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401-3375-67L 7 Credits DR , MSC D-MATH

Homogeneous Dynamics and Applications

Lecturers & Examiners: Prof. Dr. Manfred Einsiedler
VVZ CR n/a

Last Updated: 2026-06-01 11:30:59

Abstract

We will build the theory of homogeneous dynamics and discuss applications to number theory, e.g. counting results concerning quadratic forms. We will assume (the willingness to learn the) basics of dynamics, but will build the theory of homogeneous dynamics from scratch.

Objective

Basic theory of homogeneous dynamics, including the mixing property arising from the Howe-Moore theorem. Unipotent dynamics: horospherical case, non-divergence, Ratner's ground-breaking results, proofs of special cases, and some corollaries. Applications will include counting results in the spirit of Duke-Rudnick-Sarnak and Eskin-McMullen, the connection to Diophantine approximations, classification of abstract factors in the sense of ergodic theory, and others.

Content

We will start with the basic theory of homogeneous dynamics: construction of quotients, connection to geometry of numbers, Howe-Moore and its dynamical interpretation as the mixing property. After a brief introduction we will also start using the language of algebraic groups, which will be useful in seeing closed orbits as algebraic or number theoretic objects. In unipotent dynamics we will discuss the horospherical case and non-divergence results in greater detail. However, for Ratner's ground-breaking results we will only give proofs in special cases. Applications will include counting results in the spirit of Duke-Rudnick-Sarnak and Eskin-McMullen, the connection to Diophantine approximations, classification of abstract factors, and time permitting other topics as well. Most of these applications will only need basic results from homogeneous spaces or special cases of Ratner's theorems that we were able to prove before.

Resources

Lecture Notes

We will followhttps://tbward0.wixsite.com/books/homogeneous(which will be updated along the way).

Learning Materials (Links)

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 30 minutes
The exam is only offered in the Examination Sessions Winter 2026 and Summer 2026.30 minutes preparation and 30 minutes exam (one candidate prepares during the 30 minutes oral exam of the previous candidate).

Course Components

Type Title Time & Place Hours
lecture with exercise Homogeneous Dynamics and Applications
notice the rescheduling: Thu 14-16 instead of Wed 10-12
  • Thu 14:15-16:00 (HG D 7.2)
  • Fri 10:15-12:00 (HG D 1.1)
4 h weekly

Offered In