Generic PhenomenaIgor Rivin

St Andrews, Scotland

October 16, 2014

What is our goal in doing mathematics?

Grothendieck view

Top down: Understand things in maximal

generality

Thurston view: bottom up

Understand examples

Samuel Beckett, Mercier and Camier.

Now we must choose, said Mercier.

Between What? said Camier.

Between ruin and collapse, said Mercier.

Could we not somehow combine them? said Camier.

Yes, we can!Pascal’s view - study collections of events.

The program:

Play a game (“throw the dice”)

Figure out what the typical outcome is.

See if the outcome you have gotten is actually typical (“are the dice loaded?”)

Warning: none of the three parts are easy.

Extended example

What does an integer matrix look like?

Or, more generally, what does a collection of integer matrices look like?

Make it a little more precise…

(with apologies to semigroup theorists): groups are easier for us peasants, so

Take a lattice in some nice group (SL(n, Z), Sp(2n, Z) come to mind).

Pick a random matrix

How???

Since the group is infinite, not so easy, so need to approximate infinity.

Pick a random matrix

One method: our groups are finitely generated, so take random words of length N, see what properties they have, and does anything interesting happen when N goes to infinity.

Pick a random matrix

Another method: matrices are just NxN tuples of real (or complex, if you prefer) numbers, so you can define their size (as elements of some Euclidean space), so pick them uniformly at random from balls of size N.

WARNING!

Seemingly hard problem: how DO you pick a random INTEGER matrix uniformly at random from all the matrices of bounded norm.

IR: 2014(!) (Math. Comp., to appear): there are approximation algorithms which work well in low dimensions (using lattice reduction).

But high dimensions (and general groups) remain hard.

OK, suppose we picked them.

Now what?

First result

The characteristic polynomial of a “random” matrix is irreducible (over Q).

(Just like a “random” polynomial with integral coefficients)

(But not like a random integer)

Can make it better

The Galois group of the characteristic polynomial is the full symmetric group (with probability approaching 1 exponentially fast in N)

(Both results: IR 2008, DMJ)

Are the dice loaded?

In other words, can we compute the Galois group?

Are the dice loaded?

In general, computing Galois group is hard!

However, checking that it is equal to a “large” group like the symmetric group is easier.

Now move to the “even more noncommutative” setting

Take a bunch of “random” matrices in (say) SL(n, Z).

What can we say about the group they generate?

How do we roll the dice?

Now it’s easy, we just do it separately for each matrix.

Although we can do it for some of the matrices, and leave others fixed - the results are more-or-less the same.

And?

A “random” finitely generated subgroup is Zariski-dense (satisfies no “spurious” polynomial relation). (IR 2010)

A random finitely generated subgroup is a free group (R. Aoun 2012)

A random finitely generated subgroup is infinite index in the ambient lattice.

Are the dice loaded?

In other words, how do we check that our random subgroup has“generic” behavior?

Well…

Are the dice loaded?

Zariski-density - can be checked quickly(ish) (IR 2014) - check that some element has Galois group Sn, and there there is an element of infinite order which does not commute with it.

Are the dice loaded?

Is the group infinite index?

In higher rank (so, SL(n, Z), for n>2: conjecturally undecidable, though there are tricks which work in some special cases.

Are the dice loaded?

In rank one, can show (Elena Fuchs + IR) under technical conditions that the Hausdorff dimension of the limit set goes to 0, and this is (at least in principle) computable.

Are the dice loaded?

Checking that the group is free - again, seems to be undecidable, but who can say…

In rank 1, can construct the Dirichlet domain, so the computation is finite, but no complexity bounds!

Just the beginning

Random graphs? (much studied, but still lots of very interesting questions are completely open).

Random surfaces? (see above, and also below)

Random 3-manifolds? (a lot known, see IR 2014, but not really understood.

Random n-manifolds? (completely open)

Random varieties? (exciting progress recently of Sarnak/Wigman, but totally open).