February 8, 2012

Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Posted by Alexander Hoffnung

This past October Alistair Savage and I organized a workshop on Category Theoretic Methods in Representation Theory at the University of Ottawa. The event was generously supported by the Fields Institute.

Following the success of the October workshop, Oded Yacobi, Chris Dodd, and I decided to hold another workshop, this time with a focus on researchers who are still very early on in their careers. The Fields Institute has again offered funding and this time will host the event as well.

We would like to draw your attention to the upcoming Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory to be held May 7-9, 2012 at the Fields Institute in Toronto.

Keep reading below the fold and see the workshop website for more on the content of the workshop, registration information, and applications for financial support.

This three day workshop will bring together young (mainly postdoctoral and graduate student) researchers in representation theory and related fields with a focus on exciting new developments in algebraic and geometric methods.

The meeting will incorporate both a graduate student workshop and a research level conference. The workshop component will consist of a series of three lectures by Ben Webster (Northeastern) explaining S-duality. In addition to these lectures, there will be research talks given by the leading young researchers in representation theory and related fields.

The goals of this workshop are:

• to highlight algebraic-geometric and categorical methods in representation theory which have made important contributions in recent years,
• to learn about the program on S-duality initiated by Braden, Licata, Proudfoot, and Webster,
• to provide an opportunity for young researchers in the field to learn of and present cutting-edge research, and to foster interaction between Canadian and American researchers working in this area.

There will be approximately 12 talks, each 50 minutes in length, spread out over Monday, Tuesday and Wednesday, with ample time in between talks for discussion. Three of these talks will be reserved for Ben Webster’s lectures.

Here is the current list of speakers:

• Ben Cooper (University of Virginia)
• Ben Elias (MIT)
• David Jordan (UT Austin)
• Carl Mautner (Harvard)
• Weiwei Pan (Göttingen) *
• Jaimie Thind (University of Toronto)
• Peter Tingley (MIT) *
• Ben Webster (Northeastern)
• Xinwen Zhu (Harvard) *

(* tentative)

We will be able to provide limited support for some graduate student and postdoc participants. Registration and applications for support will be open soon at the website. Please don’t hesitate to contact one of us if you have any questions.

We hope to see you there!

Posted at February 8, 2012 7:28 PM UTC

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Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

to learn about the program on S-duality initiated by Braden, Licata, Proudfoot, and Webster,

I have only briefly looked at the slides

Ben Webster, Representation theory and a strange duality for symplectic varieties, 2010 (pdf)

On slide 80 it says

My collaborators originally dubbed this “symplectic duality” but it seems that for talking with physicists in the audience, the name should be shortened to “S-duality.”

[…]

Our examples coincide with a notion of duality in physics; they are the Higgs branches of mirror dual 3-dimensional gauge theories.

I have trouble locating the reference that says this in more detail. Is this related to

• Tudor Dimofte, S-duality and mirror symmetry in Chern-Simons theory (talk)

• Tudor Dimofte, Sergei Gukov, Chern-Simons Theory and S-duality (arXiv:1106.4550)

?

Posted by: Urs Schreiber on February 9, 2012 9:03 AM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

I can’t speak to this myself at all at the moment, but hopefully after hearing what Ben has to say, I can come back and connect the dots.

Posted by: Alex Hoffnung on February 12, 2012 7:15 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

But I suppse there is some canonical entry point for

the program on S-duality initiated by Braden, Licata, Proudfoot, and Webster,

?

What are the standard references? I can’t seem to find any on the workshop website. (?)

Posted by: Urs Schreiber on February 12, 2012 9:36 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Ah, good point. Things should run much more smoothly if the website for the workshop points people towards at least an overview of the subject.

I think a reasonable place to start is the brief overview given in Nick Proudfoot’s research statement, which states some of the proven and conjectural examples of symplectic dualities.

From there it probably makes sense to have a look at the first paper, Gale duality and Koszul duality put out by Braden, Licata, Proudfoot, and Webster as a group. Here, amongst other duality results, they show that Gale dual polarized arrangements have associated symplectic dual hypertoric varieties.

I am sure that hypertoric varieties are not universally familiar to people, so it probably makes sense to list Proudfoot’s A survey of hypertoric geometry and topology to either read first or use as a point of reference while working through the $S$-duality papers.

With Christopher Phan, the four authors put out Localization algebras and deformations of Koszul algebras, which tells us that symplectic dual pairs of algebraic symplectic manifolds have equivariant cohomology rings that are dual as localization algebras.

I don’t know how much this helps right away, but thank you for pointing out the need for some information in this direction on the website. I will take a closer look at the papers just mentioned, the most recent paper Hypertoric category O, and at least some of the host of papers relevant to these works and to the conjectural statements within. I’ll give a suggested reading list on the workshop website, and hopefully some guidance on navigating the papers.

Posted by: Alex Hoffnung on February 12, 2012 11:51 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Urs-

We’re working on the whole standard reference thing (or having a reference period). It’s pretty hard to write a paper on a subject when all you have is conjectures, so we’ve been marshaling our evidence over the past few years in order to really have something to say when we lay out the general picture. We now have drafts of two 60 page long papers on the subject, so it’s a little unfortunate that there’s still nothing for the public to read, but the big picture bits are (as usual) a bit hostage to the niggling details that need to be worked out before anything as rash as posting on the arXiv.

I think in terms of understanding the big picture, Nick’s research statement is a good start. I’ll work on putting together a better reading list for the conference website.

Posted by: Ben Webster on February 17, 2012 5:27 AM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

From Proudfoot’s research statement

The meaning of the relationship between symplectic duality in mathematics and mirror duality in physics remains mysterious; understanding it is one of my main goals in the coming years.

Mirror symmetry, according the the nLab entry,

At least in some cases … can be understood as a special case of T-duality.

Above Urs mentions the link to S-duality, and work connecting S-duality and mirror symmetry.

Is there expected to be some big story which accounts for these four dualities and their relationships?

Posted by: David Corfield on February 17, 2012 8:28 AM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

Thanks Alex, thanks Ben! That helps.

I did see the articles with “Koszul” in the title on the arXiv, but couldn’t tell from their abstracts (and had no time for more than that, alas) whether they are relevant here.

Thanks to the pointers to Nick Proudfoot’s research statement. That helps, too.

Posted by: Urs Schreiber on February 17, 2012 2:34 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

I started writing a page on the nLab to serve as a reference about “symplectic duality.” It’s still a little messy at the moment, but I think it is human readable, and I’d be happy to receive any comments, enquiries or emendations.

Posted by: Ben Webster on February 17, 2012 9:41 PM | Permalink | Reply to this

Re: Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory

I started writing a page on the nLab to serve as a reference about “symplectic duality.”

Thanks, Ben! That’s great. Most useful.

I have gone through the page and done some minor editing. Mostly I have added further hyperlinks and fixed instki syntax here and there (for instance, unfortunately instiki cannot handle math in hyperlinks). Also added a “floating context TOC” in the top right (“symplectic geometry”) and a “References”-section at the very end (which you may, in turn, want to edit).

The analog of what in Wikipedia would be the corresponding “talk page” is on the $n$Forum here.

If you find the time to look at the $n$Forum at all, more important right now would maybe be your attention to the discussion here under “Koszul”. As you’ll see there, I slightly intervened with your creation of an entry titled “Koszul”, because we try to stick to the convention that entries are titled by singular nouns. But even apart from this we eventually need to harmonize a bit with existing Koszul-entries such as Koszul duality. Most of these are somewhat neglected and would deserve some attention.

Posted by: Urs Schreiber on February 18, 2012 1:47 PM | Permalink | Reply to this

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