Ars Mathematica

The fourth Carnival of Mathematics is up.

This article at the website for the Atlas of Lie Groups and Representations provides more details for last week’s highly touted announcement of the computation of the Kahzdan-Lusztig-Vogan polynomials for E8. It sounds like they were as surprised as anyone that the announcement made such a big media splash.

Alexandre Borovik is reporting that Paul Cohen has died. Cohen of course proved that the continuum hypothesis is independent of the axioms of set theory, and that the axiom of choice was independent of the other axioms.

Gödel had already established that both the continuum hypothesis and axiom of choice were consistent, in the sense if you could derive a contradiction by adding them as axioms then you could derive a contradiction in set theory without them. Gödel accomplished this by defining a certain minimal model of set theory, the constructible universe, and showing that in this model both axioms hold. Cohen then completed the proof of independence by showing that you can construct a model of set theory in which both axioms are false. To do so, he had to invent a new technique: forcing.

His Wikipedia biography has the oddest snippet:

His twin sons Steven and Eric played the Dancing Twins on the TV show Ally McBeal.

Is that really true?

I have seen references here and there to a new paper on ArXiv, The Riemann Hypothesis, by Tribikram Pati, that claims to derive a contradiction from the Riemann Hypothesis. I caution anyone who’s seen it to not to get too excited. The gap between thinking you have a proof and actually having a proof can be large, and it’s particularly large for a famous problem. Since ArXiv if for preprints and not-necessarily-polished work, people can and do upload papers that they have to retract later.

The argument in the paper is fairly elementary, using only basic real and complex analysis, so if anyone is interested in seeing if they can find a flaw in the argument I’d recommend giving it a try.

Terence Tao has a thoughtful post that explains why proving existence results for Navier-Stokes equations is so hard.

The Atlas of Lie Groups and Representations, a project to compute all unitary representations of real semisimple Lie groups, has successfully computed the Kazhdan-Lustig-Vogan polynomials of the split real form of the exceptional Lie group E₈, a calculation. This result is so exciting that the American Institute of Mathematics
put out a press release comparing it to the Human Genome Project.

I can’t even imagine what an announcement like this looks like from the outside. I don’t know the significance of the result myself, though the classification of all unitary representations of Lie groups is obviously an important one. For a more detailed descriptions, see this post at the n-Category Cafe, and the ensuing discussion.

For the most bizarre math curriculum ever, check out the course listings at Maharishi University of Management, founded by Maharishi Mahesh Yogi.

Via Cosmic Variance.

I’ve seen multiple claims on the Internet that yesterday was Pi day. Was there a memo that I didn’t get? Who decided it was pi day?

I do remember quite clearly when I was in elementary school and the teacher announced that it was impossible to write pi as a fraction. I was shocked.

I was going to agitate for other deserving constants, such as Euler’s constant should get their own day, but honestly, I love every real number equally. They all deserve a day of their own.

David Corfield highlights two articles by Samson Abramsky: one on Temperley-Lieb algebras and the other on concurrency.

The concurrency article, What are the fundamental structures of concurrency? We still don’t know!, discusses the profusion of formalisms for representing concurrency. Abramsky passes on the following anecdote:

The mathematician André Weil apparently compared finding the right definitions in algebraic number theory — which was like carving adamantine rock—to making definitions in the theory of uniform spaces (which he founded), which was like sculpting with snow.

This post by Mark Chu-Carroll made me wonder something. Compared to people in the sciences or computers, are mathematicians disproportionately less likely to like science fiction? I’ve only known personally known a few that were big science fiction fans, while virtually every science blogger seems to be a big fan. Are science bloggers somehow a non-representative sample, or is this a genuine phenomenon?