401-3370-67L Homogeneous Dynamics and Counting Problems
Semester | Autumn Semester 2019 |
Lecturers | P. Yang, further speakers |
Periodicity | non-recurring course |
Language of instruction | English |
Comment | Number of participants limited to 12. Registration to the seminar will only be effective once confirmed by the organisers. Please contact Link. |
Abstract | Introductory seminar about the connection between counting problems and mixing properties for group actions. We discuss linear groups, Haar measures, measure preserving actions, ergodicity, the theorem of Howe-Moore and use these concepts to count integer points on certain affine varieties. |
Objective | |
Content | The goal behind the Gauss circle problem is to describe the asymptotics of the number of integer points in a given ball in Euclidean space as the radius of the ball goes to infinity. In this course we will study similar problems such as counting the number of integer matrices of a given determinant in large balls. In 1993 Duke, Rudnick and Sarnak solved counting problems of this kind by proving equidistribution of certain orbits in homogeneous spaces. Shortly thereafter, Eskin and McMullen gave an approach to proving the desired equidistribution result by exploiting mixing properties of certain group actions. In this seminar we develop the tools required for understanding the connection between mixing and counting for a selected number of explicit cases. Exercises are an integral part of the seminar. |
Lecture notes | References will be provided. |
Literature | Main references: M. Einsiedler, T. Ward Ergodic Theory with a view towards number theory, Springer. Further references will be provided. Additional references: W. Duke, Z. Rudnick, and P. Sarnak. Density of integer points on affine homogeneous varieties. Duke Math. J. Volume 71, Number 1 (1993), 143-179. A. Eskin and C. McMullen. Mixing, counting, and equidistribution in Lie groups. Duke Math. J. Volume 71, Number 1 (1993), 181-209. |
Prerequisites / Notice | The students are expected to have mastered the content of the first two years taught at ETH. The seminar is mainly intended for Bachelor students. |