Hermann Weyl once said “In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.” Discuss.
Vakil on Gromov-Witten Theory
One of the stranger developments of recent years is the influence of physics on algebraic geometry. A dramatic recent example is Gromov-Witten theory, which draws its inspiration from quantum field theory, but can be used to study the moduli space of complex algebraic curves.
A moduli space is a space that parameterizes all objects of a certain type. The classic example is the projective line, which classifies lines in the plane: each line in the plane corresponds to one and only one point in the plane. The moduli space of curves classifies complex-algebraic curves. The space itself is a geometric object, but its structure turns out to be very complicated, and recent progress has relied on these ideas from physics.
Ravi Vakil has posted a survey article on the subject, The moduli space of curves and Gromov-Witten theory, to Arxiv. He also has an older, more elementary article from the June/July 2003 Notices of the AMS, The moduli space of curves and its tautological ring.
In the air
I admit to not reading other mathy blogs on any sort of schedule, so when I followed the not even wrong link, the previous entry caught my eye.
Just remember I already called it.
Yau on geometric analysis
Shing-Tung Yau has posted a new survey article to ArXiv, Perspectives on geometric analysis. The article is pitched at a very advanced level, but gives some of the flavor of the frontiers in differential geometry.
Further discussion can be found at Not Even Wrong.
March Notices
The March Notices of the AMS is out.
Complexity Zoo
Scott Aaronson and Greg Kuperberg have put together a website, the Complexity Zoo, that describes 443 (!) computational complexity classes.
Baez Week 226
Week 226 of This Week’s Finds in Mathematical Physics is up. This week is pretty far from his usual topics, as John Baez talks about cryptography, pseudorandom number generators, and P versus NP.
Karoubi on K-theory
Max Karoubi has posted to Arxiv K-theory. An elementary introduction. It’s as elementary as can be expected (which means it’s not all that elementary).
MOND in the news again.
Who are you…who who, who who.
I wanted to take a sounding of the Ars Mathematica readership, if for no other reason than me being nosy. It seems to me that we have a pretty broad readership (from Ph.D.’s to high school students), but I for one would like to get a better idea of the distribution.
Let me be the first to divulge info.
Walt, Robbie and I all went to grad school at the University of Washington – Robbie does/did Ergodic Theory, I am somewhere in the intersection of algebraic topology, combinatorics and logic, and – as far as I have been able to ascertain – Walt knows everything.
So sound off if you would…