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October 11, 2009

MSRI Workshops on Knot Homology Theories

Posted by Alexander Hoffnung

I promise some entries with mathematical content soon, but today I wanted to draw attention to some upcoming workshops affiliated with the MSRI semester on knot homology theories . The two workshops are:

  • The Connections for Women Workshop,

organized by Eli Grigsby, Olga Plamenevskaya, Katrin Wehrheim, and what is called

  • The Introductory Workshop,

organized by Aaron Lauda, Robert Lipshitz, Dylan Thurston.

I’ll leave any actual information regarding the content until below the fold. Just so people who do not already love knot homology theories keep on reading, I will quote the organizer’s invitation:

Both workshops will feature a mix of survey, introductory, and research talks, which are aimed at non-experts. One goal is to familiarize workers in one of the fields represented with the tools and ideas of the other fields.

They seem to have some funding for graduate students and fresh Ph.D.’s, so this is a good opportunity for graduate students to get some exposure to some exciting topics.

The Connections for Women workshop, scheduled for January 21-22, 2010, will focus on:

  • positioning knot homology theories in a broader mathematical context,

and

  • emphasizing connections to contact/symplectic geometry, quantum topology, and representation theory.

There will be opportunities (through invited research talks and a poster session) for young women working in the field to discuss and present their research. Invited speakers include Shelly Harvey, Keiko Kawamuro, Effie Kalfagianni, Gordana Matic, Dusa McDuff, Heather Russell, and Vera Vertesi.

More details can be found here .

Here is some clarification on the idea of a women’s workshop from the organizers:

The Connections for Women workshop is absolutely open for men to attend, and we hope they will find it valuable and comfortable to do so. The only catch: the funds MSRI gives us to support graduate students, etc., are earmarked for females.

I didn’t immediately notice any men on the list of participants. So, if you are going to be around for other parts of the program or other workshops, do not be shy to sign up for this as well. They have some really great speakers!

The Introductory workshop, scheduled for January 25-29, 2010, will introduce the main branches in the study of knot homology theories. It will consist of three mini-courses:

  • one on knot Floer homology and related topics;
  • one on the various approaches to Khovanov and Khovanov-Rozansky homology;
  • and one on categorification of quantum groups.

There will also be several stand-alone lectures. Invited speakers include Sabin Cautis, Matt Hedden, Andras Juhasz, Scott Morrison, Lenhard Ng, Catharina Stroppel, and Ben Webster.

More details can be found here .

I know about some of these topics and would love to learn more about others. I would be interested to know if someone could provide more details on the stand alone talks. I think some of the speakers are probably reading this.

Posted at October 11, 2009 11:09 PM UTC

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13 Comments & 0 Trackbacks

Re: MSRI Workshops on Knot Homology Theories

Well, here’s one speaker who’s reading. I can’t be all that specific about what I’ll say, since I haven’t planned the talks yet, and exactly what I say may depend on what people prove over the next couple of months.

Posted by: Ben Webster on October 12, 2009 4:32 AM | Permalink | Reply to this

Floer Homology Theories

Does anyone know of a `Floer homology for dummies’?

Posted by: jim stasheff on January 28, 2010 2:21 PM | Permalink | Reply to this

Re: Floer Homology Theories

There is a nice book:

  • Donaldson: “Floer homology groups in Yang–Mills theory”

Here is the pdf version of the review in the ZMATH database.

I’m a bit lost at guessing what you would consider a “dummy” , but I hope that this book will be of interest to one or maybe even two readers of this blog.

From the introduction:

Floer’s original motivation for introducing his groups - beyond the intrinsic interest and beauty of the construction - seems to have been largely as a source of new invariants in 3-manifold theory, refining the Casson invariant which had been discovered shortly before. It was soon realised however that Floer’s conception fitted in perfectly with the ‘instanton invariants’ of 4-dimensional manifolds, which date from much the same period. Roughly speaking, the Floer groups are the data required to extend this theory from closed 4-manifolds to manifolds with boundary. From another point of view the Floer groups appear, formally, as the homology groups in the ‘middle dimension’ of an infinite-dimensional space (the space of connections modulo equivalence) associated to a 3-manifold. This picture is obtained by carrying certain aspects of the Morse theory description of the homology of a finite-dimensional manifold over to infinite dimensions. All of this is closely related to ideas from quantum field theory - indeed, one of Floer’s startingpoints was the renowned paper of Witten, [49], which inter alia forged a link between quantum mechanics and Morse theory - and the connection with mathematical physics permeates the whole subject.

Ref. 49 is this: [49] E. Witten, Supersymmetry and Morse theory Jour. Differential Geometry 17 661–692 1982

Posted by: Tim van Beek on January 29, 2010 9:06 AM | Permalink | Reply to this

Re: Floer Homology Theories

Sounds like what my friend Larry Conlon needs. Many of the technical tools are at his command. I’ll encourage him to participate directly at the cafe.

Posted by: jim stasheff on January 29, 2010 1:50 PM | Permalink | Reply to this

Re: Floer Homology Theories

There is more than one thing called “Floer homology.” My understanding is that they are all infinite dimensional Morse theories in the situations where the gradient flow doesn’t exist but the Morse complex does.

Posted by: Eugene Lerman on January 29, 2010 5:59 PM | Permalink | Reply to this

Re: Floer Homology Theories

This is sort of why I have been hesitant to reply, but of course, the week of introduction to Heegaard-Floer homology is just coming to an end and the lectures from MSRI should be available online shortly.

Posted by: Alex Hoffnung on January 29, 2010 6:05 PM | Permalink | Reply to this

Re: Floer Homology Theories

That was my understanding too: Floer’s construction is an infinite-dimensional variant of Morse theory that applies to two a priori different contexts:

- to define symplectic invariants for pairs of Lagrangian submanifolds of a symplectic manifold, this is sometimes called Lagrangian Floer homology,

- to give topological invariants for three-manifolds, this is sometimes called instanton Floer homology.

The book by Donaldson is about the latter, naturally, since his interest lies partly in - ugh - Donaldson invariants.

Posted by: Tim van Beek on January 31, 2010 11:47 AM | Permalink | Reply to this

Re: Floer Homology Theories

Not to mention the Floer homology that is the basis of various proofs of the Arnold conjecture (roughly speaking it “counts” periodic orbits of a Hamiltonian vector field) or contact homology (which “counts” Reeb orbits and provides invariants of contact structures) or Symplectic Field Theory (which generalizes both) or …

Posted by: Eugene Lerman on February 1, 2010 1:33 AM | Permalink | Reply to this

Re: Floer Homology Theories

Cool! Maybe someone should draw a chart of the different notions and their interconnections, but that would not be me, because my knowledge seems to be both incomplete and outdated.

Larry Conlon mentions “sutured Floer homology”, and skimming the material I still have I did not find any mention of that - but a quick online search showed that some authors date the invention of that back to 2006. By that time I had already dropped out of academia - now here is a good excuse if there ever was one.

But seriously: If there is a book or an expository article about “sutured Floer homology” then I did not find it.

Posted by: Tim van Beek on February 1, 2010 8:44 AM | Permalink | Reply to this

Re: Floer Homology Theories

I think sutured Floer homology was introduced by Juhasz in: A. Juhász, Holomorphic discs and sutured manifolds, Algebraic and Geometric Topology 6
(2006), 1429-1457. The paper that got my attention was: S. Friedl, A. Juhasz and J. Rasmussen, THE DECATEGORIFICATION OF SUTURED FLOER HOMOLOGY
(arXiv:0903.5287v2 [math.GT] 8 Jun 2009).

Posted by: Larry Conlon on February 1, 2010 6:20 PM | Permalink | Reply to this

Re: Floer Homology Theories

These are exactly the papers that I came up with, I’ll post the links:

which refers to

and since I look up TQFT from now and then I also came up with:

Maybe a kind soul picks us up

Posted by: Tim van Beek on February 2, 2010 9:10 AM | Permalink | Reply to this

Sutured Floer Homology

John Cantwell and I are curious to learn about knot Floer homology and sutured Floer homology. There is a “sutured Floer norm”(Friedl, Juhasz and Rasmussen) which is related to, but not equal to, the “sutured Thurston norm” (introduced by Scharlemann as a “generalized Thurston norm”). This latter norm is useful for analyzing our “foliation cones” (Proceedings of the KirbyFest, Geometry and Topology Monographs, 1999) and we wonder if the Floer norm might also be a useful tool in this study? Our work originated in efforts to completely classify Gabai’s depth one foliations of disk decomposable knot and link complements.

As outsiders, we find the literature on Heegard Floer homology quite daunting. Where does this come from, where is it going and why should I care? Raoul Bott once remarked to me that, when mathematicians write papers, they almost always lie! They give you an organized logical account while telling you nothing about the exciting process, intuitions and insights by which their theory was discovered.

Posted by: Larry Conlon on January 31, 2010 12:39 AM | Permalink | Reply to this

Re: Sutured Floer Homology

Beeing an outsider myself the best place to start I know of is
- “Floer Homology, Gauge Theory and Low-Dimensional Topology”, Proceedings of the Clay Mathematics Institute 2004 Summer School.

The first two papers are an introduction to Heegard Floer Homology.

…when mathematicians write papers, they almost always lie!

All scientists do in a certain sense! See for example:

- Karin Knorr Cetina: “Die Fabrikation von Erkenntnis” (not available in English, it would seem),

- Karin Knorr Cetina: “Epistemic Cultures “

Posted by: Tim van Beek on January 31, 2010 9:20 AM | Permalink | Reply to this

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