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Woven tennis

Poster

People have asked for copies of the school poster. You can download a low-resolution image (for viewing online) or a high-resolution image (for printing). (Thanks to Sukant Saran (Tata Institute) for making the poster.) Speakers may also request two printed copies to be sent to them by post.

Schedule

The plan is to have five lectures per day, from Friday 23 July to Saturday 7 August. The current schedule is here. (Go forward to July. You can select "week" rather than "month" on the upper right corner.)

Extended summaries, prerequisites and recommended reading

Yuri Bilu

I will assume just some very basic background in geometry of numbers (Minkowski's "strong" theorem would suffice), but I will state will state it in any case. Nothing really more, everything else is totally elementary.

Manfred Einsiedler

For the course on effective equidistribution by Einsiedler the student should prepare the following material: a) Bochner's theorem for positive definite functions on R^2 ( as used for the spectral theory of unitary actions of R^2 -- will be needed to analyze the unitary actions of SL_3(R) ). b) Basic notions of hyperbolic geometry, connection between the hyperbolic plane and SL_2(R). c) Basic notions of ergodic theory, ergodicity and mixing. d) Definitions of horocycle flow and geodesic flow, Mautner phenomenon. e) Semisimplicity of finite-dimensional linear representations of SL_2(R).

Alexander Gorodnik

Diophantine approximation and homogeneous dynamics

The basic question in Diophantine approximation is how well a real number can be approximated by rationals with a given bound on denominators. It turns out that Diophantine properties of real numbers can be encoded by the dynamics of the geodesic flow on a quotient of the hyperbolic plane, and, more generally, according to the Dani correspondence, Diophantine properties of real vectors can be encoded using dynamics on the space of lattices. In this course we discuss how the methods from the theory of dynamical systems are utilised to prove some of the deep results in Diophantine approximation. In particular, we will explain a connection between the dynamics of unipotent flows and the Baker-Sprindzuk conjecture, which was proved by Kleinbock and Margulus using dynamical techniques. In the second part the course, we discuss "nonabelean" Diophantine approximation, which is concerned with the approximation properties on more general varieties, and its connection the generalised Ramanujan conjectures on spectrum of automorphic representations.

Prospective students are suggested to read introductory chapters in the classical treatments of Diophantine approximation:

  1. J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Math., vol. 45, Cambridge Univ. Press, Cambridge, 1957,
  2. W. Schmidt, Diophantine approximation, Springer-Verlag, Berlin and New York, 1980,
  3. V. Sprindzuk, Metric theory of Diophantine approximations, John Wiley & Sons, New York-Toronto-London, 1979.

However, we don't assume familiarity with these references and discuss related notions and results as we proceed. Basic knowledge of the theory of dynamical systems, which can be learned from the introductory chapters of

  1. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, vol. 269, Cambridge University Press, Cambridge, 2000,
  2. 2. M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, Springer Graduate Texts in Mathematics, 2010,
would be also useful.

Some of the topics covered in this course are surveyed in:

  1. D. Kleinbock, Some applications of homogeneous dynamics to number theory, in: Smooth ergodic theory and its applications, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, pp. 639–660,
  2. G. Margulis, Diophantine approximation, lattices and flows on homogeneous spaces, in: A panorama of number theory or the view from Baker’s garden, Cambridge Univ. Press, Cambridge, 2002, pp. 280–310.

Philippe Michel

The audience should have had some superficial knowledge on modular forms (as discussed for instance in Sarnak's little book "Some applications of Modular forms" -note that chapter 4 of that book contains a proof of Duke's theorem but we will not follow exactly this path- or in Iwaniec's book "Topics in classical automorphic forms"). We will try to present some of the material in the language of adeles which in first approximation may seem a bit forbidding but which is eventually very convenient. For this it will be good to read the first seven section of Knapp's article (Introduction to the Langlands program, PSPM Vol 61).

H A Helfgott

Growth and expansion in linear algebraic groups

Let A be a subset of a finite group G. If G is abelian, the set A· A = {x· y: x,y ∈ A} can be or can fail to be much larger than A the study of these two possibilities is a central classical question in additive combinatorics. If G is not abelian (or even nilpotent), matters start to look different: for example, I proved a few years ago that all subsets A that generate G = SL(2,Z/pZ) and that are not too large must grow rapidly under multiplication in G.

Since then, a great deal of ground has been covered; the result has been generalised to all simple groups of Lie type, and there is plenty of work on solvable groups as well. The work of Bourgain, Gamburd and Sarnak has established strong consequences on expander graphs and their applications.

Detailed topics:

  1. Background: problems in growth, diameter and expansion
  2. Basic techniques
    1. Non-commutative arithmetic combinatorics; Ruzsa distance
    2. Growth by group actions: Pivoting; Sum-product phenomena, incidence and linear relations
  3. Properties of growth in algebraic groups
    1. Groups, subgroups and orbits
    2. Escape from subvarieties
    3. Dimension and size
    4. Applications: growth in solvable groups
  4. Growth in simple groups
    1. Characters and tori
    2. Pivoting in simple groups
  5. From growth to expanders
    1. Expanders: sets and spectral gaps
    2. Growth of sets and flattening of measures
    3. High multiplicity and spectral gaps
  6. Applications of expanders
    1. Random walks
    2. The affine sieve and Apollonian packings