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Kadanoff Writes Tribute to WIlson

After the recent passing of Kenneth Wilson, his long-time collaborator Leo Kadanoff has written a tribute and published it on the arXiv.

Kenneth Geddes Wilson, 1936-2013, An Appreciation

Weekend Reading

Mate Choice and the Origin of Menopause: Computational biologists can be unconvincing, sometimes.

All the Lonely People: Rise in suicie rates linked to reduction in communities?

Brands Are Imprinted on Our Brains

Mandarin by the Numbers: More puns in Mandarin, this time with numbers.

Why War: Einstein and Freud’s Little-Known Correspondence on Violence, Peace, and Human Nature

Weekend Reading

Ever wondered why chickens have no penis? So have other people, and we’re closer to the answer.

Mandarin is rich in puns. Just beware if you hear someone saying “grass mud horse”.

Security of your data and cloud based services.

On being an octopus

Terrence Tao on the prime-tuples conjecture

Weekend Reading

No flat-earth in ancient cosmology, but the rain falls upward the other side of the equator:

One should avoid being a Glasshole:

A strange story of Wikipedia revenge-edits from ‘Qworty’

Topological order at finite temperature

Real Complete Fields, Natural Numbers, and Higher-Order Logics

When encountering Gödel’s incompleteness theorems, usually towards the end of an introduction,  we find statements that explain that the real numbers are complete, and Gödel’s incompleteness theorems do not apply.  Incompleteness of the natural numbers is then argued to not contradict this since the natural numbers are not contained in the real numbers. And this is it, often with no embellishment. Confusion ensues.

Natural numbers not contained in the real numbers? Surely the natural numbers are a subset of the real numbers? Well, yes they are, and this is close to the point of the matter. Gödel’s theorems are a result in first-order logic, and as such, we can quantify only over the objects in a theory. Second-order logics allow us instead to quantify over functions, predicates, and most relevantly here, over sets of numbers. The natural numbers are contained in the real numbers in a second-order theory. The containment here is not a set-theoretic containment, but a containment in the sense of being definable in a first-order theory. Mathematicians should beware when reading logic texts.

“So? Why not just work in second-order logic?”, a practical but impatient person bellows from the back. The answer depends on what kind of mathematician you are. You may think since number theorists are concerned with properties of integers, that they would be particularly aggrieved by incompleteness; some of them are, but often, usually analytic number theorists, actually work over the complex numbers. Think analytic continuations of zeta functions, and such. Here, one can do mathematics without giving Gödel a thought.

Complications from incompleteness arise in arithmetic geometry, however. Maybe I will write more on this later, but in algebraic geometry, the methods developed are roughly the same for every dimension, and only get more complicated. This has to be the most understated use of “only” ever recorded; 3-folds are significantly more difficult to classify than algebraic surfaces. For arithmetic geometry, the methods developed depend essentially on the dimension.

So, are there any concrete examples where incompleteness is manifest? This will be the subject of the next post, and will discuss the Paris-Harrington theorem.

Last Week on the arXiv (15th-19th April, 2013)

This is the first what really ought to be a series of posts updated weekly. It will contain links to papers that have been uploaded on the arXiv that I find interesting, and would like to find out more about. Presumably this will be achieved by reading the content of the paper, which in all likelihood I won’t have done by the time of the post. Well, here it goes:

Resonant delocalization on the Bethe strip

Localization is a huge topic in statistical mechanics, and one I’ll likely write about in the future. The motivation for study is physical. Philip Anderson was the first, it seems, to think about these kind of questions in relation to disordered systems; such phenomena are called Anderson localization, and is quite a general feature of things that behave like waves. Even light can be localized [Anderson localization of light ]

The associated mathematics is also interesting. The study is that of random differential operators; in practise, we have some model with a random potential which represents disorder in some system. Localization here is the same as the appearance of point-spectra, which correspond to bound states. With bound states present, diffusion cannot take place, and waves are effectively confined to a finite volume.

Delocalization is, as you’d expect, largely the opposite of this. Waves become extended, and can’t be contained in some small part of the system. For resonant delocalization, the idea is that fluctuations can sometimes enable resonances that can set-up extended states between distant sites. Aizenman and Warzel have shown this for tree-graphs [Resonant delocalization for random Schrödinger operators on tree graphs]. This work extends Aizenman and Warzel’s results to tree-like graphs with loops; more precisely, this is a cartesian product of a tree and a finite-graph.

Polya’s random walk theorem

Polya’s random walk theorem is simply stated, and a foundational result in the theory of random walks: take an infinite grid in d dimensions (we’d write this as \mathbb{Z}^d, usually. Polya’s theorem tells us for which d a random walk is guaranteed, in the sense that the probability of such an event is 1, to return to the point where it started, in which case we call it recurrent, and for which d the random walk is able to go an arbitrarily large distance away from the origin, where we call it transient. The spoiler, as everyone knows, so being disqualified from being a spoiler, is the random walks are recurrent for d=1,2, and transient for d>2

The paper is a nice exercise in special function theory, with good old classical analysis and asymptotics. An integral is given at the end whose divergence or convergence decides whether the walk is transient or recurrent;  there is something overwhelming pleasant about that.

Recurrent random walks are in some sense a type of localization; we can’t move very far from where we started. I’m not exactly sure who first linked the two, and I can’t find any nice papers giving an overview, but it is more than folklore.

Quantum ergodicity on large regular graphs

Quantum ergodicity is a natural generalization of classical ergodicity, where instead phase-space being explored uniformly by trajectories of particles, high-energy eigenstates of operators fill space uniformly. In technical terms, the measure of distribution should converge to the uniform Liouville measure for most eigenstates. If this is the only possible limit, we have quantum unique ergodicity. The high-energy states requirement mean that it really is a semi-classical phenomena, but we’ll let semantics slide.

This work extends the results of Lindenstrauss on quantum ergodicity for arithmetic surfaces [Invariant measures and arithmetic quantum unique ergodicity] to large regular graphs with accompanying large girth. This prohibits the appearance of short-cycles, and so the graph can be viewed as a tree, as long as one doesn’t look too far ahead. I’ve not read the paper, but it is high on my to-read-list. Not exactly good-old analysis, but certainly good analysis nonetheless.