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October 30, 2014

Maths, Just in Short Words

Posted by David Corfield

Guest post by David Roberts

How much maths can you talk about if you could just use short words? Late one night, somewhere between waking and sleeping, the cartoon proof of Löb’s theorem and Boolos’ explanation of Gödel’s second incompleteness theorem (Paywall! But the relevant part is all on the first page, if you can see that) teamed up to induce me to produce a proof of Cantor’s theorem of the existence of more than one infinite cardinal, using only words of one syllable (or, “just short words”). I then had the idea that it would be interesting to collect explanations or definitions or proofs that used only words of one syllable, and perhaps publish them in one go. Below the fold, I give my proof of Cantor’s theorem, comments and criticisms welcomed. I was aiming for as complete a proof as I could, whereas a number of other people (more on that below) went for simplicity, or as short as they could, so the style is different to the others I mention.

Posted at 12:41 PM UTC | Permalink | Followups (27)

October 22, 2014

Where Do Probability Measures Come From?

Posted by Tom Leinster

Guest post by Tom Avery

Tom (here Tom means me, not him — Tom) has written several times about a piece of categorical machinery that, when given an appropriate input, churns out some well-known mathematical concepts. This machine is the process of constructing the codensity monad of a functor.

In this post, I’ll give another example of a well-known concept that arises as a codensity monad; namely probability measures. This is something that I’ve just written a paper about.

Posted at 2:29 PM UTC | Permalink | Followups (22)

October 17, 2014

New Evidence of the NSA Deliberately Weakening Encryption

Posted by Tom Leinster

One of the most high-profile ways in which mathematicians are implicated in mass surveillance is in the intelligence agencies’ deliberate weakening of commercially available encryption systems — the same systems that we rely on to protect ourselves from fraud, and, if we wish, to ensure our basic human privacy.

We already knew quite a lot about what they’ve been doing. The NSA’s 2013 budget request asked for funding to “insert vulnerabilities into commercial encryption systems”. Many people now know the story of the Dual Elliptic Curve pseudorandom number generator, used for online encryption, which the NSA aggressively and successfully pushed to become the industry standard, and which has weaknesses that are widely agreed by experts to be a back door. Reuters reported last year that the NSA arranged a secret $10 million contract with the influential American security company RSA (yes, that RSA), who became the most important distributor of that compromised algorithm.

In the August Notices of the AMS, longtime NSA employee Richard George tried to suggest that this was baseless innuendo. But new evidence published in The Intercept makes that even harder to believe than it already was. For instance, we now know about the top secret programme Sentry Raven, which

works with specific US commercial entities … to modify US manufactured encryption systems to make them exploitable for SIGINT [signals intelligence].

(page 9 of this 2004 NSA document).

Posted at 3:18 PM UTC | Permalink | Followups (40)

‘Competing Foundations?’ Conference

Posted by David Corfield

FINAL CFP and EXTENDED DEADLINE: SoTFoM II `Competing Foundations?’, 12-13 January 2015, London.

The focus of this conference is on different approaches to the foundations of mathematics. The interaction between set-theoretic and category-theoretic foundations has had significant philosophical impact, and represents a shift in attitudes towards the philosophy of mathematics. This conference will bring together leading scholars in these areas to showcase contemporary philosophical research on different approaches to the foundations of mathematics. To accomplish this, the conference has the following general aims and objectives. First, to bring to a wider philosophical audience the different approaches that one can take to the foundations of mathematics. Second, to elucidate the pressing issues of meaning and truth that turn on these different approaches. And third, to address philosophical questions concerning the need for a foundation of mathematics, and whether or not either of these approaches can provide the necessary foundation.

Date and Venue: 12-13 January 2015 - Birkbeck College, University of London.

Posted at 2:09 PM UTC | Permalink | Followups (6)

October 5, 2014

M-theory, Octonions and Tricategories

Posted by John Baez

Quite a witches’ brew, eh?

Amazingly, they seem to be deeply related. John Huerta has just finished a paper connecting them… and this concludes a series of papers that makes me very happy, because it fulfills a long-held dream: to connect physics, division algebras, and higher categories.

Posted at 10:25 PM UTC | Permalink | Followups (15)

The Atoms of the Module World

Posted by Tom Leinster

In many branches of mathematics, there is a clear notion of “atomic” or “indivisible” object. Examples are prime numbers, connected spaces, transitive group actions, and ergodic dynamical systems.

But in the world of modules, things aren’t so clear. There are at least two competing notions of “atomic” object: simple modules and, less obviously, projective indecomposable modules. Neither condition implies the other, even when the ring we’re over is a nice one, such as a finite-dimensional algebra over a field.

So it’s a wonderful fact that when we’re over a nice ring, there is a canonical bijection between {\{isomorphism classes of simple modules}\} and {\{isomorphism classes of projective indecomposable modules}\}.

Even though neither condition implies the other, modules that are “atoms” in one sense correspond one-to-one with modules that are “atoms” in the other. And the correspondence is defined in a really easy way: a simple module SS corresponds to a projective indecomposable module PP exactly when SS is a quotient of PP.

This fact is so wonderful that I had to write a short expository note on it (update — now arXived). I’ll explain the best bits here — including how it all depends on one of my favourite things in linear algebra, the eventual image.

Posted at 12:20 PM UTC | Permalink | Followups (19)