In which I stop by NYU and get categorified by Kazhdan

I had the good luck to be in New York on Friday when David Kazhdan gave an unscheduled lecture at NYU about categorification and representations of finite groups.  For people like me, who spend most of our days dismally uncategorified, the talk was a beautiful advertisement for categorification.

Actually, the first twenty minutes of the talk were apparently a beautiful advertisement for the Langlands program, but I got lost coming from the train and missed these.  As a result, I don’t know whether the results described below are due to Kazhdan, Kazhdan + collaborators, or someone else entirely.  And I missed some definitions — but I think I can transmit Kazhdan’s point even without knowing them.  You be the judge.

It went something like this:

Let G be a reductive split group over a finite field k and B a Borel. Then C[G(k)/B(k)] is a representation of G(k), each of whose irreducible constituents is a unipotent representation of G(k).  (Note:  the definition of “unipotent representation” is one that I missed but it comes from Deligne-Lusztig theory.)

When G = GL_n, all unipotent representations of G appear in C[G(k)/B(k)], so this procedure gives a very clean classification of unipotent representations — they are precisely the constituents of C[G(k)/B(k)].  Equivalently, they are the direct summands of the center of the Hecke algebra C[B(k) \G(k) / B(k)].  For more general G (e.g. Sp_6, E_8) this isn’t the case.  Some unipotent representations are missing from C[G(k)/B(k)]!

Where are they?

One category-level up, naturally.

(see what I did there?)

OK, so:  instead of C[B(k)\G(k)/B(k)], which is the algebra of B(k)-invariant functions on G(k)/B(k), let’s consider H, the category of B-invariant perverse l-adic sheaves on G/B.  (Update:  Ben Webster explained that I didn’t need to say “derived” here — Kazhdan was literally talking about the abelian category of perverse sheaves.)  This is supposed to be an algebra (H is for “Hecke”) and indeed we have a convolution, which makes H into a monoidal category.

Now all we have to do is compute the center of the category H.   And what we should mean by this is the Drinfeld center Z(H).  Just as the center of an algebra has more structure than the algebra structure — it is a commutative algebra! — the Drinfeld center of a monoidal category has more structure than a monoidal category — it is a braided monoidal category.  It’s Grothendieck Group K_0(Z(H)) (if you like, its decategorification) is just a plain old commutative algebra.

Now you might think that if you categorify C[B(k)\G(k)/B(k)], and then take the (Drinfeld) center, and then decategorify, you would get back the center of C[B(k)\G(k)/B(k)].

But you don’t!  You get something bigger — and the bigger algebra breaks up into direct summands which are naturally identified with the whole set of unipotent representations of G(k).

How can we get irreducible characters of G(k) out of Z(H)?  This is the function-sheaf correspondence —  for each object F of Z(H), and each point x of G(k), you get a number by evaluating the trace of Frobenius on the stalk of F at x.  This evidently yields a map from the Grothendieck group K_0(Z(H)) to characters of G(k).

To sum up:  the natural representation C[G(k)/B(k)] sometimes sees the whole unipotent representation theory of G(k), but sometimes doesn’t. When it doesn’t, it’s somewhat confusing to understand which representations it misses, and why.  But in Kazhdan’s view this is an artifact of working in the Grothendieck group of the thing instead of the thing itself, the monoidal category H, which, from its higher categorical perch, sees everything.

(I feel like the recent paper of Ben-Zvi, Francis and Nadler must have something to do with this post — experts?)

More MALBEC: Niyogi on geometry of data, Coen on abstract nonsense

Tuesday, April 21 — tomorrow! — brings the third lecture in the MALBEC series:  Michael Coen, of computer sciences and biostat, talks on “Toward Formalizing “Abstract Nonsense”,” in Computer Sciences 1221 at 4pm.  Here’s the abstract:

The idea of a category — a set of objects sharing common properties
— is a fundamental concept in many fields, including mathematics,
artificial intelligence, and cognitive and neuroscience.  Numerous
frameworks, for example, in machine learning and linguistics, rest
upon the simple presumption that categories are well-defined.  This is
slightly worrisome, as the many attempts formalizing categories have
met with equally many attempts shooting them down.

Instead of approaching this issue head on, I derive a robust theory of
“similarity,” from a biologically-inspired approach to perception in
animals.  The very idea of creating categories assumes some implicit
notion of similarity, but it is rarely examined in isolation.
However, doing so is worthwhile, and I demonstrate the theory’s
applicability to a variety of natural and artificial learning
problems.  Even when faced with Watanabe’s “Ugly Duckling” theorem or
Wolpert’s stingy cafeteria (serving the famous “No Free Lunch”
theorems), one can make significant progress toward formalizing a
theory of categories by examining their often unstated properties.

I demonstrate practical applications of this work in several domains,
including unsupervised machine learning, ensemble clustering, image
segmentation, human acquisition of language, and cognitive
neuroscience.

(Joint work with M.H.Ansari)

Delicious food will follow the talk, as if this delicious abstract isn’t enough!

On Friday,  Partha Niyogi gave a beautiful talk on “Geometry, Perception, and Learning.”  His work fits into a really exciting movement in data analysis that one might call “use the submanifold.”  Namely:  data is often given to you as a set of points in some massively high-dimensional space.  For instance, a set of images from a digital camera can be thought of as a sequence of points in R^N, where N is the number of pixels in your camera, a number in the millions, and the value of the Nth coordinate is the brightness of the Nth pixel.  A guy like Niyogi might want to train a computer to distinguish between pictures of horses and pictures of fish.  Now one way to do this is to try to draw a hyperplane across R^N with all the horses are on one side and all the fish on the other.  But the dimension of the space is so high that this is essentially impossible to do well.

But there’s another way — you can take the view that the N-dimensionality of the space of all images is an illusion, and that the images you might be interested in — for instance, some class of images including all horses and all fish — might lie on some submanifold of drastically smaller dimension.

If you believe that manifold is linear, you’re in business:   statisticians have tons of tools, essentially souped-up versions of linear regression, for fitting a linear subspace to a bunch of data.  But linearity is probably too much to ask for.  If you superimpose a picture of a trout on a picture of a walleye, you don’t get a picture of a fish; which is to say, the space of fish isn’t linear.

So it becomes crucial to figure out things about the mystery “fish manifold” from which all pictures of fish are sampled; what are its connected components, or more generally its homology?  What can we say about its curvature?  How well can we interpolate on it to generate novel fish-pictures from the ones in the input?  The work of Carlsson, Diaconis, Ghrist, etc. that I mentioned here is part of the same project.

And in some sense the work of Candes, Tao, and a million others on compressed sensing (well-explained on Terry’s blog) has a similar flavor.  For Niyogi, you have a bunch of given points in R^N and a mystery manifold which is supposed to contain, or at least be close to, those points.  In compressed sensing, the manifold is known — it’s just a union of low-dimensional linear subspaces parametrizing vectors which are sparse in a suitable basis — but the points are not!