Colloquium talk at St Andrews on what mathematics is (for me).

1. Generic Phenomena
Igor Rivin

2. St Andrews,
Scotland
October 16, 2014

3. What is our goal in doing
mathematics?

4. Grothendieck
view
Top down: Understand things
in maximal generality

5. Thurston view:
bottom up
Understand examples

7. Now we must choose, said Mercier.
Between What? said Camier.
Between ruin and collapse, said Mercier.
Could we not somehow combine them? said
Camier.
Samuel Beckett, Mercier and Camier.

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

9. 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?”)

10. Warning: none of the three
parts are easy.

11. Extended example
What does an integer matrix look like?
Or, more generally, what does a collection of
integer matrices look like?

12. 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).

13. Pick a random matrix
How???
Since the group is infinite, not so easy, so need
to approximate infinity.

14. 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.

15. 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.

16. 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.

17. OK, suppose we picked
them.
Now what?

18. 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)

19. 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)

20. Are the dice loaded?
In other words, can we compute the Galois
group?

21. 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.

22. 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?

23. 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.

24. 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.

25. Are the dice loaded?
In other words, how do we check that our
random subgroup has“generic” behavior?
Well…

26. 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.

27. 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.

28. 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.

29. 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!

30. 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).