We invite mainly Phd/MSc students. We offer a number (limited, available according to the order of applications) of free accommodation places in the Banach Center guest rooms and, if there is a need, a cheap accommodation elsewhere. There will be no registration fee. We enclose an application form at the end of the advertisement.
The first goal of this course is to describe the objects of extensive recent and on-going research in rigidity theory for group actions, including necessary background from Lie theory and algebraic number theory. After that we prove model results in differentiable rigidity and measure rigidity which illustrate some of the methods used in this area. Along the way we will introduce necessary background from ergodic theory and hyperbolic dynamics.
The course will be accessible to students with a solid background in real analysis, advanced linear algebra and basic geometry/topology, including elementary properties of differentiable manifold. Basic acquaintance with ergodic theory will be helpful but not strictly necessary. No previous knowledge of Lie groups, algebraic number theory of hyperbolic dynamics is required.
Chapters from forthcoming book
SYLLABUS 1. PRELIMINARIES (approx. one lecture) 1.1. Differentiable, topological and measure-preserving group actions. Functorial constructions: restriction, Cartesian product, factor, suspension, natural extension, skew product. 1.2. Linear actions of higher rank abelian groups. Roots, Lyapunov exponents and Weyl chambers, hyperbolic and partially hyperbolic actions. 1.3. Elements of Lie groups. Lie algebra, exponential map. Examples of linear Lie groups. Definitions and examples of lattices in Lie groups 2. PRINCIPAL CLASSES OF ALGEBRAIC ACTIONS (approx. three lectures) 2.1. Automorphisms, homogeneous and affine actions. Definitions and first examples 2.2. Automorphism of the torus. Equivalent forms of ergodicity and partial hyperbolicity conditions. Harmonic analysis method for studying dynamical and ergodic properties of automorphisms of the torus. 2.3 Algebraic centralizer of a toral automorphism. Connection with units in algebraic number fields. Dirichlet unit theorem. Linear algebra over the rationals and over the integers. 2.4. Commuting hyperbolic automorphisms of the torus. Genuine higher rank condition. Examples: Cartan actions, symplectic actions. 2.5. Partially hyperbolic actions by toral automorphisms. Examples of genuinely partially hyperbolic actions. Dimension restrictions. Peculiarity of dynamical behavior. 2.6. Background on nilpotent groups and nilpotent Lie groups. Compact nilmanifolds. Examples of hyperbolic actions by automorphisms of nilmanifolds. 2.7. First examples of homogeneous actions on factors of simple Lie groups. Classical geodesic and horocycle flows on surfaces of constant negative curvature as homogeneous flows on factors of SL(2,R). 2.8. The central example in the theory of higher rank abelian group actions: left translations by the diagonals on facts of SL(n,R) for n>2 - the Weyl chamber flow. 2.9. Contrast between dynamical properties of the geodesic flow (n=2) and Weyl chamber flow (n>2). 3. ELEMENTS OF RIGIDITY THEORY (approx. two lectures) 3.1. Structural stability and differentiable rigidity. Proof of structural stability of hyperbolic automorphisms of the torus. Infinitely many moduli for differentiable conjugacy. 3.2. Invariant measures for hyperbolic automorphisms and for higher-rank actions. Furstenberg times 2, times 3 problem. 3.3. Model problem in differentiable rigidity: local rigidity for Cartan action (two commuting hyperbolic automorphisms) on three-dimensional torus. 3.4. Model problem in measure rigidity: the only positive entropy ergodic measure for Cartan action on three-dimensional torus is Lebesgue. -------------------------------------------------------------------- APPLICATION FORM: ALGEBRAIC ACTIONS OF HIGHER RANK ABELIAN GROUPS AND INTRODUCTION TO RIGIDITY Feb/March, 2007 Name: E-mail: University: Status: PhD student/MSc student/Postdoc (please choose) If you are a PhD/MSc student, please provide a recommendation letter from your supervisor Please sketch your background in the topic of the course and/or enclose your scientific CV.Please fill out the form and send it before the application deadline of 22 January 2007 to the e-mail address.