Ottawa

I recently got back to London, Ontario from a trip to Ottawa, the first purpose of which was to attend the Ottawa Mathematics Conference. The other purpose was to visit family and friends, many of whom happen to be located there, which is one reason it’s taken me a week or so to get around to writing about the trip. Now, the OMC was a general-purpose conference, mainly for grad students, and some postdocs, to give short talks (plus a couple of invited faculty from Ottawa’s two universities - the University of Ottawa, and Carleton University - who gave lengthier talks in the mornings). This is not a type of conference I’ve been to before, so I wasn’t sure what to expect.

From one, fairly goal-oriented, point of view, the style of the conference seemed a little scattered. There was no particular topic of focus, for instance. On the other hand, for someone just starting out in mathematical research, this type of thing has some up sides. It gives a chance to talk about new work, see what’s being done across a range of subjects, and meet people in the region (in this case, mainly Ottawa, but also elsewhere across Eastern and Southern Ontario). The only other general-purpose mathematics conference I’ve been to so far was the joint meeting of the AMS in New Orleans in 2007, which had 5000 people and anyone attending talks would pick special sessions suiting their interests. I do think it’s worthwhile to find ways of circumventing the various pressures toward specialization in research - it may be useful in some ways, but balance is also good. Particularly for Ph.D. students, for whom specialization is the name of the game.

One useful thing - again, particularly for students - is the reminder that the world of mathematics is broader than just one’s own department, which almost certainly has its own specialties and peculiarities. For example, whereas here at UWO “Applied” mathematics (mostly involving computer modelling) is done in a separate department, this isn’t so everywhere. Or, again, while my interactions in the UWO department focus a lot on geometry and topology (there are active groups in homotopy theory and noncommutative geometry, for example), it’s been a while since I saw anyone talk about combinatorics, or differential equations. Since I actually did a major in combinatorics at U of Waterloo, it was kind of refreshing to see some of that material again.

There were a couple of invited talks by faculty. Monica Nevins from U of Ottawa gave a broad and enthusiastic survey of representation theory for graduate students. Brett Stevens from Carleton talked about “software testing”, which surprised me by actually being about combinatorial designs. Basically, it’s about the problem of how, if you have many variables with many possible values each, to design a minimal collection of “settings” for those variables which tests all possible combinations of, say, two variables (or three, etc.). One imagines the variables representing circumstances software might have to cope with - combinations of inputs, peripherals, and so on - so the combinatorial problem is if there are 10 variables with 10 possible values each, you can’t possibly test all 10 billion combinations - but you might be able to test all possible settings of any given PAIR of variables, and much more efficiently than just an exhaustive search, by combining some tests together.

Among the other talks were several combinatorial ones - error correcting codes using groups, path ideals in simplicial trees (which I understand to be a sort of generalization to simplicial sets of what trees are for graphs), heuristic algorithms for finding minimal cost collections of edges in weighted graphs that leave the graph with at least a given connectivity, and so on. Charles Starling from U of O gave an interesting talk about how to associate a topological space to an aperiodic tiling (roughly, any finite-size region in an aperiodic tiling is repeated infinitely many times - so the points of the space are translations, and two translations are within of one another if they produce matching regions about the origin of size - then the thing is to study cohomology of such spaces, and so forth).

The talk immediately following mine was by Mehmetcik Pamuk about homotopy self-equivalences of 4-manifolds, which used a certain braid of exact sequences of groups of automorphisms (among other things). I expected this to be very interesting, and it was certainly intriguing, but I can’t adequately summarize it - whatever he was saying, it proved to be hard to pick up from just a 25 minute talk. I did like something he said in his introduction, though: nowadays, if a topologist says they’re doing “low-dimensional” topology, they mean dimension 3, and “high-dimensional” means dimension 4. This is a glib but indicative way to point out that topology of manifolds in dimensions 1 and 2 is well understood (the connected components are, respectively, circles and n-holed tori), and in dimension 5 and above have been straightened out more recently thanks to Smale.

There were some quite applied talks which I missed, though I did catch one on “gravity waves”, which turn out not to be gravitational waves, but the kind of waves produced in fluids of varying density acted on by gravity. (In particular, due to layers of temperature and pressure in the atmosphere, sometimes denser air sits above less dense air, and gravity is trying to reverse this, producing waves. This produces those long rippling patterns you sometimes see in high-altitude clouds. Lidia Nikitina told us about some work modelling these in situations where the ground topography matters, such as near mountains - and had some really nice pictures to illustrate both the theory and the practice.)

On the second day there were quite a few talks of an algebraic or algebra-geometric flavour - about rings of algebraic invariants, about enumerating lines in special “blow-up” varieties, function fields associated to hyperelliptic curves, and so on - but although this is interesting, I had a harder time extracting informative things to say about these, so I’ll gloss over them glibly. However, I did appreciate the chance to gradually absorb a little more of this area of math by osmosis.

The flip side of seeing what many other people are doing was getting a chance to see what other people had to say about my own talk - about groupoids, spans, and 2-vector spaces. One of the things I find is that, while here at UWO the language of category theory is widely used (at least by the homotopy theorists and noncommutative geometry people I’ve been talking to), it’s not as familiar in other places. This seems to have been going on for some time - since the 1970’s if I understand the stories correctly. After MacLane and Eilenberg introduced categories in the 1940’s, the concept had significant effects in algebraic geometry/topology, homological algebra, and spread out from there. There was some deep enthusiasm - possibly well-founded, though I won’t claim so - that category theory was a viable replacement for set theory as a “foundation” for mathematics. True or not, that idea seemed to be one of those which was picked up by mathematicans who didn’t otherwise know much about category theory, and it seems to be one that’s still remembered. So maybe it had something to do with the apparent fall from fashion of category theory. I’ve heard that theory suggested before: roughly, that many mathematicians thought category theory was supposed to be a new foundation for mathematics, couldn’t see the point, and lost interest.

Now, my view of foundations is roughly suggested in my explanation of the title of this blog. I tend to think that our understanding of the world comes in bits and pieces, which we refine, then try to stick together into larger and more inclusive bits and pieces - the “Atlas” of charts of the title. This isn’t really just about the physical world, but the mathematical world as well (in fact I’m not really a Platonist who believes in a separate “world” of mathematical objects - though that’s a different conversation). This is really just a view of epistemology - namely, empirical methods work best because we don’t know things for sure, not being infinitely smart. So the “idealist”-style program of coming up with some foundational axioms (say, for set theory), and deriving all of mathematics from them without further reference to the outside doesn’t seem like the end of the story. It’s useful as a way of generating predictions in physics, but not of testing them. In mathematics, it generates many correct theorems, but doesn’t help identify interesting, or useful, ones.

So could category theory be used in foundations of mathematics? Maybe - but you could also say that mathematics consists of manipulating strings in a formal language, and strings are just words in a free monoid, so actually all of mathematics is the theory of monoids with some extra structure (giving rules of inference in the formal language). Yet monoid theory - indeed, algebra generally - is not mainly interesting as foundations, and probably neither is category theory.

On the whole, it was an interesting step out of the usual routine.

 

Musing, and a Tale of Two Kauffmans

Writing sizeable chunks of math blog takes longer than I expected. Here are a few non-intensive things that occurred to me.

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While I was walking home from the UWO campus, I was reminded of the nature of Canada in late November: everything, from sky to plantlife to earth, is in shades of grey, brown, ochre, and the occasional desaturated greenish-whatever. Autumn leaves have pretty much stopped falling, and are on the ground turning greyish versions of whatever colours they were before. There are whole vistas of bare branches, dead underbrush, and so on.

Which seems dreary for a while, until you’re immersed in it, as I am on the particular route I walk home, along London, Ontario’s Thames River (not to be confused with the River Thames in London, England), which is lined with parks. Then, after a while, all the subtle differences in shading and texture start to jump out at you more and more, until brownish moss on a tree under overcast late-afternoon light is vibrant green, a patch of snow is glowing bluish white, the occasional flicker of sunset through the cloud cover is warm pumpkin-orange, that one particular bush’s leaves look startlingly red… and then you see something artificial, like someone’s nylon jacket, or a kid’s plastic play-structure, and their colours look implausibly oversaturated, like a badly photoshopped picture.

Which got me to thinking about fine distinctions that seem drab outside their context - the way these colours look at first. Or nitpicky, like having 30 different words for “cold” and the different qualities it can have, or recognizing 15 different types of snowflake from a distance. Coming back to Canada after several years in California, I noticed all this specialized knowledge I’d forgotten about, and seems terribly arcane outside its native habitat. It occurred to me that this is how mathematics probably seems to outsiders - like physicists, or statisticians… (I jest)

For instance: I often have the experience of using the term “categorification” in describing something I’m doing - often in scare-quotes, followed by some kind of explanation - only to have it echoed back as “categorization”, and wonder whether to risk pedantry and explain that they’re not the same thing at all. “Categorification, not to be confused with…”

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On another note, I went looking for this paper by Carter, Kauffman and Saito, on a kind of invariant of 4-manifolds which generalizes 3D Dijkgraaf-Witten invariants, on the supposition that it would be closely related to some things I’ve been thinking about, from a diagrammatic point of view I’ve not paid much attention to in the last year or so. As I was looking through seach results, I noticed a paper from about 10 years ago by Kauffman and Smolin with an interesting sounding title, A Possible Solution to the Problem of Time in Quantum Cosmology. Since Lee Smolin has written on linking topological field theory and quantum gravity, I guessed it would also be interesting to look at. Only after reading the first few pages did I notice that the first listed author was not Louis Kauffman, who studies knot theory (and things tangent thereto), but Stuart Kauffman, who studies biocomplexity and complex systems.

I happen to be interested in the work of both Kauffmans - more immediately and professionally that of Louis, but I also read a couple of Stuart’s more accessible books, “At Home in the Universe”, and “Investigations” - and since the paper was short, I finished reading it. The basic premise is that the configuration space for 4D quantum gravity may not be constructible by any finite procedure (classifying spin networks, they say, might present a problem; doing path integrals over all 4-manifold topologies certainly does). So the “problem of time”, that there’s no role for time in describing dynamics in terms of paths through a configuration space, wouldn’t make sense - at least for a constructivist. (Or indeed a constructivist, though of course they shouldn’t be confused.) One thing that threw me off in noticing which Kauffman was involved was that part of this portion of the argument was about classifying knots.

That cleared itself up when they got to the part proposing a solution - that the total space of possible states isn’t a-priori given, but time re-enters the situation as the universe evolves, at each time step having some amplitude to move into each configuration in a (newly defined!) space called the adjacent possible. Having read Stuart K.’s books, this was when I realized my mistake - he describes this concept in “Investigations” in the context of a biosphere, or an economy, where a theorist also doesn’t have an explicit description of all possible future states given in advance.

It seems like this idea has a lot in common with type theory as a solution to Russell’s paradox: the collection of all sets isn’t a set, and so to get at it, sets are generated starting with nothing in successive stages. Whether this also doubles as a solution to the problem of time, I don’t know. In any case, it’s an interesting idea. It definitely would be a problem to have to do path integrals over a space of all topologies for 4-manifolds, when these can’t be classified, so some sort of suggestions are definitely a good thing here.