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May 13, 2008

Ottawa

Posted by Jeffrey Morton under category theory, conferences, meta, musing, philosophical, talks
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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 \epsilon of one another if they produce matching regions about the origin of size \frac{1}{\epsilon} – 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.

 

April 29, 2008

Recent Talk – Enxin Wu on Algebra Deformations

Posted by Jeffrey Morton under algebra, cohomology, quantization, talks
1 Comment 

I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice – gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa.

Enxin is one of those guys who seems to really understand – it’s tempting to say grok- algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me – you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder.

The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are A_{\infty}-algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras – the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra A by new things, valued in the bigger algebra A[[t]] of formal power series in t with coefficients in A, so that the original structure you started with is just the constant part that appears when you set t=0. (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket – in other words, algebras of observables of a physical system).

Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation – and there certainly is a lot – but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me.

So let’s get more specific.

The point is to replace a multiplication operator m : A \otimes A \rightarrow A with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra (A,m) (where m : A \otimes A \rightarrow A is the multiplication for A) is (A[[t]],m_t), where the new multiplication m_t is defined (by linearity) by its action on elements of A, which works like this:

m_t(a,b) = \sum_{i=0}^{\infty} {\alpha_i}(a,b){t^i}

for some operators \alpha_i : A \otimes A \rightarrow A. Then there are a bunch of conditions on the \alpha that are needed to make m_t associative. There’s one condition for each power of t, since the coefficients in the associator should be zero:

\sum_{i+j=n\\i,j>0} \alpha_i( (\alpha_j \otimes 1) - (1 \otimes \alpha_j)) = 0

The n=0 condition just says that \alpha_0 is associative – so it’s the m from the original algebra, which you get back when t=0.

Then given an algebra A, you can create the deformation category \mathcal{D} of A whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero n=0 term are invertible (a consequence of the Lagrange theorem) this \mathcal{D} is actually a groupoid. Then the question is to classify the isomorphism classes of deformations – that is, \Pi_0(\mathcal{D}). One can easily imagine that there might be no nontrivial deformations of some algebra – that is, every one is isomorphic to the deformation where all the \alpha_i are trivial except \alpha_0 = m. So when does this happen? More generally, how can one classify the deformations up to isomorphism?

The answer has to do with Hochschild cohomology, which is related to a complex you can make from A. Taking C^n(A) = hom(A^{\otimes n},A), the space of n-ary multilinear operations on A, you build this complex:

0 \stackrel{d_0}{\longrightarrow} C^0(A) \stackrel{d_1}{\longrightarrow} C^1(A) \stackrel{d_2}{\longrightarrow} \dots

where the differential maps are d_n : C^n(A) \rightarrow C^{n+1}(A) defined by an alternating sum:

d(f)(a_1, \dots, a_n) = a_1 f(a_2, \dots, a_{n+1}) + \sum_{i=1}^{n} (-1)^i f(a_1, \dots, a_i a_{i+1}, \dots, a_{n+1}) + (-1)^{n+1} f(a_1, \dots,a_n) a_{n+1}

(Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that d^2 = 0, you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.)

Then you take the Hochschild cohomology groups in the standard cohomology way: HH^i = \frac{ker(d_{i+1})}{Im(d_i)}. A cohomology class in one of these groups is a class of multilinear maps from n copies of A to A (up to a factor which is d_n of something). As usual with cohomology, they describe obstructions to something – to exactness. Exactness, in this setting, would mean that A has no interesting deformations at the n^{th} level.

What does “level” mean here? Well, for example, at level 2 we’re talking about maps A \otimes A \rightarrow A, such as the multiplication map. In fact, we have d_3(m) = 0 for an associative algebra – you can check that d(m) is twice the associator a_1(a_2a_3) - (a_1a_2)a_3, which is zero. So m is a cochain. Is it a coboundary? Sure – it’s d_2(1). So m is in the trivial class in HH^2(A). The point then is that it turns out that if this is the only class – if HH^2(A) = 0 – then there are no interesting deformations of the multiplication of A in the sense described above. The groupoid $\mathcal{D}$ has just one object. (One thing that occurs to me is that this makes it a group – which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if A = \mathbb{C} (as an algebra over \mathbb{R}), there are no nontrivial deformations: HH^2(\mathbb{C}) = 0.

What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to A_{\infty}-algebras. Whereas for an associative algebra A, the associator $a(x,y,z) = x(yz) – (xy)z$ is zero, in general an A_{\infty}-algebra will have an associator map a : A^{\otimes 3} \rightarrow A (that is, a \in C^3 in the complex above), which might not be zero, but which is d_3(m).

This is the beginning of a story relating A_{\infty}-algebras to weak \infty-categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity – the Pentagon identity – but in general you’d just ask for a “pentagonator” (something in C^4), and so on (this is where those seminar notes above help me think in pictures, by the way). An A_{\infty}-algebra is a vector space equipped with maps at all these levels – described by Stasheff’s associahedra – satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of A_{\infty}-algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of A is a deformation not in A[[t]], but in the quotient A[[t]]/(t^2=0). This only needs two maps, \alpha_0 , \alpha_1. The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in A[[t]] – if HH^3(A) = 0, then any infinitesimal deformation can be extended to a full deformation.

All in all, I thought the talk was interesting – it tied in much more closely to things I already knew about TQFTs and higher categories than I’d expected. I’ll be really impressed if he can condense it into a 25-minute version…