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# BMS scolloquium

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My BMS colloquium in November 2011

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### BMS scolloquium

1. 1. Probability, Geometry, and algorithms <ul><li>(for matrix, mostly, groups) </li></ul>
2. 2. Igor Rivin <ul><li>Temple University and BMS </li></ul>
3. 3. How do we make a simple object <ul><li>Should be linear. </li></ul><ul><li>Should be abelian </li></ul><ul><li>Should be continuous </li></ul>
4. 4. Why is linear good? <ul><li>x+x = 5 -- easy </li></ul>hard
5. 5. Why is continuous good? <ul><li>Easy: </li></ul><ul><li>Hard: </li></ul>
6. 6. Abelian is good <ul><li>Umm, we will see examples later. </li></ul>
7. 7. Now, apply the above to groups: <ul><li>What is the simplest group? Has to be abelian, continuous, linear?! </li></ul><ul><li>What is more linear than a line? </li></ul><ul><li>So, our first group is R </li></ul>
8. 8. How easy is <ul><li>Well, the full study of R </li></ul>is known as harmonic analysis
9. 9. Obviously too easy. <ul><li>Now, let us remove continuity. We have the lattice of integers Z. </li></ul><ul><li>The study of Z is known as number theory , also much too easy a subject, but here is a mildly interesting result: </li></ul>
10. 10. The Prime Number Theorem <ul><li>If we pick a number uniformly at random between 1 and N, the probability that the number is prime is approximately 1/log(N). </li></ul><ul><li>To make the above really precise requires the Riemann Hypothesis. </li></ul>
11. 11. (Some confusion) <ul><li>Notice that Z is an ADDITIVE group, so talking about primes confuses the issue, since now we are treating it as a multiplicative semigroup. </li></ul>
12. 12. Another continuous, linear, abelian group <ul><li>R n . Again, this can be studied via harmonic analysis. It has its own integral lattice Z n . </li></ul><ul><li>Z n also has a semigroup structure, where the primitive elements are points (a, b, ..., c) where gcd(a, b, ..., c) = 1. </li></ul>
13. 13. Analogue of prime number theorem? <ul><li>Yes! The probability that a lattice point in a ball of radius M in is “primitive” approaches 1/ζ(n) (this in Z n ). </li></ul><ul><li>Note that this does NOT go to zero as M goes to infinity </li></ul><ul><li>To make it very precise is MUCH harder than the Riemann hypothesis. </li></ul>
14. 14. The study of Z n <ul><li>Is known as the geometry of numbers. In particular, it studies the group of automorphisms of the integer lattice, known as GL(n, Z ), which we will get to shortly. </li></ul>
15. 15. Now, let’s remove commutativity <ul><li>The simplest class of non-abelian Lie groups is (arguably) (P)SL(n, R ): the group of nxn matrices with determinant one. This is a Lie group of real rank n-1 -- the rank of a Lie group is the dimension of its maximal torus (maximal abelian subgroup). By analogy with the abelian case, we next define a lattice in SL(n) (or any Lie group, for that matter): </li></ul>
16. 16. What is a lattice? <ul><li>Definition: A discrete subgroup Γ of a Lie group G is called a LATTICE, if the coset space H=G/Γ has finite measure. A lattice is called uniform if H is compact. </li></ul><ul><li>(this is defined by analogy with the integer lattices we looked at before). </li></ul>
17. 17. Geometry of lattices: examples <ul><li>PSL(2, R ) is also known as the isometry group of the hyperbolic plane H 2 . The quotients of H 2 by lattices in PSL(2, R ) are finite area hyperbolic orbifolds, ( surfaces if there is no torsion). </li></ul>
18. 18. Here is an example
19. 19. Here is another
20. 20. Everyone’s favorite lattice <ul><li>PSL(2, Z ) -- matrices with integer entries (on the right is the fundamental domain in the upper half plane). </li></ul>
21. 21. More generally <ul><li>The group SL(n, Z ) is the group of automorphisms of Z n . </li></ul>
22. 22. More favorites <ul><li>Principal congruence subgroup Γ 0 (N): the kernel of the natural map </li></ul><ul><li>modN: SL(n, Z ) ⟼⟼SL(n, Z/ N Z ) </li></ul>
23. 23. Why “principal”? <ul><li>In general a congruence subgroup is the preimage of some subgroup of SL(2, Z/ N Z ) under modN. </li></ul>
24. 24. (Hard) exercises <ul><li>The map modN is always surjective </li></ul><ul><li>SL(n, Z /N Z ) x SL(n, Z /M Z ) = SL(n, Z /(MN) Z ) if M, N relatively prime. </li></ul><ul><li>(special case of “strong approximation”) </li></ul><ul><li>(good reference: M. Newman, Integral Matrices) </li></ul>
25. 25. Congruence subgroup property (CSP) <ul><li>Lattices in SL(n, Z ) have the Congruence Subgroup Property , which means that any lattice is a congruence subgroup. (Mennicke, Milnor-Bass-Serre, ‘60s) </li></ul>
26. 26. Is a random matrix in SL(n, Z ) irreducible? <ul><li>What is “a random matrix”? </li></ul><ul><li>What is “irreducible”? </li></ul>
27. 27. Second question first... <ul><li>Irreducible means that the characteristic polynomial is irreducible over Z , which is equivalent to saying that there is no invariant submodule of Z n . </li></ul>
28. 28. First question is harder: <ul><li>Arithmetic answer: pick a matrix uniformly at random from the set of those matrices in SL(n, Z ) whose elements are bounded in absolute value by N. </li></ul><ul><li>Open question: how do you do this efficiently? As far as I know, this is not even known for n=2. </li></ul>
29. 29. <ul><li>Combinatorial answer: pick some generating set, look at all words of length bounded by N. </li></ul><ul><li>Problem: the answer (potentially) depends on the choice of generating set, the probability of picking a given matrix A can be quite different from the probability of picking some other matrix B. </li></ul>
30. 30. <ul><li>(IR ’06): The probability that a matrix is reducible approaches zero polynomially fast in N in the first model, exponentially fast in the second model. </li></ul><ul><li>Why the difference? In the first model the number of matrices of size bounded by N is polynomial -- </li></ul>Luckily, in both cases the answer is the same
31. 31. <ul><li>(the above is a nontrivial result, for SL(2) -- M. Newman, for SL(n) -- Duke/Rudnick/Sarnak) </li></ul><ul><li>In the second model, the number of different elements of length bounded by N is exponential. </li></ul>
32. 32. Experimental results
33. 33. Generating sets <ul><li>Standard generating set is that of elementary matrices. </li></ul><ul><li>Hua and Reiner discovered in 1949(?) that SL(n) is generated by only two matrices, so this is the other generating set. </li></ul>
34. 34. How are results like this proved? <ul><li>work with the FINITE groups SL(n, Z /p Z ), and show that a random walk on that group must become equidistributed, and then analyse the morphism from the group to the set of characteristic polynomials. </li></ul><ul><li>Then use “strong approximation” as the analogue of “Chinese remaindering” to lift the result to SL(n, Z ). </li></ul>
35. 35. To get convergence rates <ul><li>Use the fact that SL(n, Z ) has property T for n > 2, and property𝞽 𝝉𝜏with respect to representations with finite image. </li></ul><ul><li>What does this mean? </li></ul><ul><li>Trivial representation is isolated in the space of representations. </li></ul>
36. 36. Property T, continued <ul><li>So a sort of a spectral gap condition. Holds for all lattice in semi-simple groups of rank greater than 1, because holds for the Lie groups, and lattices are “close” to the mother Lie group. </li></ul>
37. 37. Going even further from continuous <ul><li>The results work also for “thin” Zariski dense subgroups, using strong approximation and the expansion properties of Cayley graphs of finite projections of the special linear group. </li></ul><ul><li>Zariski-dense: not contained in an algebraic subvariety. </li></ul>
38. 38. Is a group Zariski dense? <ul><li>Amazing fact: For a Zariski dense subgroup modP is surjective except for finitely many exceptions (Matthews-Vaserstein-Weisfeiler 1987) and conversely, if modP is onto for some prime > 2 (T. Weigel, 200?), then the subgroup is Zariski-dense. </li></ul><ul><li>The number of exceptions in MVW is effectively computable, supposedly (I have never seen an effective bound). </li></ul>
39. 39. But <ul><li>Convergence estimates for Zariski-dense subgroups are not very effective. </li></ul>
40. 40. Unlike for lattices <ul><li>Where the convergence estimates are explicit (IR ’07, Kowalski ’08) </li></ul><ul><li>But (BIG OPEN QUESTION) it appears to be undecidable when a subgroup of SL(n, Z ) given by generators is of finite index, unless n=2, though it is not easy even then. </li></ul><ul><li>Congruence Subgroup Property ought to help, but not clear how. </li></ul>
41. 41. Why undecidable? <ul><li>It is known that the “generalized word problem” or “membership problem” is undecidable for SL(n, Z ) for n>3 (decidable for n=2, OPEN for n=3) -- reduces to Post Correspondence, since the groups contain F 2 xF 2 . </li></ul><ul><li>Membership problem: does a group generated by A, B, C, ..., D contain a given matrix M? </li></ul>