TCC Course (Spring 2009)

Flows on homogeneous spaces and applications

Course Objectives: In this course we discuss some aspects of dynamics on the space of unimodular lattices in the Euclidean space. This subject lies on crossroads between the theory of dynamical systems, number theory, and representation theory. However, we don't assume any background in these fields and rely only on basic knowledge of analysis and algebra.

In this concete but rich setting, we plan to introduce from scratch some basic concepts of the theory of dynamical systems and explan how they are related to classical problems in number theory.

This course should be of interest to students working in dynamical systems, number theory, and analysis on homogeneous spaces.

We cover some of the following topics:

• Examples of flows (geodesic and horocycle flows) and basic notions of ergodic theory,
• Howe-Moore theorem on decay of matrix coefficients,
• Counting integral points on varieties using dynamics,
• Elements of the Ratner theory of unipotent flows,
• Oppenheim conjecture on distribution of values of quadratic forms,
• Elements of the theory of Diophantine approximation and its dynamical interpretation,
• Baker-Sprindzhuk conjecture on Diophantine approximation on manifolds,
• Actions of higher-rank abelean groups and Furstenberg's conjecture,
• Littlewood conjecture on simultaneous Diophantine approximation.

Time: Fri. 9-11am; See TCC Timetable

Lecture notes:

• Lecture 1: Problems in number theory and dynamical systems and space of lattices
• Lecture 2: Reduction theory and applications
• Lecture 3: Howe-Moore theorem and counting lattice points
• Lecture 4: Mahler compactness criterion and applications to Diophantine approximation
• Lecture 5: Nondivergence of unipotent flows and Borel density theorem
• Lecture 6-7: Oppenheim conjecture and dynamics of unipotent flows
• Lecture 8: Diophantine approximation and dynamics of higher rank abelean groups: Littlewood and Furstenberg conjectures

Homework Problems:

• Homework problem set

Course Assessment: The course will be assessed by problem sheets that will be posted here.

References:

• 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.
• D. Kleinbock, Some applications of homogeneous dynamics to number theory. Smooth ergodic theory and its applications (Seattle, WA, 1999), 639--660, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001.
• D. Kleinbock, Metric Diophantine approximation and dynamical systems.
• E. Lindenstrauss, Some examples how to use measure classification in number theory. Equidistribution in number theory (A. Granville and Z. Rudnick, eds.), 261-303, Springer, 2007.
• G. Margulis, Diophantine approximation, lattices and flows on homogeneous spaces. A panorama of number theory or the view from Baker's garden (Zurich, 1999), 280-310, Cambridge Univ. Press, Cambridge, 2002.
• A. Starkov, Dynamical systems on homogeneous spaces. Translations of Mathematical Monographs, 190. American Mathematical Society, Providence, RI, 2000.
• A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture. Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 117-134.
• D. Witte Morris, Ratner's theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2005.