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.

 

A Problem Arises

In “The Fabric of Reality”, David Deutch gives a refutation of solipsism. I’m not entirely sure it works - all he really tries to do is to show that the difference between solipsism and realism is more nearly a mere semantic distinction than is generally assumed. But in any case, along the way, there’s an anecdote about a solipsist professor lecturing his (imaginary?) class merely to help him clarify his ideas. The idea being that, even if the imaginary students don’t really exist, it helps to clarify the professor’s own ideas by lecturing to them, answering questions, and so forth. In this view, you don’t really understand your own opinions - let alone justifiably believe in them - unless you’ve argued for them against a variety of possible criticisms. (J.S. Mill gave a defense of full-fledged freedom of speech, even for grossly offensive and even “dangerous” opinion, on this ground.)

I mention this because, when I told Dan about the blog, he seemed dubious about blogging as a way of communicating math. It’s certainly more solipsistic than a usenet newsgroup, or a mailing list. Those are channels devoted to a particular subject, with many participants. A blog, comments notwithstanding, is mainly a channel devoted to one voice, on many particular subjects. It’s true that half the point of communicating ideas is to get feedback on them from other people. You make your thinking part of one of those great processes like cathedral-building - ad-hoc, gradual, and (significantly) collective. Even so, relatively solipsistic channels are not entirely pointless.

To wit: by working through my theorems about transporting 2-vectors through spans - both for this blog, and for my talk at Groupoidfest, I discovered some problems. Nobody pointed them out, but discovering them was a consequence of approaching the material again from a new angle, with an audience in mind.

The problem is a conceptually important one - mistaking an n-dimensional space for a 1-dimensional space. I’m fairly sure, for various reasons, that the theorem that there is a 2-functor is still true, but the proof I have in my thesis (in the special case where the groupoids are flat connection groupoids on spaces) has a problem. Since that affects the Part 4 of “Spans and Vector Spaces” which I was going to post, I’ll put that off for a while as I get the proof straightened out.

Here is the issue in a nutshell, however:

The proof I have involves a construction of a functor by a particular method, which I’ve been describing in the last three posts. The final step I was going to describe involved what the contstruction does for 2-morphisms - spans between spans. (There is more to the proof, but the remainder is technical enough to be fairly unenlightening - basically, to be a 2-functor, there need to be specified natural isomorphisms replacing the equations for preserving identities and composition in the definition of a functor, and these have to obey some equations which need to be checked.)

The construction given in my thesis is supposed to give a way to take a span of spans of groupoids, and give a natural transformation between a pair of 2-linear maps. But a 2-linear map can be written as a matrix of vector spaces, and a natural transformation is then written as a matrix of linear operators which act componentwise. So one way to look at the problem is to construct a linear map between vector spaces from a span of groupoids.

That is, we have spans and . Picking basis objects for and (namely, objects and , plus representations of their automorphism groups) gives a subgroupoid of of , consisting of those objects which are sent to and under the maps in the span. It also gives a vector space which is built as a colimit of some vector spaces associated to these objects. Assuming is skeletal, this works out (as I described before) to for each of the in question. The same holds for .

Now suppose we have a span-of-spans making the obvious diagram commute. Then because of that commutation, we also have a span of groupoids over each of the choices of objects, and so then the question becomes, partly, how to get a linear map between the vector spaces we just constructed. If you have bases for all the vector spaces here, it’s not too bad: vectors can be seen as complex-valued functions on the basis. We can push these through the span just as we’ve been talking about in the last few posts here: first pull back a function along one leg by composition, then push forward along the other leg. The push-forward will involve a sum over some objects, and some normalizing factors having to do with the groupoid cardinalities of the groupoids in the span.

However, I won’t go too far into detail about this, because the construction I actually outlined doesn’t adequately specify the basis to use. In fact, it will really only work if all the vector spaces is one-dimensional. Then there is a basis for the combined space which just consists of all the objects . I’d hoped that Schur’s lemma (that intertwiners from to itself, or from to itself, have to be multiples of the identity) would get out of this problem, but I’m not sure it does. So there is a problem with the construction I was trying to use.

As I say, I’m fairly sure the theorem remains true - it’s just the proof needs fixing, which I don’t expect to be too hard. However, I’ll refrain from getting sidetracked until I know I have it worked out.

Instead, next time I’ll describe some of the things I learned at Groupoidfest 07 when I presented a talk on this stuff. (At first I was nervous, having discovered this flaw while preparing the talk - but then, a lot of people were talking about work-in-progress, so I don’t feel too bad now. Plus, the meeting was a lot of fun.)

 

The Theoretical Atlas - Title

You may be wondering about the title: “Theoretical Atlas”. Both words have a double meaning here.

First, Atlas: originally, this was the name of a Titan in Greek mythology, who was condemned by Zeus to stand at the Western edge of the world and hold up the sky on his shoulders forever. The Western edge of the Greek world - the Mediterranean - is indeed where the Atlas mountains are found, in the Maghreb. Also named for him is the Atlantic Ocean (and, therefore, Atlantis, a continent once speculated to be located somewhere in it). You can see a picture of the Atlas mountains in the banner at the top of this blog’s main page.

So one meaning comes from a notion that tends to crop up fairly often when one talks about the project of finding a quantum theory of gravity. This is the prospect of a complete unified theory of physics, a Theory of Everything (TOE), or some such name. People peering into the mist of our limited knowledge sometimes seem to see prospects of a single theory that unifies every aspect of the physical world in one single model - all forms of matter, energy, forces, gravity, etc. The name “M-theory” is popular in some circles for this idea - an as-yet undiscovered theory which might go beyond what string theory can do today. Other prospects have been proposed, but the image I have is of a single, immensely powerful theory, holding up the entire world on the strength of its explanatory power - a theoretical Atlas holding up this enormous burden.

But this great Atlas of a theory has never been written down - alas. For myself, I’m quite skeptical if it even could be: why should there be a short, pithy idea that encodes the whole huge, complex, endlessly surprising universe? Even if we had a theory which accounted for all particles and forces in nature, would that be a theory of everything? The point of a theory, after all, is to help us understand things: we’d still need, at the very least, a theory to explain how chemistry emerges from physics, what life is and how it can come into being - all just to account for even our most basic experience. Then there are whole areas of the world that open up from there. So this great single Atlas of an idea that accounts for the entire world of experience is, as they say, just a theory. It’s a (merely) theoretical Atlas.

(Of course, this use of the phrase “just a theory”, often used to dismiss the insights of Darwin, and much less prominently used any other way, is simply wrong. The meaning of “theory” depends on context, but it always means something more than a mere guess. Still, as I said before, I’m not going to worry TOO much about being wrong now and then - and the more accurate hypothetical Atlas just didn’t sound as good.)

The other meaning of the word “atlas” has to do with maps. The other element of the banner above mentions the Bellman’s map from The Hunting of the Snark. It had no markings on it at all - “purely conventional signs”. But mathematics is all about using purely conventional signs as a reference point in describing the features of the world. The Bellman’s map showed no land - only sea - and so it left out not only the conventional reference points, but also anything definite to refer to.

A “theory” can be seen as a way of taking some standard, pre-existing structure, and trying to “map” it onto the features we see in the real world. In a way, a literal map is an example of a theory: it imposes a regular grid of coordinates on some convoluted shape, which is itself a model of some territory off elsewhere in the world. It’s an artificial imposition - but it allows us to find our way around. Assuming it’s accurate enough, and we know how to read it.

In the case of a literal atlas, we have a collection of - usually flat, generally rectangular - drawings of the surface of a sphere (more or less). Each one is a little bit distorted, because the Earth isn’t flat (no, no, I know - that’s just a theory - but I think it’s accurate enough). In the study of manifolds, these are called “charts” - each one is a map from some open subset of Rⁿ to a subset of the manifold. Generally - and, for instance, on the surface of the Earth - one chart won’t be enough. You need several charts, and an understanding of how they fit together. The collection of charts is an atlas, and one imagines a big book filled with these charts, each one imposing a rectilinear grid of coordinates onto some underlying terrain. “Transition maps” tell you how they fit together to cover the whole surface.

So the other meaning of theoretical atlas is the notion that we may need many theories to properly account for the world. Each one may describe some part of it fairly well - maybe with a bit of distortion, but certainly not so much that it doesn’t help to find our way around. None by itself explains everything - but given enough, and some knowledge of how to manage the transition between the domain of one theory and the domain of another, they can tell us a lot. This is my image of what our researches into physics, and the world in general, are aiming at: an atlas of theories that covers everything.

Mind you, I realize that such an atlas, like the other kind of Atlas, is purely theoretical.

Oh, all right: hypothetical.

 

The Theoretical Atlas - An Apology

Here is an apology - with apologies to the Unapologetic Mathematician…

One inspiration for starting this blog is the fact that Dr. Baez has a great abundance of stuff on the Web. Some of the better-known include the ever-popular This Week’s Finds in Mathematical Physics, and the newer n-Category Café, which is a group venture together with Urs Schreiber and David Corfield. Between the three of them, they write on “math, physics, and philosophy”. That’s more or less what I propose to do here.

Why the redundancy?

The n-Category Café has turned out to be a very productive way of sharing ideas informally over long distances, and without being too confined by a narrow topic or the strictures of publishability. The participants have also adopted the ethic that it’s better to share ideas than keep them secret until they’re perfected. One essential reason is that science, math, and philosophy are cultural products - discussion is like oxygen for culture. This is a lesson that has been learned many times in the past, and, I suspect, will have to be learned many times again in the future. Publication, peer review, giving public talks - the whole essence of research is communicating ideas. Of course, you need to develop good ideas to communicate, but the point is to share and discuss them. One more voice in a conversation like that may be a drop in a bucket, but it’s not redundant.

So I aim for this to be my particular drop in our great collective bucket. I’ll relate things that I’ve been thinking about; explain things I’ve figured out; express confusion over things I haven’t; describe the experience of starting a research career; muse; investigate; and, if possible, not bloviate. And I won’t worry too much about being incomplete, tentative, or even (a little bit) wrong. That’s all part of investigating things.

This is as much “apology” (in the sense of a justification of one’s actions - quite the opposite of what we moderns usually mean by “apologize”) as I suppose the minor nuisance of starting yet another blog really requires.

 

The Theoretical Atlas - Introduction

Hello!

This is Entry #1 of this blog, the “Theoretical Atlas”, in which introductions are made. It is the first of a few “meta” entries, setting out where we’re going.

Introductions

My name is Jeffrey Morton. I am a mathematician interested in physics - and life. I’m starting this blog as I start out on my first postdoctoral position after finishing my Ph.D with John Baez at UCR.

The starting point for my research to date was my interest in efforts to find a quantum theory of gravity. It started innocently enough, learning some differential geometry, which led into General Relativity; that, in turn, led into studying quantum field theory in curved spacetimes. There many limitations on what you can say about that: general relativity and quantum field theory are based in very different mathematical vocabularies (not to mention grammar). Relativity theory describes gravity in terms of the geometry of spacetime - which is quite definite. Quantum field theory, on the other hand, describe matter in a very different way, in which the observed values of physical quantities can be any of the eigenvalues of certain linear operators on some Hilbert space. In particular, it doesn’t always predict a definite, specific value for the concentration of mass in any given location. But it’s mass-energy, in particular places, which supposedly creates gravity. These theories have a hard time talking to each other unless you carefully limit how much they interact, or assume a great deal of symmetry.

I encountered this fact when I was studying the Einstein-Dirac-Maxwell equations for my M.Sc. and an interest in confronting it led me to UCR. What I learned from Dr. Baez once there pushed my interests in several other directions. As one might expect: if the problem is that two theories use very different language, a step to reconciling it is to develop a new language which can handle both of them. This is where category theory entered the story. Category theory is a very general mathematical language, which can be applied to many subjects within mathematics, and thence to their many and various applications. One way to state the essential idea is that it takes both “things” and “relations” between things (in a very general way) as fundamental concepts. So far, I’ve been thinking about various ideas regarding how this can show up in physics.

Now I have introduced myself and two of the the main conceptual characters, what about the blog?