J. S. Milne, author of Etale Cohomology, has a terrific set of lectures notes up at his website www.jmilne.org, which run the gamut from algebraic geometry to algebraic number theory (admittedly, not a very wide gamut). His notes on class field theory are particularly nice, but he also has extensive lecture notes on etale cohomology (I’m not sure how different these are from the book), modular forms, abelian varieties, and some more elementary topics such as elliptic curves.
Author Archives: Walt
Goodstein Revisited
In the comments to Programming Language and Logic Links, citylight asked for an example of what proof theory is good for. I wanted to sketch an example using Goodstein sequences.
Proving that Goodstein sequences always eventually go to zero requires transfinite induction up to the ordinal \epsilon_0. Gentzen proved, using transfinite induction up to \epsilon_0, that Peano arithmetic was consistent. By Godel’s second incompleteness theorem, not consistent system can prove its own consistency, so transfinite induction up to \epsilon_0 is not expressible in Peano arithmetic, and Goodstein’s theorem is independent of it.
More details can be found at Fast Growing Functions and Unprovable Theorems.
Figure Eight Orbit
The three body problem in physics is the study of the trajectories of three bodies mutually attracted by (Newtonian) gravity. Unlike the two body problem, the three body problem cannot be solved in general, but some specific solutions are known. An article in the Notices of the AMS reports that a new (well, new circa 2001) solution has been found where all three bodies travel in the same orbit — a figure eight.
The figure eight orbit is stable, which means that it is robust under small perturbations. Since any actual physical system will be perturbed slightly by the gravitational pull of other bodies, stability is a prerequisite for a solution to actually be found in nature. So the figure eight orbit may someday be observed by astronomers.
Math FAQ
Back when I was a grad student, I would get asked weird questions about what I did all day. One day, after someone asked me if I worked on adding really big numbers together or something, I wrote a Math FAQ to answer questions about what mathematicians did. The link is to a copy of the FAQ that is being maintained by Sarah Marie Belcastro, a math professor at Xavier.
Programming Language and Logic Links
These thread on Lambda the Ultimate has several interesting links to online papers and books about the links between logic and functional programming languages.
What’s interesting is that, with one giant exception (category theory), the mathematics used is among the least fashionable. Most mathematicians can go through their entire careers without learning anything about proof theory and intuitionistic logic, and I think the reason is that both undermine the naive model of mathematical foundations that most mathematicians carry around in their heads. Mathematicians hate thinking about foundations. Whenever a famous open problem turns out to be equivalent to the Continuum Hypothesis, it’s like a family member died, or worse joined a cult.
Proof theory is disconcerting because it treats mathematican proofs as purely syntactic. Mathematicians, whatever their actual philosophy, adopt a working philosophy of Platonism: symplectic manifolds and 7-spheres and von Neumann regular algebras all exist in some nebulous “out there”. While mathematicians occasionally argue that mathematics is just the formal manipulation of symbols, in practice they think of a 7-sphere as an actual object.
In proof theory, mathematics really is just a formal manipulation of symbols. The more elementary parts of proof theory consist of proving one method of representing proof symbolically is the same as another. More advanced proof theory consists of studying topics such as proof normalization, where it is shown that proofs can be systematically rewritten in a particular form. Here are some further links to proof theory texts.
Intuitionistic logic is another field more prominent in computer science than in mathematics. Intuitionistic logic unnerves mathematicians by removing the law of the excluded middle: that a statement is true, or its negation is true. In classical logic, every statement can be (in principle) assigned a value of either true or false. To do the same for intuitionistic logic, some statements must be assigned intermediate truth values (in fact, infinitely many intermediate values become possible). Most mathematicians regard intuitionism as a historical curiosity not particularly of study.
Intuitionism is attractive to computer scientists, because whether or not its axioms correctly model truth, they do model knowability. The law of excluded middle doesn’t apply to knowability. A statement that is not known to be true may also be not known to false. Curry-Howard correspondence between logical formulas and function types has insired study of even weaker logical systems.
Utrecht Superstring Experiment
Last week Slashdot linked to an article on Physicsweb that reports on an experiment at the Institute of Theoretical Physics to create superstrings in the lab.
I finally found a chance to look at the original article on ArXiv, and Physicsweb’s article turns out to be misleading. The proposed experiment would simulate a superstring, not actually create one. This is a perfectly legitimate idea, but condensed matter physics is the Tinkertoys of quantum mechanics: you can build almost anything. A simulation of a superstring is a long way away from showing that superstrings really exist.
Recovering Archimedes
Scientists are at the Stanford Linear Accelerator Center (SLAC) are using a particle accelerator to help recover a lost work by Archimedes. The Archimedes Palimpsest is a parchment book that once contained copies of Archimedes’ works, but was later erased to be reused as a prayer book. The palimpsest contains the only copy of Method of Mathematical Theorems and only copy of On Floating Bodies in the original Greek.
SLAC’s website also has a press release about the current state of research into the existence of the pentaquark, a theoretical particle made up of five quarks (all known particles made up of quarks only take two or three). The verdict? Not proven.
Goodstein Sequences
Goodstein sequences are integer sequences with a very surprising property.
Start with any number. Rewrite the number as a sum of powers of 2. In turn, rewrite the exponents as powers of 2, then the exponents of exponents, etc. For example, we’d write 33 as:
.
To compute the next value in the sequence, replace every 2 with a 3, and subtract 1. The next value in the Goodstein sequence for 33 is:
which is equal to 22876792454961.
For the next step, we replace the 3s with 4s and subtract 1, etc. This sequence continues to increase very rapidly, right?
Wrong. Goodstein proved that for any starting value, the sequence eventually goes to zero. Even more surprisingly, the proof relies on properties of infinite ordinal arithmetic. The theorem cannot have an appreciably more elementary proof: the result is independent of the Peano axioms for arithmetic.
A minicourse on Goodstein sequences and some related examples can be found at this online course: Fast-Growing Functions and Unprovable Theorems
Y combinator
This is the best explanation of the Y combinator I’ve ever seen.
Algebraic Surfaces
The Algebraic Surface Homepage has many interesting pictures of algebraic surfaces. A few examples of algebraic surfaces might be familiar from high school: spheres, ellipsoids, hyperboloids. Higher degree surfaces are much more exotic, though. For exmple, check out the septic with 99 singularities.
Another page with galleries is the Cubic Surface Homepage.