Woah, look at that! The seventh Carnival of Mathematics is up at nOnoscience.
Author Archives: Walt
Sixth Carnival of Mathematics
The sixth Carnival of Mathematics is up at Modulo Errors.
Dispersive Wiki
If you happen to be an expert on dispersive PDEs, the Dispersive Wiki could use your help.
Bulletin of the AMS, Vol. 44, No. 1
I neglected to put up a post about the first issue of the Bulletin of the AMS for 2007. Here’s some of the highlights:
- K. Soundararajan describes a recent result of Goldston, Pintz, and Yildirim on gaps between primes. The article provides a taste of what analytic number theory is like.
- Jaikumar Radhakrishnan and Madhu Sudan provide an introduction to Dinur’s proof of the PCP theorem. The PCP theorem, one of the most surprising complexity theory results of the last twenty years. Dinur’s proof is not the original, but provides a new approach to the topic.
- Svetlana Katok and Ilie Ugarcovici explain the relationship between symbolic dynamics and differential geometry. Symbolic dynamics fits perfectly in the computer era, so it always seemed bizarre to me that it was invented purely as a technique in differential geometry.
Now to write up the second issue before the third issue comes out…
Light Posting
Sorry posting has been light. I just moved, which is filled with many non-math-related activities such as unpacking, but also many math-related activities, such as wondering where all of your books went. The chaos is almost to an acceptable level, so posting should pick back up.
The Life Cycle of a Theoretical Physicist
Commenter h recommended Borcherd’s lecture notes on QFT. I’ve only just begun reading them, but his Life Cycle of a Theoretical Physicist, which begins the introduction, is incredibly funny.
May Notices
The May Notices of the AMS features Francesco Mezzadri’s How to Generate Random Matrices from Classical Compact Groups. The mysteriously titled If Euclid Had Been Japanese, by Bill Casselman, discusses an interesting question: what points in the plane are constructible by origami folds rather than ruler and compass? (I’d never heard of this before, but Wikipedia has a page on the subject.)
In this month’s What is…?, Valentin Poénaru answers the question What is an infinite swindle? I’d expected the article to be about the Eilenberg swindle, which is an example of a ring without invariant basis number. Instead, Poénaru describes exotic examples of spaces that are constructed recursively, such as the Whitehead manifold and Casson handles.
Time Is Short
People of Earth,
Now that this has happened, time is short. I can only assume that the inhabitants of the planet around Gliese 581 have left us unmolested up to this point is that as long as we did not know of their existence we were no threat. Now that the word is out, I can only assume that their long-prepared invasion fleet is under way. Fortunately, we have 20.5 years until word reaches them, and another 20.5 before their planet-killing machines can acheive Earth orbit, so we must use the 41 years of peace left to us to prepare. I am not a crank.
Sincerely,
Walt
Combinatorial Nullstellensatz
In this thread on his weblog, Terry Tao mentioned an exciting paper by Noga Alon. The paper explains a result that Alon calls the Combinatorial Nullstellensatz. The ordinary Nullstellensatz relates algebra and geometry over algebraically closed fields. Consider the set of common zeroes of a system of (multivariate) polynomial equations over the field. Then a polynomial vanishes on that set if and only if a power of the polynomial is in the ideal generated by the system of equations.
It is easy to see that no such simple result holds over non-algebraically closed fields. Alon is able to prove an analogue of the Nullstellensatz in a very special case, so special that it is not of particular interest of itself. But he is able to use it to give new short proofs of many existing results, such as the Chevalley-Warning and Cauchy-Davenport theorems, as well as many results in combinatorics.
Guide to QFT Textbooks
I’ve come across this guide to quantum field theory textbooks. Since QFT is on my to-do list of things to learn before I die, this list may come in handy.