BMS scolloquium

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My BMS colloquium in November 2011

My BMS colloquium in November 2011

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  • 1. Probability, Geometry, and algorithms
    • (for matrix, mostly, groups)
  • 2. Igor Rivin
    • Temple University and BMS
  • 3. How do we make a simple object
    • Should be linear.
    • Should be abelian
    • Should be continuous
  • 4. Why is linear good?
    • x+x = 5 -- easy
    hard
  • 5. Why is continuous good?
    • Easy:
    • Hard:
  • 6. Abelian is good
    • Umm, we will see examples later.
  • 7. Now, apply the above to groups:
    • What is the simplest group? Has to be abelian, continuous, linear?!
    • What is more linear than a line?
    • So, our first group is R
  • 8. How easy is
    • Well, the full study of R
    is known as harmonic analysis
  • 9. Obviously too easy.
    • Now, let us remove continuity. We have the lattice of integers Z.
    • The study of Z is known as number theory , also much too easy a subject, but here is a mildly interesting result:
  • 10. The Prime Number Theorem
    • If we pick a number uniformly at random between 1 and N, the probability that the number is prime is approximately 1/log(N).
    • To make the above really precise requires the Riemann Hypothesis.
  • 11. (Some confusion)
    • Notice that Z is an ADDITIVE group, so talking about primes confuses the issue, since now we are treating it as a multiplicative semigroup.
  • 12. Another continuous, linear, abelian group
    • R n . Again, this can be studied via harmonic analysis. It has its own integral lattice Z n .
    • Z n also has a semigroup structure, where the primitive elements are points (a, b, ..., c) where gcd(a, b, ..., c) = 1.
  • 13. Analogue of prime number theorem?
    • Yes! The probability that a lattice point in a ball of radius M in is “primitive” approaches 1/ζ(n) (this in Z n ).
    • Note that this does NOT go to zero as M goes to infinity
    • To make it very precise is MUCH harder than the Riemann hypothesis.
  • 14. The study of Z n
    • 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.
  • 15. Now, let’s remove commutativity
    • 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):
  • 16. What is a lattice?
    • 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.
    • (this is defined by analogy with the integer lattices we looked at before).
  • 17. Geometry of lattices: examples
    • 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).
  • 18. Here is an example
  • 19. Here is another
  • 20. Everyone’s favorite lattice
    • PSL(2, Z ) -- matrices with integer entries (on the right is the fundamental domain in the upper half plane).
  • 21. More generally
    • The group SL(n, Z ) is the group of automorphisms of Z n .
  • 22. More favorites
    • Principal congruence subgroup Γ 0 (N): the kernel of the natural map
    • modN: SL(n, Z ) ⟼⟼SL(n, Z/ N Z )
  • 23. Why “principal”?
    • In general a congruence subgroup is the preimage of some subgroup of SL(2, Z/ N Z ) under modN.
  • 24. (Hard) exercises
    • The map modN is always surjective
    • SL(n, Z /N Z ) x SL(n, Z /M Z ) = SL(n, Z /(MN) Z ) if M, N relatively prime.
    • (special case of “strong approximation”)
    • (good reference: M. Newman, Integral Matrices)
  • 25. Congruence subgroup property (CSP)
    • Lattices in SL(n, Z ) have the Congruence Subgroup Property , which means that any lattice is a congruence subgroup. (Mennicke, Milnor-Bass-Serre, ‘60s)
  • 26. Is a random matrix in SL(n, Z ) irreducible?
    • What is “a random matrix”?
    • What is “irreducible”?
  • 27. Second question first...
    • Irreducible means that the characteristic polynomial is irreducible over Z , which is equivalent to saying that there is no invariant submodule of Z n .
  • 28. First question is harder:
    • 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.
    • Open question: how do you do this efficiently? As far as I know, this is not even known for n=2.
  • 29.
    • Combinatorial answer: pick some generating set, look at all words of length bounded by N.
    • 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.
  • 30.
    • (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.
    • Why the difference? In the first model the number of matrices of size bounded by N is polynomial --
    Luckily, in both cases the answer is the same
  • 31.
    • (the above is a nontrivial result, for SL(2) -- M. Newman, for SL(n) -- Duke/Rudnick/Sarnak)
    • In the second model, the number of different elements of length bounded by N is exponential.
  • 32. Experimental results
  • 33. Generating sets
    • Standard generating set is that of elementary matrices.
    • Hua and Reiner discovered in 1949(?) that SL(n) is generated by only two matrices, so this is the other generating set.
  • 34. How are results like this proved?
    • 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.
    • Then use “strong approximation” as the analogue of “Chinese remaindering” to lift the result to SL(n, Z ).
  • 35. To get convergence rates
    • Use the fact that SL(n, Z ) has property T for n > 2, and property𝞽 𝝉𝜏with respect to representations with finite image.
    • What does this mean?
    • Trivial representation is isolated in the space of representations.
  • 36. Property T, continued
    • 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.
  • 37. Going even further from continuous
    • 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.
    • Zariski-dense: not contained in an algebraic subvariety.
  • 38. Is a group Zariski dense?
    • 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.
    • The number of exceptions in MVW is effectively computable, supposedly (I have never seen an effective bound).
  • 39. But
    • Convergence estimates for Zariski-dense subgroups are not very effective.
  • 40. Unlike for lattices
    • Where the convergence estimates are explicit (IR ’07, Kowalski ’08)
    • 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.
    • Congruence Subgroup Property ought to help, but not clear how.
  • 41. Why undecidable?
    • 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 .
    • Membership problem: does a group generated by A, B, C, ..., D contain a given matrix M?