Workshop at ICTP June 2012

April 18, 2012

I am co-organizing a two-week activity this June at ICTP. The general theme of the school is that of computational algebra and number theory. The second week the school will include a more specific AIM workshop on hypergeometric motives. This workshop is a natural continuation of that in Benasque in 2009. I will be posting in this blog some comments on the matter. We will have a more formal site to organize the activities of the workshop shortly.

One of the main goals of the AIM workshop is to be able to explicitly compute the $L$-function of hypergeometric motives. This means computing numerically enough of the ingredients of the $L$-function in detail so as to, for example, check that it satisfies the expected functional equation. In fact, testing the functional equation is a good way to find a few of the ingredients, say the conductor, that might be missing. Once we have a large computable set of higher degree $L$-functions (i.e., with Euler factors of degree bigger than two say) we can begin exploring numerically various conjectures about their zeros, special values, etc.

So, what is a hypergeometric motive? The best place to start is the Legendre family of elliptic curves

$E_t: y^2=x(x-1)(x-t),$

Deuring computed the number of points of this family over the finite field $\mathbb F_p$ and showed that modulo $p$ is given in terms of the polynomial

$(-1)^{(p-1)/2}\sum_{n=0}^{(p-1)/2} \binom{2n}{n}^2t^n.$

Igusa later pointed out that this polynomial satisfies the same hypergeometric differential equation as a period of $E_t$ over the complex numbers.

On one hand we have for $t \in \mathbb Q$ a two dimensional vector space $H^1(E_t,\mathbb Q_l)$ with an action of Galois and on the other the two dimensional space of solutions to the hypergeometric differential equation. These are two incarnations of an abstract object: a “motive”.

Now the point is that we can start with *any* hypergeometric equation with appropriate parameters and it will yield a family of motives parameterized by $t$. (Because we have a rigid system the hypergeometric equation is determined by very simple discrete data; in other words, there is no moduli.) For each rational value of $t$ we get an associated $L$-function. These are what we want to be able to calculate.

The key fact is that though it is not entirely clear how to describe the motive (for example, how do we come up with the Legendre family of elliptic curves starting from the differential equation?) this description is not strictly necessary! Indeed, there is a formula like that of Deuring’s above that gives the trace of Frobenius on the motive. This finite field analogue of the hypergeometic series (due to N. Katz) is quite computable and it actually gives the trace as a $p$-adic number and not just modulo $p$. With this we can compute the Euler factors of our $L$-series at least for all but finitely many primes $p$ up to some bound.

The challenge that remains is how to compute the rest of the data of the $L$-function. Namely, i) the Euler factors for the remaining primes, ii) the Gamma factors, iii) the conductor and iv) the sign of the functional equation.

To complete this program some further description of the motive would in fact be helpful. But there isn’t necessarily a unique or canonical way to exhibit the motive. Different ways of presenting it bring their own advantages. To my mind a motive is somewhat like what genes are in biology. The same gene can appear in the DNA of very different life forms.

An example we worked out with Henri Cohen is a good illustration of this point. The motive has weight zero and corresponds to a Galois extension of $L/\mathbb Q(t)$ with Galois group the Weyl group of $F_4$ of order $1152$. The $L$-function of the motive is the Artin $L$-function attached to the reflection representation (of dimension $4$). As it turned out the motive also sits naturally in $H^2$ of an affine cubic surface $S$ with an Eckardt point. This surface has $24$ lines that are permuted by the Galois group. This description gives us an explicit degree $24$ extension $K/\mathbb Q(t)$ with Galois closure $L/\mathbb Q (t)$. From here it is not too hard to give the associated Artin $L$-function directly.

Taxi rides

June 1, 2011

I am not sure if it is an Austin phenomenon or not but I have had a few interesting taxi rides back and forth to the airport over the years.

I’ll leave aside the late night taxi ride where the driver, a young woman, told a few off-color jokes (“you are not a prude man, are you?”); including one that I am pretty sure was popular when I was in primary school. (Let’s just say that it involved traveling by camel in the desert.)

More curious was the guy with the fourth dimension. It started innocently enough: “what’s your business?” or something like that. When I said I taught math (often a risky move) he went off about the fourth dimension and extra-terrestrial beings living among us. He actually showed me a pile of issues of a magazine called “The Fourth Dimension”. I also recall something about a meeting of extra-terrestrials in Washington DC the following month. And this was just a few minutes into the ride.

I then made a really reckless move. I mentioned that in fact, I was currently working with some physicists (this was at the time of my collaboration with Ph. Candelas and X. de la Ossa) and that in string theory the thinking was that the universe was actually ten dimensional.

The response was a chilly silence. I distinctively recall thinking: the driver must be saying to himself “Now, there is a nut in this taxi … and it is not me”.

What’s new in the world of addition?

May 5, 2011

I gave a talk at the Mathematics Teachers’ Circle of Austin on the distribution of carries in adding numbers in a given base. (The full title “Dizzying the memory of arithmetic” is supposed to be a play of words on a phrase in Macbeth…)

The question is this. Take x_1,…,x_k some n digit numbers, say in base 10. When you add them with the usual elementary school algorithm some of the columns of k digits will contribute a carry, a number in the range 0 to k-1, to the next column. What is the distribution of these carries for random x_i for large n?

The question has an interesting answer, which I won’t spoil but refer you to the original beautiful paper by J. Holte (mathscinet).

Surprisingly, Diaconis and Fulman found a connection between this question and card shuffling, Foulkes characters, symmetric functions,…

I love it when we discover something new and interesting in things right under our noses.

Limiting distribution of Betti numbers I

April 27, 2011

I gave a talk in the algebra seminar at Georgia Tech on April 11, 2011 with this title. The basic question is this: how are the Betti numbers of a discrete family of algebraic varieties distributed as we approach large values of the parameters? After appropriately shifting and scaling, is there a limiting distribution?

The question arose from our joint work with Hausel on the geometry of character varieties. We later found and instance of this phenomenon already discussed by Reineke archiv (see below).

I presented three examples.

1. The Grassmanian G_n,k. Here for large n and k the Betti numbers are distributed as a Gaussian. No surprise perhaps; by Poincare duality the distribution is symmetric and by the Hard Lefschetz theorem it increases up to the middle of the range and then decreases.

2. The Hilbert scheme of n points on the plane. Here the Betti numbers are distributed as partitions of n according to length (by Goettsche’s work). There is a limiting distribution as proved by Erdos and Lehner. It is the Gumbel distribution exp(-exp(-x)) that appears as an universal distribution for max(x_1,…,x_n) of independent, identically distributed random variables x_i (given the known height of a river for the past 10 years how high could we expect it to get this year?) You can see the case n=500 and the corresponding limit case in the notes of the talk. The distribution is no longer symmetric.

3. The toric hyperkaehler variety (Hausel-Sturmfels archiv) associated to the complete graph K_n. Here the Betti numbers are given by the coefficients of the reliability polynomial of K_n. These are known to have a limiting distribution: the Airy distribution (the same as that appearing in Reineke’s case mentioned above.) This distribution appears in a number of different other contexts (hashing algorithms, area of Brownian motion, large graphs with fixed genus, etc.)

It is remarkable that the distributions in 2 and 3 (Gumbel and Airy respectively) are in fact very close (scaling and shifting appropriately) to each other but are definitely not the same. (In the early stages we thought that perhaps we always got the Gumbel distribution; it makes you wonder how much to read in the comparison of continuous graphs to discrete data.)

Is there some kind of universality?

After the talk Stavros Garoufalidis mentioned that the Airy distribution also appears in various matrix models in Physics.

Back and short

January 8, 2009

I’ve lived in the US now for over 22 years. I have encountered two dollar bills exactly three times (I kept them all). The first time was around Harvard Square as change in a store. It was so strange.

I’ve seen a few dollar coins too.  Usually all of them at once unfortunately. At least 15 years ago, if you bought a NJ transit train ticket in a machine at the station and paid with a 20 dollar bill the change came clanking down in a stream of dollar coins. They are heavy.

And just the other day reading a post at the The Oil Drum encountered the word copacetic for the first time in my life, according to World Wide Words a uniquely American slang word meaning “fine, excellent, going just right”. (Not much use for it these days.)

On my way back from Vancouver once the US customs officer that checked my passport, after finding out I was a mathematician, told me: [thinking pause] “I hope things add up for you”.

I hope the post does.

Teaching Technology

September 1, 2007

I started teaching this semester using moodle. I am pretty happy with it. I think there is a lot of potential to make teaching more productive. In the process I got really interested in the whole issue of technology and teaching. Last week there was an article in the Washington post on the issue describing how education technology is becoming a fast-growing business. At the very end it mentions Moodle as a free (open source) alternative to Blackboard and how UCLA has decided to switch to using it. I find this a significant move; UCLA is not exactly a small college, we are talking about a pretty big operation. I look forward to seeing how things go.

A quick search uncovered this interview with Ruth Sabean, the assistant vice provost for educational technology at UCLA, where she discusses the decision. It is interesting and so is the posted discussion with readers at the end. In particular, I was led to this insightful post at e-Learning, a blog by Michael Feldstein from Oracle, who claims that “The monolithic closed source LMS [Learning Management Operating System] is dead meat.”

Incidentally, UT uses Blackboard, which according to the Washington Post article “was founded in 1997 by a few 20-somethings who quit comfortable jobs to start the company. The dot-com boom swept up Blackboard, and it weathered the subsequent bust before going public. Last year, it had sales of \$180 million.”

Dead meat or not the question for me is: are we really improving the way we teach? or are we just using new tools to make the usual easier? We all know that writing a paper in TeX (and who doesn’t?) does nothing to its content, neither does giving a powerpoint talk.

On a related topic what I would really like to see is a web environment for collaboration in Mathematics. There are a number of systems out there that could be adapted for this purpose but I have yet to find one I am really satisfied with. Sakai looked promising but it doesn’t seem quite ready for what I have in mind yet.

I’d love to have something like this: a browser based interface (so that one could login from anywhere, whatever the operating system is) that allows one to up/down-load files, keeps an easy-to-use history of the files as they evolve, create webpages, link to documents, etc. It should have a broad array of communication tools: voice, video, whiteboard, chat, forums, etc. For mathematics the chat should have an automatic TeX formatting filter that would allow people to type formulas in TeX while talking via Skype, say. Finally, a powerful and smart search feature (within the system as well as the archive, numdam, JSTOR, GDZ, etc.) including, if they so allow it, other people discussions.

It seems to me that if technology is going to take us to a higher level of doing research it will only do so by increasing the opportunities for the random associations that fuel it. Powerful and smart searching is crucial for this. Am I the only one annoyed by, say, the searching capabilities of mathscinet? Unless you know an author’s name exactly chances are you’ll never find the paper you’re looking for.

As an example, the following funny thing happened to me recently. I was looking for material on a few integrals that have value a rational multiple of pi squared, which Coxeter talks about in the preface to his book “Twelve geometrical essays”. These arose from some volume calculations and are in one of his earliest papers. I quickly found the paper by Wagner, Peter, “Solution to a problem posed by H. S. M. Coxeter”, C. R. Math. Rep. Acad. Sci. Canada 18 (1996), no. 6, 273–277 (related to a different integral actually). The review in Mathscinet has the phrase: ” The method seems to be due to Tortellini in Crelle’s Journal 34 and is explained in Dirichlet’s Bestimmte Integrale.”

I was intrigued both by the alluded method and the name of its author, Tortellini. I had never heard of a mathematician of that name (of any era). A search in mathscinet with the author’s name gave nothing. At least the reviewer, H. W. Guggenheimer, had included the reference to Crelle’s Journal volume 34. A search in GDZ (not that straightforward either actually, I searched for Crelle in title, then clicked on the Journal’s actual name “Journal für die reine und angewandte Mathematik”) yielded the table of contents of volume 34 showing a paper by Barnaba Tortolini! Talk about a Freudian slip.

Actually, who was Barnaba Tortolini?

I find myself spending a lot of time searching for something (on the web, my hard disk or my office), which I know I have found before… (For example, I had to reconstruct the above Tortellini/Tortolini story all over again.) I could, of course, be more organized but wouldn’t it be great to have your computer help you out?

Benasque

July 25, 2007

While I wait for my brain strings to settle after a long summer shuttling about I make a quick core dump before it’s all gone.

I spent two weeks in Benasque, Spain for the workshop P-adic analysis, Periods and Physics . We ended up being a very small group of participants, unfortunate in a way but the result was a very charming workshop. And what a place! Benasque is in a valley in the Spanish Pyrenees not far from the border with France, a small mountain town where you walk everywhere, surrounded by amazing mountains full of unbelievably beautiful hikes. String theorist have been coming here every two years for quite a while. We should have some more math workshops in Benasque!

I’m becoming somewhat obsessed with technology and how it’s changing our daily life in ways we hardly have time to think about, let alone understand. I grew up in Argentina at a time when phones were a luxury; apartments had them or not. Here I was in Benasque sitting under a tree facing a soccer field and some towering green mountains, talking to my brother in Argentina on Skype through a wireless connection to my laptop; it was right about 3pm, before the Benasque kids came out of their siesta to play and my brother had to take his kids to school.

Since writing the post on communicating mathematics today I became more and more convinced that we, the mathematics community, are not exploiting the available technology as much as we could.

Talking about this in Benasque my friend Maria mentioned Moodle a open source management system for courses. It is now installed at UT and I will give it a try for my graduate class this Fall. It looks really well done, with lots of customizing possible: forums, chats, blogs, quizzes, you name it.

I plan to use its “workshop” assignment module, which allows students to grade anonymously a number of other students homework. My goal is to give students the chance to read and assess somebody else’s mathematics and write a report about it. It is after all what we working mathematicians spend quite a bit of our time doing. Hopefully the anonymous feature will also give them a flavor of the referee system for publishing papers (without unleashing too much of the nasty sadism that this can involve). I am all in favor of assigned work for graduate classes in any case. To paraphrase a rock fashion designer: “If your pants don’t hurt it ain’t rock and roll”.

Of course technology can be a mirage too. In a memorable story John Tate was once asked how he dealt with papers when he was a graduate student, an era without photocopying machines. John, without missing a bit, answered: “We read the papers”.

On the other hand, as Philip Candelas remarked, it is likely that at the time one could be on top of pretty much everything being published in a given topic.

I’ve been looking around for some open source software to manage conferences, seminars, lectures, etc. Found some things but they’re not quite it yet. Any recommendations?

Finally, a shameless plug. My book “Experimental Number Theory” has finally appeared, published by Oxford University Press. It contains many computer scripts for the wonderful Number Theory package PARI-GP on hopefully interesting mathematics. It was a lot of fun to write and I learned a great deal in the process. In particular, I encountered my currently favorite elementary math problem (due to D. Knuth): A certain baseball player has a batting average of .334. At least how many times did he bat? (No, is not 3.)

The solution goes back to an algorithm of Gosper included in Hakmem , the remarkable Hacker’s memorandum of the early 70’s.

Of course, you can also check my book…