401-3370-67L  Homogeneous Dynamics and Counting Problems

SemesterAutumn Semester 2019
LecturersP. Yang, further speakers
Periodicitynon-recurring course
Language of instructionEnglish
CommentNumber of participants limited to 12. Registration to the seminar will only be effective once confirmed by the organisers. Please contact Link.


AbstractIntroductory 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
ContentThe 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 notesReferences will be provided.
LiteratureMain 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 / NoticeThe students are expected to have mastered the content of the first two
years taught at ETH. The seminar is mainly intended for Bachelor students.