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June 16, 2009

Accessible Even to a Philosopher

Posted by David Corfield

Edward Frenkel has a paper out today – Gauge Theory and Langlands Duality – which sets out from André’s Weil’s letter to his sister.

This is a remarkable document, in which Weil tries to explain, in fairly elementary terms (presumably, accessible even to a philosopher), the “big picture” of mathematics, the way he saw it. I think this sets a great example to follow for all of us.

Martin Krieger provided a translation of the letter. As for its accessibility, I can say that it did inspire chapter 4 of my book. Let’s hope Frenkel’s paper can also be inspirational. At first glance, however, it looks tough going.

Serge Lang was not so enamoured with Weil’s letter (as explained here) for reasons of inaccuracy. He also gives his translation of the sentence where Weil comments on its accessibility:

Maybe you will believe you understand the beginning: you will understand nothing after that.

Now let’s see whether I can do any better with Gauge Theory and Langlands Duality.

Posted at June 16, 2009 1:04 PM UTC

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understanding

Contrast

Maybe you will believe you understand the beginning: you will understand nothing after that.

with the author of `The Cloud of Unknowing’
who advises if you understand nothing at the beginning, push on after that.

Posted by: jim stasheff on June 16, 2009 1:18 PM | Permalink | Reply to this

Re: Accessible Even to a Philosopher

Don’t you just love it, David, when people say “accessible even to a philosopher”?

This is a beautiful paper. I think I’ll get a lot out of it.

I would like to get some intuitive understanding of the conjectured “much stronger, categorical version of the geometric Langlands correspondence” at equation (3.1) on page 13.

Why should OO-modules on the moduli stack of flat GG-bundles be secretly ‘the same’ as DD-modules on the moduli stack of G *G^*-bundles?

(Here ‘the same’ means an equivalence of derived categories. I’m using G *G^* to mean the Langlands dual group — nonstandard notation which is supposed to make you think of a categorified and nonabelian version of the Fourier transform.)

I know roughly what all the words mean…

Posted by: John Baez on June 16, 2009 2:35 PM | Permalink | Reply to this

Re: Accessible Even to a Philosopher

Note who is being thanked for funding this research…

Posted by: Eugene Lerman on June 18, 2009 1:47 AM | Permalink | Reply to this

Re: Accessible Even to a Philosopher

Ah, I wonder if this is to do with the DARPA 23 Mathematical Challenges.

Number 17:

Geometric Langlands and Quantum Physics

How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?

Is there a way of finding out who the successful bidders were?

Posted by: David Corfield on June 18, 2009 8:50 AM | Permalink | Reply to this

Re: Accessible Even to a Philosopher

No, it’s not one of the 23 Challenges. The Challenges are still open up for bids (or are being rebid). This one is a different program:

Focus Areas in Theoretical Mathematics

It’s co-managed by Ed Frenkel and Kari Vilonen. Look here for more info.

Posted by: Eugene Lerman on June 18, 2009 5:10 PM | Permalink | Reply to this

Re: Accessible Even to a Philosopher

David writes:

Is there a way of finding out who the successful bidders were?

I know how to find out who got NSF grants — anyone can do it. But I don’t know about DARPA.

In any case, a quick trawl shows that for the DARPA Mathematical Challenges, proposal abstracts are due July 25th of this year, while the ‘closing date’ (whatever that means) is September 25th. So, it’s premature to ask who got this money.

But it’s not premature to guess who will.

Posted by: John Baez on June 18, 2009 1:21 PM | Permalink | Reply to this

Re: Accessible Even to a Philosopher

“accessible even to a caveman philosopher”?

Some papers open up making sense, then slip away like water through my fingers. I set them on the “try again” pile.

Some papers start out with stuff that I don’t know or care about, but, skipping to the end, conclude with something that I know from other contexts. So I review those contexts and start at the beginning again.

There are even a few papers whose beginnings and ends make no sense to me, but something in the middle catches my eye – often an intriguing diagram – and I flip through the pages (or scroll down) looking for another mountain to poke its way over sea level as an island I can comprehend.

Longer papers (such as Bulletin of AMS) are sometimes neatly divided into sections, where one section works for me, and the others seem forced or off-agenda.

Anyone else read nonlinearly like this?

There is a threhhold level of (1) makinkes sense; (2) connects with something I’m working on; where I email the URL or citation to a co-author, and ask if it works for them, too. Sometimes these help us in arguments for some paper we’ve been stuck on writing, or are proposing to write.

That pages are consecutively numbered is almost besides the point. Papers are manifolds. I crawl around on their borders trying to comprehemd the local structure, or co-crawl the coborders to
-mprehend the global structure.

Posted by: Jonathan Vos Post on June 17, 2009 3:57 PM | Permalink | Reply to this

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