Reinhard Diestel has a new edition of his graduate text, Graph Theory available on his website. The book is a dense but broad introduction to the field.
Author Archives: Walt
Pioneer Anomaly
The Pioneer 10 and 11 spacecraft have experienced an unexplained drag that has caused them to travel slower than predicted, a phenomenon known as the Pioneer anomaly. Attempts to explain it using normal physics have failed, which leads people to speculate that it will require brand-new physical theories to explain. I personally hope that this is like the discovery of radioactivity by Roentgen in 1895 — a first initial glimpse into a new world.
Water Mechanics
Despite the fact that in theory it is entirely reducible to quantum mechanics, chemists do not have a mathematical model of water molecules that completely explains its behavior.
Update. Sigfpe has more thoughts at his blog.
Crawley-Boevey on Quivers et al
If you’re interested in recent developments in abstract algebra, an excellent place to look is the homepage of William Crawley-Boevey. He provides lecture notes covering quivers (which we’ve discussed before), the cohomological approach to central simple algebras, and invariant theory.
NUMDAM
The NUMDAM Project is digitizing the archives of French mathematical journals, and placing them online. Some of the journals are quite venerable — the Annales Scientifiques de l’École Normale Supérieure was founded in 1864. Another highlight is the prestigious Publications Mathématiques de l’IHÉS.
Elementary Version of Riemann Hypothesis
I was looking at the Wikipedia entry for Harmonic number, where I spotted a rather surprising reformulation of the Riemann hypothesis.
The Riemann hypothesis was already known to be equivalent to a not-very-complicated statement about the distribution of primes. Let π be the number of primes less than n. Then the Riemann hypothesis is equivalent to:
for all ε > 0. This fact, which goes back at least to Riemann, is the main reason why the Riemann hypothesis is of interest. In 2002, Jeffrey Lagarias found an even more elementary statement.:
where Hn is the nth Harmonic number (the sum of reciprocals less than or equal to n). It almost looks you could solve it, doesn’t it?
The Standard Model
The Standard Model is the name particle physicists give to their unified theory of electromagnetism, and the weak and strong nuclear forces. The Standard Model is an example of a gauge theory (unrelated to the gauge integral). Gauge theories are parametrized by Lie groups. Particles in gauge theories possess internal state that does not correspond to a classical observable; this internal state is described by an element in the Lie group. The group for the Standard Model is U(1) x SU(2) x SU(3).
This is a subject that I’ve always meant to learn more about, but I’ve never had the chance. Gauge theories make the subject of Lie algebra representations more vivid. For example, the part of the Standard Model that describes hadrons (particles such as protons and neutrons) is SU(3). SU(3) was found by fitting the existing hadron data to an 8-dimensional representation of the Lie algebra su(3). The 8-dimensional representation is not the smallest possible representation of su(3); there is (pretty obviously) a 3-dimensional representation. Taking that representation seriously led to the discovery of quarks.
Here are a few survey articles about the Standard Model from ArXiv:
Michael guest blogging
We’re going to start experimenting with having guest bloggers. Our first guest blogger will be our most voluminous commenter, Michael.
Urs Lang on Geometric Measure Theory
Urs Lang has posted some lecture notes on geometric measure theory.
Maharam’s Theorem
I’ve been reading Fremlin’s book, and I’ve seen a very surprising theorem that was new to me: Maharam’s Theorem. If you take an set of coins, you can define a measure space on the set of coin flips by taking the product measure. This is a probability measure: the measure of a set is the probability of a coin flip appearing in that set. Since it is a probability measure, it’s well-defined for sets of every cardinality.
You can combine any two measure spaces by taking their disjoint union; the measures are combined by addition. More generally, you can take a weighted sum. Maharam’s theorem states that every nontrivial complete measure space can be constructed from sets of coin flips by taking weighted sums. For example, counting measure is an infinite sum of flips of a single coin. Lebesgue measure on the unit interval arises from flipping an infinite number of coins.
This means that there are not very many types of (complete) measure spaces.