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August 11, 2007

This Week's Finds in Mathematical Physics (Week 255)

John Baez

I've been roaming around Europe this summer - first Paris, then

Delphi and Olympia, then Greenwich, then Oslo, and now back to

Greenwich. I'm dying to tell you about the Abel Symposium in

Oslo. There were lots of cool talks about topological quantum

field theory, homotopy theory, and motivic cohomology.

I especially want to describe Jacob Lurie and Ulrike Tillman's

talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen's

talks on string topology, Stephan Stolz's talk on cohomology and

quantum field theory, and Fabien Morel's talk on A^1-homotopy

theory. But this stuff is sort of technical, and I usually try

to start each issue of This Week's Finds with something you don't

need a PhD to enjoy.

So, here's a tour of the Paris Observatory:

1) John Baez, Astronomical Paris,

http://golem.ph.utexas.edu/category/2007/07/astronomical_paris.html

Back when England and France were battling to rule the world,

each had a team of astronomers, physicists and mathematicians

devoted to precise measurement of latitudes, longitudes, and times.

The British team was centered at the Royal Observatory here in

Greenwich. The French team was centered at the Paris Observatory,

and it featured luminaries such as Cassini, Le Verrier and Laplace.

In "week175", written during an earlier visit to Greenwich, I

mentioned a book on this battle:

2) Dava Sobel, Longitude, Fourth Estate Ltd., London, 1996.

It's a lot of fun, and I recommend it highly.

There's a lot more to say, though. The speed of light was first

measured by Ole Romer at the Paris Observatory in 1676. Later,

Henri Poincare worked for the French Bureau of Longitude. Among

other things, he was the scientific secretary for its mission to

Ecuador.

To keep track of time precisely all over the world, you need to

think about the finite speed of light. This may have spurred

Poincare's work on relativity! Here's a good book that argues

this case:

3) Peter Galison, Einstein's Clocks, Poincare's Maps: Empires

of Time, W. W. Norton, New York, 2003. Reviewed by Robert Wald

in Physics Today at http://www.physicstoday.org/vol-57/iss-9/p57.html [Broken]

I met Galison in Delphi, and it's clear he like to think about

the impact of practical stuff on math and physics.

I was in Delphi for a meeting of "Thales and Friends":

4) Thales and Friends, http://www.thalesandfriends.org

This is an organization that's trying to bridge the gap between

mathematics and the humanities. It's led by Apostolos Doxiadis,

who is famous for this novel:

5) Apostolos Doxiadis, Uncle Petros and Goldbach's Conjecture,

Bloomsbury, New York, 2000. Review by Keith Devlin at

http://www.maa.org/reviews/petros.html [Broken]

There's a lot I could say about this meeting, but I just want

to advertise a forthcoming book by Doxiadis and a computer

scientist friend of his. It's a comic book - sorry, I mean

"graphic novel"! - about the history of mathematical logic

from Russell to Goedel:

6) Apostolos Doxiadis and Christos Papadimitriou, Logicomix,

to appear.

I saw a partially finished draft. I think it does a good job

of explaining to nonmathematicians what the big deal was with

mathematical logic around the turn of the last century... and

how these ideas eventually led to computers. It's also a fun

story.

If you're eager for summer reading and can't wait for Logicomix,

you might try this other novel by Papadimitrou:

7) Christos Papadimitriou, Turing (a Novel about Computation),

MIT Press, Boston, 2003.

It's a history of mathematics from the viewpoint of computer

science, as told by a computer program named Turing to a

lovelorn archaeologist. I haven't seen it yet.

Okay - enough fun stuff. On to the Abel Symposium!

8) Abel Symposium 2007, at http://abelsymposium.no/2007

Actually this was a lot of fun too. A bunch of bigshots were

there, including a bunch who didn't even give talks, like Eric

Friedlander, Ib Madsen, Jack Morava, and Graeme Segal.

(My apologies to all the bigshots I didn't list.)

Speaking of bigshots, Vladimir Voevodsky gave a special surprise

lecture on symmetric powers of motives. He wowed the audience not

only with his mathematical powers but also his ability to solve a

technical problem that had stumped all the previous speakers! The

blackboards in the lecture hall were controlled electronically,

by a switch. But, the blackboards only moved a few inches before

stalling out. So, people had to keep hitting the switch over and

over. It was really annoying, and it became the subject of running

jokes. People would ask the speakers: "Can't you talk and press

buttons at the same time?"

So, what did Voevodsky do? He lifted the blackboard by hand!

He laughed and said "Russian solution". But, I think it's a great

example of how he gets around problems by creative new approaches.

It really pleased me how many talks mentioned n-categories, and

even used them to do exciting things. This seems quite new. In

the old days, bigshots might think about n-categories, but they'd

be embarrassed to actually mention them, since they had a

reputation for being "too abstract".

In fact, Dan Freed alluded to this in his talk on topological

quantum field theory. He said that every mathematician has

an "n-category number". Your n-category number is the largest n

such that you can think about n-categories for a half hour without

getting a splitting headache.

When Freed first invented this concept, he felt pretty

self-satisfied, since his n-category number was 1, while for

most mathematicians it was 0. But lately, he says, other

people's n-category numbers have been increasing, while his has

stayed the same.

He said this makes him suspicious. In light of the scandals

plaguing the Tour de France and American baseball, he suspects

mathematicians are taking "category-enhancing substances"!

Freed shouldn't feel bad: he was among the first to introduce

n-categories in the subject of topological quantum field theory!

He gave a nice talk on this, clear and unpretentious, leading

up to a conjecture for the 3-vector space that Chern-Simons

theory assigns to a point.

That would make a great followup to these papers on the 2-vector

space that Chern-Simons theory assigns to a circle:

9) Daniel S. Freed, The Verlinde algebra is twisted equivariant

K-theory, available as arXiv:math/0101038.

Daniel S. Freed, Twisted K-theory and loop groups, available

as arXiv:math/0206237.

Daniel S. Freed, Michael J. Hopkins and Constantin Teleman,

Loop groups and twisted K-theory II, available as

arXiv:math/0511232.

Daniel S. Freed, Michael J. Hopkins and Constantin Teleman,

Twisted K-theory and loop group representations, available as

arXiv:math/0312155.

In a similar vein, Jacob Lurie talked about his work with Mike

Hopkins in which they proved a version of the "Baez-Dolan cobordism

hypothesis" in dimensions 1 and 2. I'm calling it this because

that's what Lurie called it in his title, and it makes me feel good.

You can read about this hypothesis here:

10) John Baez and James Dolan, Higher-dimensional algebra and

topological quantum field theory, J.Math.Phys. 36 (1995) 6073-6105

Also available as arXiv:q-alg/9503002.

It was an attempt to completely describe the algebraic structure of

the n-category nCob, where:

objects are 0d manifolds,

1-morphisms are 1d manifolds with boundary,

2-morphisms are 2d manifolds with corners,

3-morphisms are 3d manifolds with corners,

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# This Week's Finds in Mathematical Physics (Week 255)

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