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Everything you learned in Calculus 2 was wrong!

I think I’ve finally figured out how to get students to sit up and pay attention during those dull sections on sequences and series. Write up on the board this proof that

Actually, it would just give students the excuse they’ve always been looking for to send me to the loony bin.

P-adic Laplacian

Question: what is the analogue of the Laplacian for ?

“The Trap: What Happened to Our Dream of Freedom” (2007)

This is the film that I now want to project onto the sides of buildings in public squares.

I wouldn’t have expected to find out that the BBC showed a three-hour documentary on the telly that covered many of the topics I’ve been thinking about and kicking around in discussion with friends for the past 10 years, but there you have it.* In addition to the philosophy, game theory, economics, politics, it was most interesting to me the way they interwove psychiatry and surveys in the first two episodes (I’m still working on the third episode). The Ladd-Lipset Survey of the American Professorate and how that came about in the late 70s, although it was not mentioned in the documentary, would have also fit into this context. Mathematical measurement of dubious value was a lietmotif of those times.

Fuchsiana redux

Here’s another little puzzle about Fuchsian groups. [A collaborator came up with it and we'd like to use it to simplify a proof in a paper, but I think it's a question that may have totally independent interest.] Is any Fuchsian group of the first kind with at least one cusp contained as a subgroup of finite index in another Fuchsian group such that has exactly one cusp?

If one is more comfortable with this terminlogy, “Fuchsian group of the first kind with at least one cusp” equals “nonuniform lattice in .

For example, obviously if is a congruence subgroup of any sort, then obvserving that suffices. This gives me the feeling if there’s a counter-example to be found, it would be among non-arithmetic lattices, but I’ve never had any sort of handle on those.

“Why has God created the exceptional groups?”

Apparently, to allow mathematicians to be the center of occasional media feeding frenzies.

Yesterday, infinite dimensional representation theory of real Lie groups was one of the more pleasant, though difficult, backwaters of mathematics.  Today I hear a Congressman (McNerney) is going to make a speech about it on the House floor.   I guess now representation theory is going to become so hot that every book about it will get stolen from the libraries, cost $200+ to buy in the bookstore, and, what’s more, every brilliant young Harvard/Princeton grad student will run into the field and scoop my problems.
You can’t win.

[Quotation in the title is attributed to Harish-Chandra.]

Fuchsiana

UPDATE: I found I introduced an incorrect -1 exponent in the course of the calculations that led to my asking this question. Therefore, the answer to the question isn’t really needed per se for anything serious. I doubt that all pairs of groups have the property described, but I think it’s kind of amusing nonetheless that the principal congruence subgroups, while not normal in the full integer subgroup, do seem to display this curious property.

Tao Times

The Times has this article up on the life and work of Terence Tao. Unfortunately, they mis-state the Green-Tao Theorem:

Dr. Tao and Dr. Green proved that it is always possible to find, somewhere in the infinity of integers, a progression of prime numbers of any spacing and any length.

If this were the theorem, it would imply the twin prime conjecture. The correct statement (as well as a link to the paper of Green and Tao) can be found here. The difference between that and what the Times article is that, for a given k, you cannot specify the spacing you want your arithmetic progression of primes of length k to have. The theorem says that a suitable spacing can be found for each k, and this spacing may never be 2, as far as the theorem itself says.

Nevertheless, Green and Tao’s work is huge progress towards the twin prime conjecture, even if it doesn’t solve it outright (yet).

In Memoriam, 1927-2005

Here’s a critique of the Times obituary.

Perhaps more interesting than either, there’s this interview online where Lang explains his method in the great detail. Also gave some insights into his philosophy and technique that were new to me.