From here to there

One of the odd perks of getting old is that one can look back to the
distant past and wonder with cold detachment just who the hell we
were.

As a kid I was obsessed with cryptography and secret messages. I even
halfheartedly tried to create a language that only my friends and I
would understand. I wanted to build a robot. I also wanted to live in
a desert island and made endless plans in my head on how I would run
my life on it. Perhaps nothing of this is too surprising in a restless
kid, reader of Jules Verne (“La isla misteriosa/The mysterious
island”, “Viaje al centro de la Tierra/Voyage to the center of the
Earth”).

So how did I get from there to becoming a researcher in pure
mathematics? Not along a straight line for sure.

I started engineering at the Universidad de Buenos Aires, aiming for
electronics, the robot thing. I found myself not caring about the
specific answer to any exercise but was fascinated instead by the
solution process. In a basic physics class the lecturer casually
mentioned the exponential, even the cosine, of a matrix. What?

In the vaulted, vast, inspiring space of the Engineering central
library (Paseo Colón!) I discovered a book that described the “dual
numbers”. Just imagine this grotesque incarnation of the complex
numbers: you add to the reals a quantity whose square is zero! How
amazing, how beautiful.

I got a TI-99 programmable calculator, like all my classmates. We
carried it around hanging from our belts looking like some kind of
geeky cowboys. I spent hours playing with the built-in functions for
matrices. Computing, for example, the limiting value of the inverse
of x – A for x near an eigenvalue of A. You could program the
calculator using at most 50 steps. What a triumph to realize two
steps of your code could be combined to gain an extra line. It warped
my programming style for years.

In the hall of that imposing building, at a booth of the university
publishing house EUDEBA, I bought Kelley’s book “General Topology”,
convinced by its preface that the only requirement was a knowledge of
the real numbers. I couldn’t read past the first page.

About then I decided to switch to a degree in math, in the second year
of University; didn’t have any idea of what I was doing, didn’t know
you could make a living doing mathematics.

In my second year of high school in Buenos Aires, mathematics meant
solving problems about triangles using a slide rule or a log table. The
teacher would bring to class a huge slide rule he would hang from the
top of the blackboard and start sliding away. That year I devised an
encryption scheme involving permutations of the five vowels:
a,e,i,o,u. I kept some notes I wrote then on this scheme, written in
the standardized three-holed sheets (hojas de carpeta Rivadavia!) we
had for all our high-school work. I never actually ever used the
encryption scheme.

The all-male group marooned on the desert island is rescued at the
very end of “The mysterious island” by the cryptic Captain Nemo, the
main character of another Jules Verne book, with his
submarine. Anachronistically apparently but a fabulous coup de theatre
that impressed me immensely nevertheless.

I look at the stuff in these notes on permutations now, more than 45
years later, and don’t really understand any of it. But it sure looks
like mathematics.

What I miss

Like a teenager keeping a journal I note in shame that my last post here is from 2012. Since then I have moved to Trieste, Italy, taking a job at the ICTP: Abdus Salam International Centre for Theoretical Physics.

Horculas en el migo. And so it is.

Looking back, it is clear that growing up in Buenos Aires shaped my view of the world, what to expect from the day to day of life. And this happens in really unexpected ways: the shape of buildings, the layout of streets and cities, the birds and the trees (it is frustrating to try to explain the peculiarities of the hornero, the little bird of the endless pampas building its mud house on top of alambrados, to someone and not being able to convey all the memories it brings or why it matters).

Coming to Italy is a bit like going back. So much of the culture feels like home, from the language to the hand gestures to the food. But perhaps the most significant connection for me is: la passeggiata.

People actually go out to walk for the sake of it, for the pleasure of strolling about with their friends, their family, not just to walk the dog, to shop or to exercise. People in normal clothes, pushing a pram, or grandma in a wheel chair, kids chasing each other around, couples holding hands. And they do so in the open air, perhaps taking in the rays of a beautiful sunny Sunday, in public spaces, sidewalks, no commerce involved.

Buona Domenica!

Workshop at ICTP June 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 -function of hypergeometric motives. This means computing numerically enough of the ingredients of the -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 -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

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

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

On one hand we have for a two dimensional vector space 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 . (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 we get an associated -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 -adic number and not just modulo . With this we can compute the Euler factors of our -series at least for all but finitely many primes up to some bound.

The challenge that remains is how to compute the rest of the data of the -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 with Galois group the Weyl group of of order . The -function of the motive is the Artin -function attached to the reflection representation (of dimension ). As it turned out the motive also sits naturally in of an affine cubic surface with an Eckardt point. This surface has lines that are permuted by the Galois group. This description gives us an explicit degree extension with Galois closure . From here it is not too hard to give the associated Artin -function directly.

Taxi rides

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?

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

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

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.