Math 201A: Homogeneous dynamical systems and some of its applications.
Send me an e-mail, we can meet and discuss math (possibly in a coffee shop).
11:55 PM--12:55 PM
Send me an e-mail.
The following is a tentative list of topics to be covered:
- Dynamics on a circle: Irrational rotations, and Furstenberg's x2 x3 theorem.
- Geometry of numbers: space of lattices in Rn, Mahler's compactness criteria, Minkowski's reduction theory.
- Ergodicity and mixing properties of flows on homogeneous spaces: Mautner phenomenon, Howe-Moore theorem
- Arithmetic groups
- Statements of Ratner's theorems and a few applications.
- Margulis's proof of Oppenheim conjecture.
- Proof of some cases of Ratner's measure classification.
will be kept to a minimum.
I will not follow a particular book, but I will post the related books and articles in the course's webpage.
Elon Lindenstrauss and
Alex Gorodnik have taught similar courses.
And our course will be fairly similar to theirs. I find their notes very useful.
Here are a few related references:
- M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces,
London Mathematical Society Lecture Note Series 269 , Cambridge University Press, Cambridge, 2000.
- M. L. Einsiedler et al. (editors), Homogeneous flows, moduli spaces and arithmetic, Clay Mathematics Proceedings, volume 10.
- M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, Springer Graduate Text in Mathematics 259.
- D. W. Morris, Ratner's theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005.
- M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer, New York, 1972.
- M. Einsiedler, Ratner's theorem on SL2(R)-invariant measures, pdf
|Notes related to lectures and supplementary materials:
- Lecture 1: three type of number theoretic problems that will be discussed during this course were mentioned:
- Counting integer/rational points: X(Q) might be small compared to X(C). So we focus on the homogeneous varieties.
- Oppenheim conjectured (proved by Margulis)
- Littlewood conjecture: its connection with homogeneous dynamical system will be discussed.
- Lecture 2: Weyl's equidistribution criteria, van der Corput's trick, and Weyl's theorem were mentioned. They are given as exercises
along with an extended hints. Then equidistribution of an orbit of an irrational rotation was proved without using uniform convergence of
Fourier expansions. The scheme of this argument is fairly similar to counting arguments that we will discuss later.
Here is my note on these lectures.
- Exercise: Weyl's equidistribution theorem.
- Furstenberg's x2 x3 theorem: Here is its proof.
I used Furstenberg's original article, and Kra's note.
- Space of lattices in Rn:
- In this note, first I start with Oppenheim conjecture to motivate the study of
all the lattices of Rn at the same time. Then closed subgroups of Rn are classified. In the next step, we focus on
lattices in Rn, i.e. finite covolume, discrete subgroups. The Chabauty topology is defined, and it is proved that this
(topological) space is homeomorphic to GL(n,R)/GL(n,Z).
- In this note, we construct a nice basis for a given discrete subgroup of
Rn. Then we use Gram-Schmidt process to construct Siegel sets. Using Siegel sets, Mahler's compactness criteria is proved.
- Exercise: Minkowski's first and second theorems.
- Haar measures:
- In this note, we discuss Haar measures, compute certain Haar measures,
define Haar measures on homogeneous spaces, and prove Weil's formula.
- Exercise: Haar decomposition.
- In this note, we prove that SL(n,Z) is a lattice in SL(n,R), and state
Borel- Harish-Chandra theorem on arithmetic groups.
- Quantitative non-divergence of polynomial maps:
- Some of the consequences of Quantitative non-divergence of unipotent flows:
In this note, following Lindenstrauss's lecture notes, you will see
- Either a lattice in Rn has a primitive flag with small covolume which is invariant under U or it goes to a compact set in a positive
portion of time (the compact set is independent of the choice of the lattice) (Dani-Margulis).
- The existence of U-minimal sets (Dani-Margulis).
- A U-invariant U-ergodic Radon measure on SL(n,R)/SL(n,Z) is a finite measure (Dani).
Unit tangent bundle of a hyperboloic surface:
In this note, the connection between the multiplication by diagonal matrices and the geodesic flow
on the 2-dim hyperbolic space is explained.
Decay of matrix coefficients:
- In this note, Mautner Phenomenon is explained, and Howe-Moore theorem for SL(n,R) is proved.
- Exercise: Irrational rotations are ergodic, but not mixing. pdf
- In this note, you see how to find the growth of number of
elements of SL(n,Z) with norm less than T (Duke-Rudnick-Sarnak). (following Gorodnik's lecture notes)
- Oppenheim conjecture: In this note, you see a proof
of the strong form of (Davenport-)Oppenheim conjecture (Margulis, Dani-Margulis).