• Lecture I: Dani Correspondence: Diophantine Approximation and Flows
• Lecture II: Khinchin Theorem and Exponential Mixing
• Lecture III: Schmidt Games, Badly Approximable Numbers, and Bounded Geodesics
• Lecture IV: Diophantine Approximation on Quadratic Surfaces and Ramanujan Conjecture

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 proof of Khinchin theorem which uses mixing of flows on homogeneous spaces and the theory of Schmidt games that allows to prove that the set of badly approximable vectors is large. In the last lecture, 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:

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

Basic knowledge of the theory of dynamical systems, which can be learned from the introductory chapters of

• 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,
• M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory", Springer Graduate Texts in Mathematics, 2010,

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

• 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,
• 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.