The proposal concerns four projects regarding the distribution of geometric and arithmetic objects on homogeneous spaces of Lie groups. They are equidistribution of Hecke correspondence, counting rational points of bounded height, distribution of rational points with given denominator and distribution of values of irrational forms. They propose to use techniques from various different fields such as harmonic analysis, dynamics of group actions, ergodic theory, automorphic forms of semisimple algebraic groups in order to prove (or disprove) the equidistribution of densely distributed objects arising in number theoretic and geometric situation.
One of big achievements in modern mathematics is the use of ergodic theory, dynamics of homogeneous spaces, arithmetic geometry and automorphic forms in solving long standing open problems in number theory. The proposed projects are focused on establishing further these connections between different disciplines of mathematics.
Understanding integral or rational solutions of Diophantine equations has long been one of the central problems in mathematics. For instance, the famous Fermat's last theorem, resolved by Andrew Wiles in the nineties, concerns with the existence of a non-zero rational solution of the equation x^3+y^3=z^3. Another famous conjecture regarding rational solutions of Diophantine equations is made by Y. Manin around 1987 and this time the conjecture is about Diophantine equations which are believed to have lots of rational points and attempts to describe the number of rational points of bounded height in terms of geometric invariants of the algebraic variety formed by the complex solutions of the Diophantine equation. As one of the outcomes of this project, in collaboration with other mathematicians, we were able to prove some cases of Manin's conjecture when the relevant algebraic variety admits a transitive action of a simple matrix group. An Apollonian circle packing is an ancient Greek construction whichis made by repeatedly inscribing circles into the triangular interstices of four mutually tangent circles in the plane, whose contsruction goes back to an ancient geometer, Apollonius of Perga (250 BC) Another important outcome of this proposal concerns with finding of a precise asymptotic formula for the number of circles of bounded curvature for Apollonian circle packings. This work was featured in a public magazine (American Scientist) by Dana Mackenzie. A key feature of an Apollonian circle packing which distinguishes itself from an arbitrary circle packing is that it has many hidden symmetries arising from the inversions of the dual circles. This group of hidden symmetries is too thin from the classical mathematical viewpoint. So our work which resulted in this finding also involved introducing many new ideas and concepts which are useful in the study of ergodic theory of a more general thin group.
