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January 3, 2007

Research Proposals

Posted by David Corfield

There’s nothing quite like a research proposal to give you a sense of some of the big stories out there. Try Geometry and Quantum Theory for what’s happening in Holland of relevance to the Café. From a couple of years back, John Rognes et al. talked about Brave New Rings, and, of course, there is John’s proposal for Higher Categorical Structures and Their Applications written with Peter May. Personal research statements are equally interesting, see for example Dror Bar-Natan’s discussion of the categorification of knot invariants, including a mention of the “big dream of categorifying all of quantum algebra”. Posted at January 3, 2007 1:23 PM UTC

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11 Comments & 2 Trackbacks

Re: Research Proposals

I’m pleased that Dror Bar-Natan is interested in categorification! Here are some snippets from his research proposal:

The second reason [for studying Khovanov homology] is much better. Generally speaking, homology is “functorial”. A map between spaces provides no relationship between their Euler characteristics, but always yields a map between their homologies. Without this we wouldn’t be proving the Brouwer fixed point theorem in the first class of every algebraic topology course; it is the primary reason why homology is interesting. The excellent news is that Khovanov homology is likewise “functorial”, for the appropriate (4-dimensional) notion of “morphisms” between (3-dimensional) knots.

[…]

The third reason [for studying Khovanov homology] is the most speculative, yet IMHO it is by far the most exciting. Nobody expected Khovanov homology. The Jones polynomial has its natural place in the world of quantum algebra and topological quantum field theories. Khovanov homology yet doesn’t. Could it be that Khovanov homology is an accident? Not really, for in 2004 came Khovanov and Rozansky and showed that the HOMFLY polynomial has a lift to a homology theory, much like Khovanov lifts Jones. So the reasonable expectation is that Jones and HOMFLY lift to homological theories because their context, or at least a part of their context, can be lifted. That context is Lie algebras, quantum algebra and quantum field theory; we can now fairly expect that these great subjects are merely the “Euler” shadows of even bigger structures. Math hardly ever gets more exciting than this. The young and smart and the old and wise are converging and they will eventually unravel these bigger structures for everybody’s joy. But it’s in the back yard of what I’ve always done and I may still have something to say before it gets too crowded.

I guess it’s time for me to start explaining how James Dolan and Todd Trimble and I have categorified the theory of quantum groups!

Posted by: John Baez on January 4, 2007 1:57 AM | Permalink | Reply to this

Re: Research Proposals

How far will you be able to parallel what was done with representations of quantum groups? Invariants for 2-tangles? Invariants for 4-manifolds?

Posted by: David Corfield on January 4, 2007 8:44 AM | Permalink | Reply to this

Re: Research Proposals

The 2-tangle invariants derived from Khovanov Homology, so far, have lead to quite trivial invariants of knotted surfaces in 4-dimensional space. This failure to find something interesting seems to stem from the intrinsically skein theoretic point of view. The double points and triple points of the knotted surfaces are smoothed to oblivion, and the neck cutting relations seem to remove anything interesting that might come from genus. See the papers Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan’s theory or Khovanov’s invariant for closed surfaces.

This does not yet mean that the whole categorification endeavor for the neoclassical invariants (Jones HOMPTFLY Kauffman) is doomed to yield trivial invariants for knotted surfaces. Indeed, I still also hope that the chain homotopy equivalences in Osvath-Szabo can be used to construct non-trivial invariants. The work of Morrison and Nieh, On Khovanov’s cobordism theory for su(3) knot homology, may also be relevant.

However, it seems (to me at least) that some cohomological information (similar to quandle cocycles) has to come into the mix. These quantities should appear out of the deformations of the algebraic structures.

I am interested in the comment about 3-manifolds in 5 space though. Please explicate.

Posted by: Scott Carter on January 16, 2007 3:33 AM | Permalink | Reply to this

Re: Research Proposals

It’s great to have you visit, Scott. Your diagram adorns the cover of my paperback, so can’t be cast aside as when it appeared on the dust jacket of the hardback.

I am interested in the comment about 3-manifolds in 5 space though. Please explicate.

Which comment are you referring to?

Posted by: David Corfield on January 18, 2007 11:34 AM | Permalink | Reply to this

Re: Research Proposals

David,
I think the item I was refering to was in the quote from Bar-Natan at the top of this thread.

Khovanov Homology is great, and one particularly nice feature is found in Bar-Natan’s exposition. This is a lesson that we all should learn: If you have something that is counted by a power of two, arrange it in a hypercube. That seems obvious, now, but prior to, everything was expanded as a binary tree.

I have been having a lot of fun with these ideas. Nothing deep, the binomial theorem makes more sense now that it is seen on the n-cube. Someone might get a kick out of a quatum version thereof.

Anyway I sort of thought that there was a claim being made about relations among the chain homotopy equivalences in KhoHo.

Posted by: Scott Carter on January 26, 2007 1:43 AM | Permalink | Reply to this

Khovanov homology

That context is Lie algebras, quantum algebra and quantum field theory; we can now fairly expect that these great subjects are merely the “Euler” shadows of even bigger structures.

Why? How?

I don’t know what Dror Bar-Natan is talking about here. But it sounds indeed fascinating.

Can anyone give a brief, rough, explanation?

I guess it’s time for me to start explaining how James Dolan and Todd Trimble and I have categorified the theory of quantum groups!

Please do!

If that also involves explaining what Khovanov homology itself is, please start with that!

I understand that Aaron Lauda will – “if time permits” – talk about Khovanov homology in Toronto - but maybe you can give us rough sneak preview of what the basic idea is.

Posted by: urs on January 4, 2007 10:35 AM | Permalink | Reply to this

Re: Khovanov homology

Bar-Natan’s a good expositor. You might try his own paper Khovanov’s Homology for Tangles and Cobordisms.

Posted by: David Corfield on January 4, 2007 10:52 AM | Permalink | Reply to this

Re: Khovanov homology

Aaron Lauda’s a good expositor too, and as you know, he’s working with Khovanov at Columbia. You might try his talk at the Fields Institute next week! He plans to explain this diagram:

 2d TQFTs -----------> Khovanov Homology for links
   |                        |
   |                        |
   |                        |
   V                        V
 2d TQFTs ----------> Khovanov homology for tangles
 (extended)                 |
   |                        |
   |                        |
   |                        |
   V                        V
  State sums ---------> Full braided monoidal
underlying TQFTs       2-category underlying
                         Khovanov homology

Since you like the first column of this diagram, you’ll love the second column!

For details, try Lauda’s paper with Hendryk Pfeiffer and also his forthcoming work with Pfeiffer and Khovanov.

Posted by: John Baez on January 4, 2007 8:42 PM | Permalink | Reply to this

Re: Khovanov homology

you’ll love the second column!

I am sure I will!

Lauda’s paper with Hendryk Pfeiffer

Ah, thanks. I missed that.

Posted by: urs on January 4, 2007 8:47 PM | Permalink | Reply to this
Read the post Khovanov Homology
Weblog: The n-Category Café
Excerpt: Khovanov homology and its generalization from links to tangles.
Tracked: January 14, 2007 5:48 AM

Re: Research Proposals

A related paper appeared today. Mednykh’s Formula via Lattice Topological Quantum Field Theories by Noah Snyder.

Abstract:

Using techniques from quantum field theory Mednykh proved that for GG any finite group and SS any orientable closed surface, there is a formula for #Hom(π 1(S),G)#Hom(\pi_1(S), G) in terms of the Euler characteristic of SS and the dimensions of the irreducible representations of GG. A similar formula in the nonorientable case was proved by Frobenius and Schur using group theory and an explicit presentation of π 1(S)\pi_1(S). Here we present a new proof of these results which uses only elementary topology and combinatorics. The main tool is the lattice topological quantum field theory attached to a semisimple algebra.

Posted by: David Corfield on March 6, 2007 12:05 PM | Permalink | Reply to this

Re: Research Proposals

Thanks for this link! This is interesting. Have downloaded it and now I am starting to read it.

Posted by: Bruce Bartlett on March 6, 2007 8:31 PM | Permalink | Reply to this
Read the post Generalising Hopf Algebras
Weblog: The n-Category Café
Excerpt: Gizem Karaali's paper 'On Hopf algebras and their generalizations', in which she describes Hopf algebras and five attempts to generalise them.
Tracked: March 20, 2007 2:02 PM

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