Geometric aspects of Diophantine approximation

Everyone knows Dirichlet's principle. But not everyone is aware of the fact that the reason for attributing this principle to Dirichlet was his proof of a rather simple but fundamental statement, which lies in the very basement of the theory of Diophantine approximation — the area of number theory studying questions arising in connection with the problem of approximating real numbers with rationals. As Minkowski showed afterwards, Dirichlet's theorem can be proved with the help of purely geometric tools. The talk is devoted to discussing geometric methods in Diophantine approximation. We shall talk about geometry of numbers, Diophantine exponents, transference principle, geometry of continued fractions and their multidimensional generalisations, mentioning also some open problems such as the Littlewood conjecture and the Cassels-Swinnerton-Dyer-Margulis conjecture.