I’ve been to two conferences in the past two weeks, and seen a lot of interesting talks. A couple of weekends ago, I was in Ottawa at the Fields Institute workshop on “Smooth Structures in Logic, Category Theory, and Physics“. There were quite a few interesting talks, on a fairly wide range of points of view, and I had some interesting conversations as well. A good workshop overall. Another report on it by Alex Hoffnung is here on the n-Category Cafe. Then this past weekend, I attended the conference “Connections in Geometry and Physics“, at the Perimeter Institute in Waterloo (also jointly sponsored by the U of Waterloo math department, and the Fields Institute), where I gave a version of my Extended TQFT talk (if you’ve seen a previous version, this is similar, but references a few more recent variations, but is mainly distinguished as the first time I caved in and decided to use Beamer – frankly I find the graphical wingdings distracting and unhelpful, but it made sense given the facilities in the venue).

One personally interesting thing was that one of the talks in Ottawa was given by John Baez, who was my advisor for my Ph.D at UCR, and one of the talks at PI was given by Niky Kamran, who was my advisor for my M.Sc. at McGill, so I got to touch base with both of them in the space of a week. It also reminded me that I’ve worked on a fairly eclectic sampling of things, since John was talking about cartesian closed categories of smooth spaces, and Niky was talking about the long-time dynamics of the Dirac equation in the neighborhood of black holes.

In the near future I’ll make a post on the Connections conference but the following was getting long enough already…

Smooth Structures

So, as the title suggests, the Fields workshop addressed the topic of “smoothness” from several points of view, with the three mentioned being only the most obvious. To begin with, “smooth” carries the connotation of “infinitely differentiable”. So for example the space $C^{\infty}(\mathbb{R})$ of smooth functions on the real line has the property that, if $f$ is any function in it, you can take the derivative $f'$, and you get another smooth function, hence can take the derivative again, and so on. So one way to characterize the space $C^{\infty}(\mathbb{R})$ is that it has a differential operator $D$ (which satisfies some algebraic properties like the product rule etc.), and is closed under $D$.

One theme explored at the workshop has to do with finding a nice general notion of “smooth space”. The smooth structure on $\mathbb{R}$ that makes this possible is the model for smooth structures on other spaces. The most familiar way goes: (1) first extend the concept to $\mathbb{R}^n$ via partial derivatives, so we know what a smooth map $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is (or between open subsets of these), and (2) define a “smooth manifold” as a (topological) space $M$ equipped with “charts” $\phi : U \rightarrow M$ for $U$ an open subset of some $\mathbb{R}^m$, satisfying a bunch of conditions. Then we can tell smooth functions on $M$ by pulling them back to $\mathbb{R}^m$ and using the familiar concept there. We can tell smooth maps $f : N \rightarrow M$ by composing with charts and their inverses and seeing that the resulting map $\phi^{-1} \circ f \circ \psi$ is smooth. This concept is great, and underlies, just to name a couple of examples, General Relativity and gauge theory, which are the basis of 20th century physics. It’s rather brittle, though, because simple operations like taking function spaces $Man(M,N) = \{ f : M \rightarrow N | f \text{ smooth } \}$, or subspaces $A \subset M$, or quotients $M/G$, give objects which are not manifolds.

John Baez talked about some of these issues in categorical language. Those three operations illustrate the general categorical constructions of exponentials, equalizers and coequalizers (or, generally, limits and colimits). The category of differentiable manifolds has important objects, but since it lacks these constructions in general, it’s not a nice category. The idea is to find a nice category – a cartesian closed one – which contains it. John described roughly how some approaches to this problem work – there were more detailed talks by Alex Hoffnung about the Diffeological spaces of Souriau, and by Andrew Stacey summarizing how the different categories are related and making a case for Frölicher spaces – but mostly focused on examples where these categories would be useful.

One interesting case deals with orbifolds (John suggested checking out this paper of Eugene Lerman, “Orbifolds as Stacks”), which were also the subject of Dorette Pronk’s talk. Dorette described some orbifolds – sometimes they arise by taking quotients of manifolds under the action of finite groups, and sometimes they only look locally as if they did. She also talked about the right way to think about maps between orbifolds, which is basically in terms of spans. That is, the right kind of “map” between orbifolds $X$ and $Y$ consists of (certain) maps into each of them from some common orbifold $Z$.

Under “logic” (and overlapping with “categories”), the first two days started off with talks by Anders Kock about Kähler differentials and synthetic differential geometry (regarding which see e.g. Mike Schulman’s intro, or the book by Anders Kock himself). SDG is a generalization of differential geometry to be internal to some topos – in particular, getting rid of the assumption that the geometry is based on some space which has an underlying set of points (which is special to the topos $\mathbf{Sets}$), and doing everything in the internal language of the topos. Kock introduced some work with Eduardo Dubuc – describing a “Fermat Theory” (an abstract characterization of a ring with a concept of partial derivatives) and showed how it fits into SDG.

Some other talks in the “logic” world included those of Rick Blute, Robin Cockett, and Thomas Erhard, about linear logic and differential categories. The basic idea is that a category can be associated to any logic, by taking formulas as objects, and (equivalence classes under rewriting rules) of proofs as morphisms. Linear logic, which is topical to quantum computation, is interesting from this point of view because it has some standard logical facts (like the deMorgan laws, which the intuitionistic logic of toposes do not entirely have) and also good categorical properties (one has a nice monoidal category). It’s a little strange if you’re used to thinking of classical logic: “propositions” are replaced by “resources” whose truth can be consumed (think of a quantum computer with a bit stored somewhere – you can move the bit, but not read and copy it); there are both “additive” and “multiplicative” versions of connectives which in classical logic go by the names “AND” and “OR” (correspondingly, there are additive and multiplicative versions of “TRUE” and “FALSE”). The relation to smoothness is that a new variation on this adds an operator to the category which behaves like differentiation. This is formally very interesting, though I haven’t really grokked what it’s good for, but apparently the derivative acts like a quantifier.  Really!?

Among other talks, Konrad Waldorf spoke about parallel transport for extended objects – basically, this is a roundabout way of studying nonabelian gerbes (a kind of categorification of bundles), not by looking at the gerbes themselves but by directly looking at parallel transport for connections on them. Kristine Bauer gave two talks about “Functor Calculus” (in particular the Goodwillie calculus), which has to do with constructing something like a “Taylor series” for a functor into topological spaces, approximating the functor by “polynomials”, and which shows up in homotopy theory.  I also have the sense that the Goodwillie calculus generalizes to topological spaces a lot of what Joyal’s species (related to “analytic” functors in a similar sense of being representable by “polynomials”), but I don’t understand this well enough at the moment to say just how.