# Dynamics Seminar 2021-2022: Difference between revisions

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''Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation'' | ''Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation'' | ||

Abstract: Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets. | |||

Abstract Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets. | |||

We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed. | We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed. | ||

===Jayadev Athreya (UW Seattle)=== | ===Jayadev Athreya (UW Seattle)=== | ||

''TBA'' | ''TBA'' |

## Revision as of 00:26, 12 September 2021

The Dynamics seminar meets in room **901 of Van Vleck Hall** on **Mondays** from **2:30pm - 3:20pm**. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Caglar Uyanik or Chenxi Wu.

## Fall 2021

date | speaker | title | host(s) |
---|---|---|---|

Sep. 13 | Nate Fisher (UW Madison) | "Boundaries, random walks, and nilpotent groups" | local |

Sep. 20 | Caglar Uyanik (UW Madison) | "Dynamics on currents and applications to free group automorphisms" | local |

Sep. 27 | Michelle Chu (UIC) | "TBA" | caglar |

Oct. 4 | Osama Khalil (Utah) | "Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation" | caglar |

Oct. 11 | Open | TBA | |

Oct. 18 | Open | TBA | |

Oct. 25 | Open | TBA | |

Nov. 1 | Open | TBA | |

Nov. 8 | Jayadev Athreya (UW Seattle) tbc | TBA | caglar and grace |

Nov. 15 | Open | TBA | |

Nov. 22 | Jonah Gaster (UWM) | TBA | caglar |

Dec. 6 | Matt Clay (Arkansas) tbc | TBA | caglar |

## Abstracts

### Nate Fisher (UW Madison)

*Boundaries, random walks, and nilpotent groups*

In this talk, we will discuss boundaries and random walks in the Heisenberg group. We will discuss a class of sub-Finsler metrics on the Heisenberg group which arise as the asymptotic cones of word metrics on the integer Heisenberg group and describe new results on the boundaries of these polygonal sub-Finsler metrics. After that, we will explore experimental work to examine the asymptotic behavior of random walks in this group. Parts of this work are joint with Sebastiano Nicolussi Golo.

### Caglar Uyanik (UW Madison)

*Dynamics on currents and applications to free group automorphisms*

Currents are measure theoretic generalizations of conjugacy classes on free groups, and play an important role in various low-dimensional geometry questions. I will talk about the dynamics of certain "generic" elements of Out(F) on the space of currents, and explain how it reflects on the algebraic structure of the group.

### Michelle Chu (UIC)

*TBA*

TBA.

### Osama Khalil (Utah)

*Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation*

Abstract: Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets.

We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed.

### Jayadev Athreya (UW Seattle)

*TBA*

### Jonah Gaster (UWM)

*TBA*

TBA.

## Archive of past Dynamics seminars

2020-2021 Dynamics_Seminar_2020-2021