Fuchsiana redux

March 23, 2007

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 \Gamma_1 of the first kind with at least one cusp contained as a subgroup of finite index in another Fuchsian group \Gamma such that \Gamma 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 \mathrm{PSL}(2,\mathbf{R})''.

For example, obviously if \Gamma_1 is a congruence subgroup of any sort, then obvserving that \Gamma_1\leq\mathrm{PSL}(2,\mathbf{Z}) 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.

Advertisements

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

March 18, 2007

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. Read the rest of this entry »

Solar Energy Links

March 17, 2007

Good place to start for the basics: Wikipedia articles on solar cells, solar power, photovoltaics.

Some blogs I’ve come across that might be worth following on this (and other renewable energy resources): The Energy Blog, Alternative Energy Blog, a guy who calls his site
“The War Against Oil” (how could I pass up linking to a title like that?).

Other organizations and companies to look at: the solar energy page of the National Renewable Energy Laboratory (which I’d never even heard of until a few days ago), a company that makes quantum dots and has some flash presentations explaining them in simple terms.

Read the rest of this entry »

Fleisher, Graffman, Istomin

Speaking of Dearly Departed Ivories, the Library of Congress has released a series of videos (webstreaming links here) of Eugene Istomin interviewing a series of panels of great musicians. As an indication of the sort of the luminaries he was able to draw together, the “Pianists” panel features Emmanuel Ax, Yefin Bronfman, Leon Fleisher, Gary Graffman, Charles Rosen, and the “Composers”, “Virtuosos”, “Conductors” and “Chamber Musicians”, are all at a similar level. Of course this is all must-see for music history geeks like myself, so I’ve been working my way through the videos, starting with “The Pianists”, then “The Composers”, and I’m now on “The Conductors”. Read the rest of this entry »

Edward Aldwell, 1938-2006

March 13, 2007

Edward Aldwell

Checking up on what Aldwell was up to these days–I remembered hearing something vague about his working on some new recordings–I found out that he died in an ATV accident a few months ago. Turns out he was getting ready to record the English Suites, but now we’ll never get to find out to find out what his interpretation might have been like, which is a real shame. Amazon lists about 6 CD albums, belonging to him, all but one of them Bach. There must be others recorded on LP and not remastered, and I’d be interested to find out what those might be. Read the rest of this entry »

Tao Times

March 13, 2007

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).

Ideological Discipline

March 10, 2007

It must have been tough, the but the Times’ reporter got through an entire article on falling output from Mexico’s oil fields without so much a mention of the finiteness of the geological resource or (God forbid!) Peak Oil. Even when the evidence of it is staring her in the face in the form of the ever more scientifically advanced and costly efforts that Pemex (the state oil company) is forced to undertake, to search out ever deeper and less accessible petroleum deposits, in an effort to replenish its reserves. Does the MSM have a Little Black Book that gives them the watchword on petroleum?

It is not surprising in the least that the Beacon for Neo-liberalism on the Hudson would instead try to blame the falling output to Big Government and Labor.

My question is more of whether this sample of propaganda represents the best thinking of the supposed “oil experts” in the financial and energy industries? Are they all this blinkered? Or do they know better and feel content to pass off this fairy-tale version onto the masses through media conduits?