- Lecturer
- Manfred Einsiedler
- Coordinators
- Pengyu Yang, Emilio Corso

The main aim of this seminar is to present some applications of homogeneous dynamics to counting problems in several settings.

The question of describing the asymptotics of the number of integer or, more generally, lattice points in growing families of subsets of Euclidean/hyperbolic manifolds was first addressed by Gauss at the beginning of the 19th century, and has attracted a great deal of mathematical attention ever since. While the problem in its most general formulation still appears intractable with currently available methods, significant advancements have emerged in the last three decades, starting with the work of Duke-Rudnick-Sarnak and Eskin-McMullen, leading to a satisfactory answer in a peculiar, yet large, class of varieties. The common feature of the methods recently employed to make progress on such questions is the use of various equidistribution results, in the context of dynamics on homogeneous spaces.

After setting up the basic language of ergodic theory of group actions and unitary representations of topological groups, we will discuss the general approach to counting developed by Eskin and McMullen, who simplified and extended an equidistribution result established a few months earlier by Duke, Rudnick and Sarnak.

Students should have mastered the contents of the first two years of the Bachelor in Mathematics taught at ETH. A basic acquaintance with functional analysis, especially unitary operators on Hilbert spaces, is helpful but not strictly required.

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.

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# Exercises

# Literature

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

1 | Emilio Corso | Introduction, overview & administration. | |

2 | Robert Meier, Xiaoying Cao | Gauss Circle Problem | [GCP] and parts of Chapter 3 in [FA] |

3 | Sebastian Vögele, Roman Gorazd | Hyperbolic plane | [EW1, Chapters 9.1 and 9.2] |

4 | Emanuel Malvetti, Clemens Bannwart | Linear groups and Haar measures | [EW1,Chapters 8.3, 9.3, C.2 and Lemma 11.31], [Folland, Chapter 11], [Knapp, § § 1,2,3 of the Introduction] |

5 | Haoyun Ying, René Pfitscher | Lattices in linear groups | [EW1,Chapters 9.3, 9.4], [EW3, Chapters 1.1.1, 1.1.2] |

6 | Santiago Misteli, Alberto Perrella | Ergodic transformations, unitary representations and ergodicity of the geodesic flow | [EW1,Chapters 2.1-2.4, 11.3] |

7 | Clemens Bannwart, Sebastian Vögele | Mautner phenomenon, mixing transformations and the Howe-Moore theorem. | [EW1,Chapters 2.7, 11.4], [EW3, 2.2-2.3] |

8 | Haoyun Ying, René Pfitscher | Equidistribution of large horocycles | notes here. |

9 | Robert Meier, Roman Gorazd | Counting primitive integer points in the Euclidean plane | [EW3, 12.1-12.3] and notes here. |

10 | Xiaoying Cao, Emanuel Malvetti | The counting method of Duke, Rudnick, Sarnak and Eskin, McMullen | [EW3, 12.3] |

11 | Santiago Misteli, Alberto Perrella | Equidistribution of large circles and Selberg's theorem | [EW3, 12.2] as well as these notes. |

12 | Pengyu Yang | 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.
- [Folland] Folland, G. (1999). Real Analysis: Modern techniques and Their Applications, second edition, John Wiley & Sons.
- [Knapp] Knapp, A. (2004). Lie Groups beyond an Introduction, second edition. Progress in Mathematics (PM, volume 140). Birkh??user. Link here.