# Brandeis Dynamics and Number Theory Seminar 2019-2020

## Seminar Information

### Description

The Brandeis Dynamics and Number Theory Seminar is a research seminar broadly showcasing modern research in ergodic theory and dynamical systems, Lie theory, representation theory, geometry, and their interactions with number theory. If you would like to give a talk, please email the organizers

### Organizers

### Time and Location

Thursdays at 3:00PM in Goldsmith 226

### Online calendar

Copy this link and add the URL to your own calendar to stay updated. You may also view the calendar online here

## Schedule of Talks

### Spring 2020

Date | Speaker | Affiliation | Title | Abstract Link |
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January 16 | Yiftach Dayan | Tel Aviv University | Random walks on tori and an application to normality of numbers in self-similar sets | Link to Abstract |

January 23 | Thomas Hille | Yale University | Effective bounds for the least solutions of homogeneous quadratic Diophantine inequalities | Link to Abstract |

February 6 | Alex Kontorovich | Rutgers University | Title TBA | Link to Abstract |

February 27 | Arie Levit | Yale University | Title TBA | Link to Abstract |

April 2 | Asaf Katz | University of Chicago | Title TBA | Link to Abstract |

April 23 | Ilya Khayutin | Northwestern University | Title TBA | Link to Abstract |

April 30 | Sebastian Hurtado-Salazar | University of Chicago | Title TBA | Link to Abstract |

### January 16 @ 3:00PM (Goldsmith 226)– Yiftach Dayan (Tel Aviv University) – Random walks on tori and an application to normality of numbers in self-similar sets

## Abstract

We show that under certain conditions, random walks on a $d$-dimensional torus by affine expanding maps have a unique stationary measure which is Haar measure. In this case, one may deduce that for every starting point in the torus, almost every trajectory of the random walk is equidistributed w.r.t. Haar measure. As an application of this result, we show that given an IFS of contracting similarity maps of the real line with a uniform contraction ratio $1/D$, where $D$ is some integer $> 1$, under some suitable condition, almost every point in the attractor of the given IFS (w.r.t. a natural measure) is normal to base $D$. Joint work with Arijit Ganguly and Barak Weiss.### January 23 @ 3:00PM (Goldsmith 226)– Thomas Hille (Yale University) – Effective bounds for the least solutions of homogeneous quadratic Diophantine inequalities

## Abstract

Let $Q$ be a non-degenerate indefinite quadratic form in $d$ variables. In the mid 80's, Margulis proved the Oppenheim conjecture, which states that if $d \geq 3$ and $Q$ is not proportional to a rational form then $Q$ takes values arbitrarily close to zero at integral points. In this talk we will discuss the problem of obtaining bounds for the least integral solution of the Diophantine inequality $|Q[x]|< \epsilon$ for any positive $\epsilon$ if $d \geq 5$. We will review historical, as well as recent results in this direction and show how to obtain explicit bounds that are polynomial in $\epsilon^{-1}$, with exponents depending only on the signature of $Q$ or if applicable, the Diophantine properties of $Q$. This talk is based on joint work with P. Buterus, F. Götze and G. Margulis.### February 6 @ 3:00PM (Goldsmith 226)– Alex Kontorovich (Rutgers University) – Title TBA

## Abstract

### February 27 @ 3:00PM (Goldsmith 226)– Arie Levit (Yale University) – Title

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### March 5 @ 3:00PM (Goldsmith 226)– Speaker (Institution) – Title

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### March 12 @ 3:00PM (Goldsmith 226)– Speaker (Institution) – Title

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### March 26 @ 3:00PM (Goldsmith 226)– Speaker (Institution) – Title

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### April 2 @ 3:00PM (Goldsmith 226)– Asaf Katz (Universit of Chicago) – Title TBA

Asaf Katz (University of Chicago)

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### April 23 @ 3:00PM (Goldsmith 226)– Ilya Khayutin (Northwestern University) – Title TBA

Ilya Khayutin (Northwestern University)

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### April 30 @ 3:00PM (Goldsmith 226)– Sebastian Hurtado-Salazar (University of Chicago) – Title TBA

Sebastian Hurtado-Salazar (University of Chicago)

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### Fall 2019

Date | Speaker | Affiliation | Title | Abstract Link |
---|---|---|---|---|

September 12 | Dmitry Kleinbock | Brandeis University | Shrinking targets in dynamics and number theory | Link to Abstract |

September 19 | Lam Pham | Brandeis University | Expansion spectral gaps in Lie groups | Link to Abstract |

September 26 | Dmitry Kleinbock | Brandeis University | Shrinking targets in number theory and dynamics | Link to Abstract |

October 10 | Lam Pham | Brandeis University | Classical results around Kazhdan’s Property $(T)$ | Link to Abstract |

October 17 | Samuel Edwards | Yale University | The horocycle flow on hyperbolic surfaces | Link to Abstract |

October 24 | No seminar | - | - | - |

October 31 | Marius Lemm | Harvard University | Global eigenvalue distribution of matrices defined by the skew-shift | Link to Abstract (Unusual time @11:00AM Goldsmith 300! joint with Everytopic) |

October 31 | Rahul Krishna | Brandeis University | An introduction to the Sarnak and Chowla conjectures | Link to Abstract (Unusual time @4:30PM Goldsmith 317) |

November 7 | Mishel Skenderi | Brandeis University | Approximation of Random Functions | Link to Abstract (Unusual time @4:30PM Goldsmith 317) |

November 14 | Claire Burrin | Rutgers University | Discrete lattice orbits in the plane | Link to Abstract (Unusual time @11:00AM Goldsmith 300! joint with Everytopic) |

November 19 | Pierre Arnoux | Université Aix-Marseille | From combinatorics on words to geometry and number theory, via continued fractions | Link to Abstract (Unusual day, time and place @ 5:00PM Goldsmith 317) |

November 21 | Jonathan Jaquette | Brandeis University | Wright’s Conjecture and the Prime Number Theorem | Link to Abstract |

December 5 | Yotam Smilanski | Rutgers University | The space of cut and project quasicrystals | Link to Abstract |