Congratulations to David Corfield, who’s obtained a permanent position at the University of Kent at Canterbury.

A couple of months ago I linked to Tribikram Pati’s purported proof that the Riemann Hypothesis was false.

Bernhard Krötz has been working his way through the paper and posting periodic updates in the comments of that post. He has completed this project, and in his opinion the proof is correct.

I would still be surprised if this is really how the Riemann Hypothesis meets its doom, but Bernhard deserves recognition for taking the time to read the paper carefully and keeping us updated.

Terry Tao has just completed detailed summaries of another series of lectures, this time by Shing-Tung Yau:

1. What is a geometric structure?
2. The Basic Tools to Discuss Geometric Structures
3. Application of the Geometric Structures to Solve Problems in Algebraic Geometry and Topology

By geometric structure, Yau means additional structure that one imposes on a topological manifold. For example, a complex-analytic structure on a manifold is the requirement that the manifold be given by glueing together copies of Cⁿ, such that the map which glues two copies together is required to be complex analytic. The main question is generally given a topological manifold, what sorts of structures can be imposed on them? The answer is frequently hard, and occasionally bizarre: there are uncountably infinite smooth structures on Rⁿ for n = 4, while for any other value of n, there is only one. (Something the previous link mentioned was new to me: some exotic R⁴s can be smoothly embedded as open sets in the usual R⁴.)

The eighth Carnival of Mathematics is up at The Geomblog. The next carnival will be hosted by JD2718.

Terry Tao has offered a detailed summary of a series of talks by Shou-wu Zhang on rational points on curves. The existence of rational points turns out to depend critically on the genus of the curve, which is a topological and complex-analytic property of the curve.

The summary of each talk provides the flavor of the subject (which is very technical):

1. Overview of rational points on curves
2. Gross-Zagier formula and Birch and Swinnerton-Dyer conjecture
3. Triple L-series and effective Mordell conjecture

John Baez draws the world’s attention to the Banff protocol, a specific guideline for which journals to boycott for overcharging for subscriptions. You can submit your name there as a public supporter of the Banff Protocol.

I’ve seen two main arguments on the question of expensive journals: the “idealist” view, and the “realist” view. The idealist point of view is that in the Internet age, scientific information should be either free or low cost. The realist point of view is that journals are expensive to produce and no one has successfully produced a free one. This is one of those (rare? common?) instances of where the idealist view is closer to a pragmatic answer than the realist view. The actual question of how expensive journals should be is a red herring from the real issue: From a purely economic point of view, where the does the value of a prestigious journal come from? It’s not from the authors, most of whom need the prestigious journal a lot more than the journal needs them. Even though prestigious journals require prestigious editors, prestigious editors like editing prestigious journals, so it’s not the editors in a vacuum. As the idealists like to point out, typesetting is now done by the authors themselves. I’m sure copyediting and publishing a journal, and shipping it to every university library in the world is not cheap, but I’m sure it’s not difficult to pay someone to do them for you.

From the economic point of view, prestigious journals are like colas: the importance is in the brand. Acta Mathematica is not essentially different from Coca-Cola. Authors, peer reviewers, and editors all essentially donate their time (I assume that editors are paid, but not much) to contribute to the value of this brand. When mathematicians allowed ownership of these brands to pass into the hands of the publishers, they were suckers. Rubes. Marks who got scammed. They let the one thing of irreplaceable value to get away. So maybe the realists are right that publishing a journal is intrinsically expensive. But the real issue is not the price of journals, but that mathematicians allowed a system to evolve where there own career self-interest forces them to help big companies make fat profits off their work, and they did so by letting ownership of their own journals to slip away.

Suresh at The Geomblog asked me to let everyone know that the deadline for the next Carnival of Mathematics is fast approaching. More information on how to submit is available here.

When Gauss called number theory the queen of mathematics, he had in mind its future usefulness to the military-industrial complex. Discuss.