Eckhard Meinrenken has been teaching a course at Toronto on Lie groups and Clifford algebras, and he has lecture notes available. This is beautiful mathematics and brings together several of my favorite mathematical topics: Clifford algebras, spinors, and representation theory of Lie groups. Some of this material is quite new, and very possibly has interesting applications in physics.
Dennis Gaitsgory, a new young tenured member of the Harvard math department, has been teaching a course on Geometric Representation Theory, and also has lecture notes available. These explain the “D-module” point of view on the subject.
My Columbia math department colleague D. H. Phong and fellow ex-Princeton grad student Eric D’ Hoker have a new paper on the arXiv called Complex Geometry and Supergeometry. It is based on Phong’s recent lecture at this year’s Current Developments in Mathematics conference at Harvard last month. D’ Hoker and Phong have been able to explicitly write down and show finiteness of superstring amplitudes at two loops, with the problem still remaining open at higher loops.
The arXiv web-site has a new look today. Gone are Paul Ginsparg’s “skull and cross-bones” icons like this.
I CANT BELIEVE IT, CLIFFORD AGLEBRAS! IN FACT I WAS JUST PERUSING “”” CLIFFORD MODULES “”” BY “”” ATIYAH BOTT AND SHIPARO “”” YESTERDAY, ITS FUN FOR ALL THE FAMILY!!1! IN MORE FACT, BTW, I REMEMBER WHEN I WAS YOUNG AND I FIRST LEARNT ABOUT UNIVERSAL PROPERTY AND FILTRATIONS AND CONJUGATION AUTOMORPHISMS, I JUMPED UP AND DOWN WITH GLEE. AND THEN I WAS AMAZED ALL OVER AGAIN ONCE I READ ABOUT GENERALIZING THEM USING DIFFERENT IDEALS, AND FRACTIONAL CLIFFORD STRUCTURES AND THEIR REPRESENTATION THEORY. IT WAS SO SUPER-LATIVE THAT IT MADE ME TINGLE ALL OVER, EXCEPT I DIDNT REALLY BECAUSE THAT WOULD BE VERY SILLY INDEED. BTW I DIDN’T REALLY JUMP UP AND DOWN WITH GLEE EITHER. HMMMMM.
PETER WHY ARE YOU TALKING ABOUT STRING THEORY AND SUPERANYTHING WITHOUT ACCOMPANYING DISPARAGING COMMENTS? A Z2 GRADING IN SPINORS IS ELEGANT BUT ON SUPERSPACES IT JUST OFFENS MY SENSIBILITIES. WHY NOT SWITCH TO VENERATING ABHAY ASHTEKAR – LQG WILL BEATS STRINGS FOREVER!!
Here’s a recent paper relating this stuff to physics:
http://arxiv.org/abs/gr-qc/0511120
It shows how the fields and dynamics of gravity and the standard model can be represented as a single Clifford bundle connection and its curvature.
Garrett, well done, it looks far more sensible than string theory.
http://insti.physics.sunysb.edu/~siegel/parodies/misanthrope.html
These Clifford algebra notes look good and are worth having. Thanks for mentioning it. I had a colleague long ago give seminar lectures on them, which i really enjoyed. Admittedly, I have used Clifford algebras only sparingly in relation to K-theory of C*-algebras (through Connes’ work). (Particularly, C^{p,q} = R^p + R^q.)
It’s very dry, what? (When did lively style go out of fashion, then?) For physics, this sort of approach is hopeless. He took a lively idea and made it seem dull. That took skill.
Why (in these papers) is it always about restating the known?
(Arun: extremely funny 🙂
-drl
Siegel has the best line of all time about ST:
“Cosmology is the opposite of string theory: Instead of starting with 10 or 11 dimensions, and compactifying most of them, you start with no dimensions (except time), and the Big Bang uncompactifies 3 of them.”
..and a candidate for the best footnote ever:
“500 – No, it’s really an exponent, not a footnote. You don’t seriously think there could realistically be 500 footnotes, do you?”
-drl
I printed out all 96 pages of the lecture notes by Meinrenken. Nice stuff. Reading them carefully could take a while though…
Peter, you said “… The arXiv web-site has a new look today. Gone are Paul Ginsparg’s “skull and cross-bones” icons …”.
Just to be nit-picking historically, the “skull and cross-bones” icons were NOT the ORIGINAL logo/icons of what is now called the Cornell arXiv. IIRC, the first web site was at http://MENTOR.LANL.GOV/ and its logo/icon was the image that I have attempted to preserve for posterity at http://www.valdostamuseum.org/hamsmith/Mentor.gif (as you can see, it is more related to Ancient Egyptian Pyramids than to the more recent ocean pirates).
Tony Smith
http://www.valdostamuseum.org/hamsmith/
String theory offers promsie, but we do have as of yet a fundamentally satisfying way to falisfy it, or prove it true. The LHC will certainly give us some answers, but sheer philosophy about the string is simply not science. Something is missing, and we do not know-yet–what we are missing.