Brandeis Dynamics and Number Theory Seminar 2019-2020

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IMPORTANT

Due to recent events, the seminar has been cancelled for the remainder of the semester. The last talk was on March 12. This page is no longer maintained and is replaced by this one.

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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

Dmitry Kleinbock and Lam Pham

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

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.

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