of smooth functions between two manifolds is not itself a manifold. Many of these approaches overcome this, and thus the smooth maps between smooth spaces form a smooth space.

The work of A. Frölicher, A. Kriegl, and P. Michor generalizes calculus in Banach spaces to convenient vector spaces, a class of vector spaces equipped with enough abstract structure to allow a definition of smooth maps. The category of convenient vector spaces and smooth maps is cartesian closed. Moreover, such a space may be viewed simultaneously as a bornological space, a locally convex space, and as an ell-infinity space, all of which have been proposed as possible settings to study smoothness. Similarly, following early work of K. Chen, J.-M. Souriau, and others, categories of smooth spaces containing the category of C-infinity manifolds have been considered as suitable abstract settings for doing differential geometry.

The development of synthetic differential geometry and smooth infinitesimal analysis by F.W. Lawvere, A. Kock, I. Moerdijk, G. Reyes, and others is based on a development of a categorical theory of infinitesimals in appropriate categories of spaces (smooth toposes). Such categories contain the category of C-infinity manifolds and C-infinity-maps.

In a different direction, recently T. Ehrhard and L. Regnier developed the differential lambda-calculus motivated by linear logic, differential calculus, and Ehrhard's work on various locally convex topological models of linear logic. This inspired the recent development of differential categories by R. Blute, R. Cockett, and R. Seely.

Finally, recent work by J. Baez and U. Schreiber in theoretical physics exploits these more abstract versions of differential geometry to avoid the technical difficulties implicit in the theory of infinite-dimensional manifolds. This leads to their notion of higher gauge theory.

In this workshop, we propose to bring experts together from these different areas to encourage further interaction in the study of smooth structures in categories, physics, and logic. Several of our invited speakers will be asked to give tutorials in order to make the subject accessible to students and other new researchers. In light of the recent interest in these topics (especially in the category theory community) there is little doubt that the workshop will act as a stimulus for further discussion and collaboration.

Invited speakers: Contributed Talks:

* John Baez (UC Riverside) * Richard Blute (Ottawa) (Tutorial)

* Kristine Bauer (UCalgary) * Robin Cockett (Calgary)

* Thomas Ehrhard (PPS Paris) * Alexander Hoffnung (UC Riverside)

* Anders Kock (Aarhus) * Dorette Pronk (Dalhousie)

* Gonzalo Reyes (UMontreal) * Konrad Waldorf (Berkeley)

* Andrew Stacey (NTNU Norway)

This is intended to be a workshop, with student participation in mind, including introductory lectures and

tutorials, with plenty of time to talk with the speakers. (Please register on the Fields Webpage).

Anyone interested in attending is requested to contact one of the organizers as soon as possible.

For official preconference registration, see: Fields Webpage for Smooth Structures Workshop

Richard Blute, 613-562-5800, ext. 3535 (rblute@uottawa.ca)