# IMPORTANT

## 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

### Time and Location

Thursdays at 3:00PM in Goldsmith 226

## Schedule of Talks

### Spring 2020

January 16Yiftach DayanTel Aviv UniversityRandom walks on tori and an application to normality of numbers in self-similar setsLink to Abstract
January 23Thomas HilleYale UniversityEffective bounds for the least solutions of homogeneous quadratic Diophantine inequalitiesLink to Abstract
February 27Arie LevitYale UniversityQuantitative Weak Uniform DiscretenessLink to Abstract
March 5Shahriar MirzadehMichigan State UniversityUpper bound for the Hausdorff dimension of exceptional orbits in homogeneous spacesLink to Abstract
March 12Joël BellaïcheBrandeis UniversityIntroduction to the dynamics of self-correspondences on curvesLink to Abstract
March 26Byungchul ChaMuhlenberg CollegeIntrinsic Diophantine Approximation of Spheres (CANCELLED)Link to Abstract
April 2Asaf KatzUniversity of ChicagoCANCELLEDLink to Abstract
April 23Ilya KhayutinNorthwestern UniversityCANCELLEDLink 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

September 12Dmitry KleinbockBrandeis UniversityShrinking targets in dynamics and number theoryLink to Abstract
September 19Lam PhamBrandeis UniversityExpansion spectral gaps in Lie groupsLink to Abstract
September 26Dmitry KleinbockBrandeis UniversityShrinking targets in number theory and dynamicsLink to Abstract
October 10Lam PhamBrandeis UniversityClassical results around Kazhdan’s Property $(T)$Link to Abstract
October 17Samuel EdwardsYale UniversityThe horocycle flow on hyperbolic surfacesLink to Abstract
October 24No seminar---
October 31Marius LemmHarvard UniversityGlobal eigenvalue distribution of matrices defined by the skew-shiftLink to Abstract (Unusual time @11:00AM Goldsmith 300! joint with Everytopic)
October 31Rahul KrishnaBrandeis UniversityAn introduction to the Sarnak and Chowla conjecturesLink to Abstract (Unusual time @4:30PM Goldsmith 317)
November 7Mishel SkenderiBrandeis UniversityApproximation of Random FunctionsLink to Abstract (Unusual time @4:30PM Goldsmith 317)
November 14Claire BurrinRutgers UniversityDiscrete lattice orbits in the planeLink to Abstract (Unusual time @11:00AM Goldsmith 300! joint with Everytopic)
November 19Pierre ArnouxUniversité Aix-MarseilleFrom combinatorics on words to geometry and number theory, via continued fractionsLink to Abstract (Unusual day, time and place @ 5:00PM Goldsmith 317)
November 21Jonathan JaquetteBrandeis UniversityWright’s Conjecture and the Prime Number TheoremLink to Abstract
December 5Yotam SmilanskiRutgers UniversityThe space of cut and project quasicrystalsLink to Abstract

## Detailed Abstracts

### September 12 @ 3:00PM – Dmitry Kleinbock (Brandeis) – Shrinking targets in dynamics and number theory

Abstract This is going to be an introduction to an active area of ergodic theory and dynamics, with applications to number theory. No background will be assumed, all are welcome.

### September 19 @ 3:00PM – Lam Pham (Brandeis) – Expansion and spectral gaps in Lie groups

Abstract I will survey recent results on various forms of expansion in groups. This is an old topic rooted in Lie theory and representation theory with many deep connections to number theory, geometry, dynamics, and even computing. I will present classical constructions due to Margulis, Kazhdan, and Lubotzky-Phillips-Sarnak and explain the breakthrough work of Bourgain and Gamburd on expansion in thin groups. Finally, I will present related works on improvements and generalizations of the Tits alternative and Kazhdan's Property $(T)$. The talk will be self-contained, with no background assumed.

### September 26 @ 3:00PM – Dmitry Kleinbock (Brandeis) – Shrinking targets in number theory and dynamics

Abstract This is a sequel to the talk I gave on Sept 12. I will provide its brief synopsis (so it's OK if you missed it) and then proceed to describe more slowly what I briefly mentioned at the end of the previous talk, give examples and applications, and highlight the remarkable connection of ergodic theory with shrinking targets to Diophantine approximation.

### October 10 @ 3:00PM – Lam Pham (Brandeis) – Classical results around Kazhdan’s Property $(T)$

Abstract In 1967, Kazhdan introduced (at the time only 20 years old) introduced a fundamental concept in representation theory, now known as Property $(T)$. I will give an introduction to Property $(T)$ with proofs of classical results and some examples and applications. With roots in Lie groups and representation theory, it has been highly influential in several other areas of mathematics, most notably geometry, number theory, dynamics, rigidity, and combinatorics. The talk should be accessible to any graduate student.

### October 17 @ 3:00PM – Samuel Edwards (Yale University) – The horocycle flow on hyperbolic surfaces

Abstract The horocycle flow on finite-volume hyperbolic surfaces is one of the most well-understood unipotent flows in homogeneous dynamics. In particular, its relation with the geodesic flow allows one to use exponential mixing to obtain “polynomially fast” effective equidistribution of all non-closed horocycle orbits. We will discuss a few aspects of the statements of effective equidistribution, and explain how similar results may be obtained for the horocycle flow on infinite-volume geometrically finite hyperbolic surfaces.

### October 31 @ 11:00AM (Goldsmith 300) – Marius Lemm (Harvard University) – Global eigenvalue distribution of matrices defined by the skew-shift

Abstract We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\binom{j}{2}\omega+j\cdot y+x \mod 1$ for irrational frequency $\omega$. We prove that the global eigenvalue distribution of these matrices converges to the corresponding distributions from random matrix theory, namely, the Wigner semicircle law for square matrices and the Marchenko-Pastur law for rectangular matrices. The result evidences the quasi-random nature of the skew-shift dynamics. This is joint work with Arka Adhikari and Horng-Tzer Yau.

### October 31 @ 4:30PM (Goldsmith 317) – Rahul Krishna (Brandeis University) – An introduction to the Sarnak and Chowla conjectures

Abstract The prime number theorem states that the number of primes $\pi(x)$ up to size $x$ is asymptotically $x/\log x$. An elementary restatement is that the Liouville function $\lambda(n)$ does not correlate with the constant function. In 2009, Sarnak suggested a far reaching conjectural generalization of this fact: that the Liouville function should not correlate with any sequence coming from a dynamical system of zero topological entropy. Recently, there has been some beautiful progress on this conjecture by Tao, Tao-Teravainen, Franzikinakis, and Franzikinakis-Host-Kra (to name a few!). I will try to explain some aspects of these recent developments.

### November 7 @ 4:30PM (Goldsmith 317)– Mishel Skenderi (Brandeis University) – Approximation of Random Functions

Abstract This talk will be based on joint work with Dmitry Kleinbock that has been motivated by several recent papers (among them, those of Athreya-Margulis, Bourgain, Ghosh-Gorodnik-Nevo, Kelmer-Yu). Given a certain sort of group $G$ and certain sorts of functions $f: \mathbb{R}^n \to \mathbb{R}$ and $\psi : \mathbb{R}^n \to \mathbb{R}_{>0},$ we obtained necessary and sufficient conditions so that for Haar-almost every $g \in G,$ there exist infinitely many (respectively, finitely many) $v \in \mathbb{Z}^n$ for which $|(f \circ g)(v)| \leq \psi(\|v\|),$ where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^n.$ As a consequence of our methods, we obtained generalizations to the case of vector-valued (simultaneous) approximation. We also obtained sufficient conditions for uniform approximation. Our methods involved probabilistic results in the geometry of numbers that go back several decades to the work of Siegel, Rogers, and Schmidt; these methods have recently found new life thanks to a 2009 paper of Athreya-Margulis.

### November 14 @ 11:00AM (Goldsmith 300)– Claire Burrin (Rutgers University) – Dicrete lattice orbits in the plane

Abstract Take a lattice in $\mathrm{SL}(2,\mathbb{R})$ and let it act linearly on the plane. Its orbits will be either discrete or dense. If we take a discrete lattice orbit; how are its points distributed in the plane? We will see examples, illustrate what makes this question challenging, and what can be said using some ideas from number theory. The latter bit is based on recent work with Amos Nevo, Rene Rühr, and Barak Weiss.

### November 19 @ 5:00PM (Goldsmith 317)– Pierre Arnoux (Université Aix-Marseille) – From combinatorics on words to geometry and number theory, via continued fractions

Abstract It is well known, since the pioneering works of Adler-Weiss, Sinai and Bowen, that hyperbolic systems have Markov partitions. Adler-Weiss exhibited such an explicit Markov partition for the $2$-torus, but Bowen proved that in higher dimensions, non-trivial Markov partitions must have fractal boundaries. We will show how one can construct explicit Markov partitions using substitutions on words; in some cases, this allows to construct conjugacies between a pseudo-Anosov automorphism on a surface of genus $g$ and a torus of dimension $g$. We will also show how one can extend this construction from one automorphism to an infinite sequence of automorphism, using multidimensional continued fraction (the case of one automorphism corresponds to a periodic orbit for this continued fraction).

### November 21 @ 3:00PM – Jonathan Jaquette (Brandeis University) – Wright’s Conjecture and the Prime Number Theorem

Abstract In 1955, Wright studied the nonlinear delay differential equations $y′(t)=− \alpha y(t−1)[1+y(t)]$ which would later become known as Wright's equation. This equation first arose from heuristic model for the distribution of prime numbers. Since then, and far beyond this motivating problem, this delay differential equation has interested dynamicist for many subsequent decades. This talk discusses two conjectures associated with this equation: Wright's conjecture, which states that the origin is the global attractor for all $\alpha \in ( 0 , \pi /2 ]$; and Jones' conjecture (1962), which states that there is a unique slowly oscillating periodic solution for $\alpha > \pi / 2$. The connection with number theory falls into the former category, and this global stability implies a 'prime number theorem' in the heuristic model -- a result proved in Wright's original paper. In particular, I will discuss my computer-assisted-proofs of these two conjectures.

### December 5 @ 3:00PM – Yotam Smilanski (Rutgers University) – The space of cut and project quasicrystals

Abstract Cut and project point sets are defined by identifying a strip of a fixed $n$-dimensional lattice (the "cut"), and projecting the lattice points in that strip to a d-dimensional subspace (the "project"). Such sets have a rich history in the study of mathematical models of quasicrystals, and include well known examples such as the Fibonacci chain and vertex sets of Penrose tilings. In the talk I will present the "space of quasicrystals", which was introduced by Jens Marklof and Andreas Strombergsson. Given a cut and project set, vary the $n$-dimensional lattice according to an $\mathrm{SL}_d(\mathbb{R})$-action. This defines a "space of quasicrystals", which, following Ratner is shown to have a homogeneous structure. Equidistribution results may be applied to establish generic properties of cut and project sets, such as a Siegel summation formula. I will describe this construction and discuss some properties and classification results. This is joint work with Rene Rühr and Barak Weiss.