### Smooth Structures in Ottawa

#### Posted by John Baez

Here at the $n$-Café we’re trying to get to the bottom of some big questions — for example, the nature of smoothness. A manifold is a kind of smooth space — but more general smooth spaces have been studied by Chen, Lawvere, Kock, Souriau and others, and these are starting to find their way into mathematical physics.

That’s just the beginning, though! Smoothness has a lot to do with derivatives. The concept of derivative can be generalized in some surprising ways. For example, it’s important in Joyal’s work on combinatorics — he explained how we can take the derivative of a structure like ‘being a 2-colored finite set’ More recently, Goodwillie introduced a concept of ‘approximation by Taylor series’ for interesting functors in homotopy theory. Even more recently, Ehrhard and Regnier introduced derivatives in logic — or more precisely, the lambda calculus.

So, it’s a great idea to have a conference on *all* these concepts of smoothness:

- Smooth Structures in Logic, Category Theory and Physics, University of Ottawa, May 1-3, 2009, organized by Richard Blute, Pieter Hofstra, Philip Scott, and Michael A. Warren.

Here’s what the organizers have to say:

Abstract categorical approaches and analogies with the differential calculus and the theory of smooth manifolds arise in a number of diverse areas of mathematics. For example, the well-known fact that the category of manifolds and smooth maps fails to be cartesian closed motivated both the theory of convenient vector spaces due to Froelicher, Kriegl, and Michor, and work on categories of smooth spaces initiated by Chen and Souriau. In topos theory, synthetic differential geometry, developed by Lawvere, Kock, Moerdijk, Reyes, and others, provides an appealing abstract setting for differential geometry using the theory of nilpotent infinitesimals. In logic, the differential lambda-calculus, due to Ehrhard and Regnier, was inspired by considerations from linear logic, differential calculus, and work on locally convex topological models of linear logic. This theory subsequently gave rise to the recent development of differential categories by Blute, Cockett, and Seely. In topology, the Goodwillie calculus, which also has connections with the study of smooth manifolds, is an example of a ‘calculus of functors’ drawing inspiration from differential calculus. And in theoretical physics, recent work by Baez and Schreiber on higher gauge theory exploits some of these more abstract versions of differential geometry in order to avoid technical difficulties implicit in the theory of infinite-dimensional manifolds.

The Logic and Foundations of Computing group at the University of Ottawa is happy to announces a workshop, supported by the Fields Institute, which aims to bring together researchers from these different areas in order to encourage further interaction in the study of smooth structures in logic, category theory and physics. In addition to the main invited lectures, several of the invited speakers will give tutorials on their areas of expertise in order to make the subject accessible to students and other new researchers in the area. The (confirmed) invited speakers are:

- John Baez (UC Riverside)
- Kristine Bauer (Calgary)
- Thomas Ehrhard (PPS Paris)
- Anders Kock (Aarhus)
- Andrew Stacey (NTNU Norway)
Some student support from the Fields Institute will be available. There will also be some time reserved in the schedule for a selection of contributed talks. Further details regarding student support and contributed talks can be found on the workshop webpage.

With best regards,

Richard Blute

Pieter Hofstra

Philip Scott

Michael A. Warren

## Re: Smooth Structures in Ottawa

Papers on differential categories can be found here.

It would be good to have someone connected with generalised species at the workshop.

Fiore writes: