# Brandeis Dynamics and Number Theory Seminar 2019-2020

## Seminar Information

**IMPORTANT:** Due to recent events, the seminar has been cancelled for the remainder of the semester. The last talk was on March 12.

### 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 |
---|---|---|---|---|

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 27 | Arie Levit | Yale University | Quantitative Weak Uniform Discreteness | Link to Abstract |

March 5 | Shahriar Mirzadeh | Michigan State University | Upper bound for the Hausdorff dimension of exceptional orbits in homogeneous spaces | Link to Abstract |

March 12 | Joël Bellaïche | Brandeis University | Introduction to the dynamics of self-correspondences on curves | Link to Abstract |

March 26 | Byungchul Cha | Muhlenberg College | Intrinsic Diophantine Approximation of Spheres (CANCELLED) | Link to Abstract |

April 2 | Asaf Katz | University of Chicago | CANCELLED | Link to Abstract |

April 23 | Ilya Khayutin | Northwestern University | CANCELLED | Link to Abstract |

April 30 | Sebastian Hurtado-Salazar | University of Chicago | CANCELLED | 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 27 @ 3:00PM (Goldsmith 226)– Arie Levit (Yale University) – Quantitative Weak Uniform Discreteness

## Abstract

I will discuss a quantitative variant of the Kazhdan-Margulis theorem generalized to probability measure preserving actions of semisimple groups over local fields. More precisely, the probability that the stabilizer of a random point admits a non-trivial intersection with a small r-neighborhood of the identity is at most $b r^d$, for some explicit constants $b,d > 0$ which depend only on the semisimple group in question. The methods of our proof involve some of the original ideas of Kazhdan and Margulis, Margulis functions as well as $(C,\alpha)$-good functions on varieties. As an application we present a unified proof of a fact that was previously known in most cases, namely that that all lattices in these groups are weakly cocompact, i.e admit a spectral gap. The talk is based on a recent preprint based on a joint work with Gelander and Margulis.### March 5 @ 3:00PM (Goldsmith 226)– Shahriar Mirzadeh (Michigan State University) – Upper bound for the Hausdorff dimension of exceptiional orbits in homogen spaces

## Abstract

Consider the set of points in a homogeneons space $X=G/\Gamma$ whose $g_{t}$-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$. This conjecture is proved when $X$ is compact or when $G$ has real rank $1$. In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: $G=\mathrm{SL}_{m+n}(\mathbb{R})$, $\Gamma=\mathrm{SL}_{m+n}(\mathbb{Z})$, and $g_t=\mathrm{diag}(e^{t/m},\ldots,e^{t/m},e^{-t/n},\ldots,e^{-t/n})$. This homogeneous space has many applications in Diophantine approximation that will be discussed in the talk if time permits. This project is joint work with Dmitry Kleinbock.### March 12 @ 3:00PM (Goldsmith 226)– Joël Bellaïche (Brandeis) – Introduction to the dynamics of self-correspondences on curves

## Abstract

Initiated by Fatou and Julia more than a century ago, the dynamical study of endomorphisms of the Riemann sphere has since grown into a huge and beautiful theory. Generalizations to endomorphisms of compact Riemann surfaces of higher genus offers little dynamical interest, however, since these endomorphisms are always (easy to describe) automorphisms. The natural generalization is to consider instead a _self-correspondence_ on a compact Riemann surface $S$, that is the data of another compact Riemann surface $S'$ and two non-constant morphisms $f,g: S' \rightarrow S$. The dynamics of that correspondence is by definition the dynamics of the _multi-valued_ map $g \circ f^{-1}$. Many self-correspondences on compact Riemann surfaces, or equivalently, on smooth complete algebraic curves over $\mathbb{C}$ (and more generally, over an algebraically closed field $k$) appear naturally in various fields of mathematics (number theory, homogeneous spaces, etc.) Despite having been considered by Fatou, the dynamics of such self-correspondences has received only scattered attention. The aim of this talk is to give an introduction to a systematic study of that dynamics, in particular the understanding of finite orbits, which are much more subtle that in the classical case.### March 26 @ 3:00PM (Goldsmith 226)– Byungchul Cha (Muhlenberg College) – Intrinsic Diophantine Approximation of Spheres

## Abstract

Let $S^1$ be the unit circle in $\mathbb{R}^2$ centered at the origin and let $Z$ be a countable dense subset of $S^1$, for instance, the set $Z = S^1(\mathbb{Q})$ of all rational points in $S^1$. We give a complete description of an initial discrete part of the Lagrange spectrum of $S^1$, in the sense of intrinsic Diophantine approximation. This is an analogue of the classical result of Markoff in 1879, where he characterized the most badly approximable real numbers via the periods of their continued fraction expansions. In addition, we present similar results for a few different subsets $Z$ of $S^1$. Finally, we report some partial results of similar type for $S^2$. This is joint work with Dong Han Kim.### April 2 @ 3:00PM (Goldsmith 226)– Asaf Katz (Universit of Chicago) – Title TBA

Asaf Katz (University of Chicago)

## Abstract

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

Ilya Khayutin (Northwestern University)

## Abstract

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

Sebastian Hurtado-Salazar (University of Chicago)

## Abstract

### 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 |