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.
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.
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.
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.
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 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:
IBM Research has put up the July challenge on their Ponder This site.
Update: Here is the problem:
Upon a rectangular table of finite dimensions L by W, we place n identical, circular coins; some of the coins may be not entirely on the table, and some may overlap. The placement is such that no new coin can be added (with its center on the table) without overlapping one of the old coins. Prove that the entire surface of the table can be covered completely by 4n coins.