- Lecturer
- Manfred Einsiedler, Manuel Luethi, Andreas Wieser

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.

Every student is assumed to give one talk in the first half and one talk in the second half of the semester, each of 45 minutes length. A short explantion on how to prepare for a talk in this seminar can be found here. The speakers for week n will be determined at the end of the seminar in week n-2.

As a preparation for your talk, we will meet with you before your talk once you have it ready. This meeting should take place around five days before the talk will take place. We are of course also willing to answer questions by email.

Week | Speakers | Topics | References |
---|---|---|---|

1 | Manuel Luethi, Andreas Wieser | Introduction, overview & administration. | Partial notes here |

2 | Janine Kilchör, Patrick Schöngrundner | Gauss Circle Problem | [GCP] and parts of Chapter 3 in [FA] |

3 | Giulia Cornali, Simone Rodoni | Hyperbolic plane | [EW1, Chapters 9.1 and 9.2] |

4 | Giulia Docimo, Emilie Epiney | Linear groups and Haar measures | [EW1,Chapters 8.3, 9.3, C.2 and Lemma 11.31], [Knapp, § § 1,2,3 of the Introduction], [Tor] |

5 | Lorraine Bersier, Beat Jäckle | Lattices in linear groups | [EW1,Chapters 9.3, 9.4], [EW3, Chapters 1.1.1, 1.1.2] |

6 | Daniel Paunovic, Milos Radosavljevic | Ergodic transformations, unitary representations and ergodicity of the geodesic flow | [EW1,Chapters 2.1-2.4, 11.3] |

7 | Horace Chaix, Giulia Cornali | Mautner phenomenon, mixing transformations and the Howe-Moore theorem. | [EW1,Chapters 2.7, 11.4], [EW3, 2.2-2.3] |

8 | Giula Docimo, Simone Rodoni | Equidistribution of large horocycles | notes here. |

9 | Lorraine Bersier, Beat Jäckle | Counting primitive integer points in the Euclidean plane | [EW3, 12.1-12.3] and notes here. |

10 | Daniel Paunovic, Patrick Schöngrundner | The counting method of Duke, Rudnick, Sarnak and Eskin, McMullen | [EW3, 12.3] as well as these notes for the first talk. |

11 | Horace Chaix, Emilie Epiney | Equidistribution of large circles and Selberg's theorem | [EW3, 12.2] as well as these notes. |

12 | Janine Kilchör, Milos Radosavljevic | Counting integer matrices with given determinant | [EW3, 12.5] |

Solving exercises is an important part of this seminar and a prerequisite for every attending student to obtain the credits. When you have solved an exercise, post it on this overleaf, so that we and other students can have a look at it. We expect every student to solve and post 3 exercises during the semester, which will be assigned by a random process. You may of course suggest exercises of your interest.

- [GCP] Einsiedler, M., & Ward, T. (2016). Diophantine Problems and Homogeneous Dynamics. In D. Badziahin, A. Gorodnik, & N. Peyerimhoff (Eds.), Dynamics and Analytic Number Theory (London Mathematical Society Lecture Note Series, pp. 258-288). Cambridge University Press. Link here.
- [FA] Einsiedler, M., & Ward, T. (2017). Functional Analysis, Spectral Theory, and Applications. Graduate Texts in Mathematics book series (GTM, volume 276). Springer. Link here.
- [EW1] Einsiedler, M., & Ward, T. (2011). Ergodic Theory with a view towards Number Theory. Graduate Texts in Mathematics (GTM, volume 259). Springer. Link here.
- [EW3] Einsiedler, M., & Ward, T. Homogeneous Dynamics and Application. Link here.
- [Knapp] Knapp, A. (2004). Lie Groups beyond an Introduction, second edition. Progress in Mathematics (PM, volume 140). Birkhäuser. Link here.
- [Tor] Tornier, S. Haar measures. Link here.