Date: 21 September 2023
Speaker: Omri Solan
Title: Birkhoff generic points on curves
Abstract: Let a_t be a diagonal flow on the space X of unimodular lattices in R^n. A point x in X is called Birkhoff generic if a_t.x equidistributes in X as t\to \infty. By Birkhoff ergodic theorem, almost every point x in X is Birkhoff generic. One may ask whether the same is true when the point x is sampled according to a measure singular to Lebesgue.
In a joint work with Andreas Wieser, we discuss the case of a generic point x in an analytic curve in X, and show that under certain conditions, it must be Birkhoff generic. This Birkhoff genericity result has various applications in Diophantine approximation. In this talk we will relate Birkhoff genericity to approximations of real numbers by algebraic numbers of degree at most n.
Date: 28 September 2023
Speaker: Zach Selk
Title: Stochastic Calculus for the Theta Process
Abstract: The Theta process, $X(t)$, is a complex valued stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums $$\sum_{n=1}^N e^{2\pi i \left(\frac{1}{2}(n^2+\beta)x+\alpha n\right)},$$ where $(\alpha,\beta)\in \mathbb R^2 \setminus \mathbb Q^2$ and $x\in \mathbb R$ is chosen at random according to any law absolutely continuous with respect to Lebesgue measure. The Theta process can be explicitly represented as $X(t)=\sqrt{t} \Theta(\Gamma g \Phi^{2 \log t})$ where $\Theta$ is an automorphic function defined on Lie group $G$, invariant under left multiplication under lattice $\Gamma$. Additionally, $g\in \Gamma \setminus G$ is chosen Haar uniformly at random and $\Phi$ is the geodesic flow on $\Gamma \setminus G$. The Theta process shares several similar properties with the Brownian motion. In particular, both lack differentiability and have the same $p$ variation and H\”older properties.
Similarly to Brownian motion, standard calculus and even Young/Riemann-Stieltjes calculus techniques do not work. However, Brownian motion is what is known as a martingale allowing for a classical theory of It\^o calculus which makes use of cancellations “on average”. The It\^o calculus can be used to prove several properties of Brownian motion such as its conformal invariance, bounds on its running maximum in terms of its quadratic variation, absolutely continuous changes in measure and much more.
Unfortunately, we show that the Theta process $X$ is not a (semi)martingale, therefore It\^o techniques don’t work. However, a new theory introduced in 1998 by Terry Lyons called rough paths theory handles processes with the same analytic regularity as $X$. The key idea in rough paths theory is that constructing stochastic calculus for a signal can be reduced to constructing the “iterated integrals” of the signal. In this talk, we will show the construction of the iterated integrals – the “rough path” – above the process $X$. Joint with Francesco Cellarosi.
Date: 5 October 2023
Speaker: Yuval Yifrach
Title: A variation on the p-adic Littlewood Conjecture
Abstract: We consider a variation on the p-adic Littlewood Conjecture where instead of using powers of one prime, we use arbitrarily large primes. We examine this conjecture from two view points: the Diophantine-approximation viewpoint and the dynamical viewpoint. Using the dynamical view point, we rephrase the conjecture in terms of Hecke neighbors and prove partial results towards the conjecture.
Namely, we prove that the Hausdorff dimension of certain exception sets is strictly smaller than 1. In addition, we show that the conjecture holds in the quadratic irrational case, and that it holds for a generic numbers according to a family of measures supported on a Lebesgue null sets.
Our tools for the proof are mainly the effective equidistribution of Hecke neighbors due to Oh Et. Al. and to expander properties of SL_2(Z/pZ) due to Bourgain-Gamburd.
This talk is based on a joint work with Erez Nesharim.
Date: 19 October 2023
Speaker: Hao Xing
Title: Equidistribution problem in the space of Euclidean sublattices
Abstract: Consider the space of covolume-one sublattices of a fixed rank m in the Euclidean space ℝd. How do the orbits behave under the action of the lattice subgroups of SL(d,ℝ) (e.g. SL(d,ℤ))? In a recent joint work with Michael Bersudsky, we established an equidistribution phenomenon of such orbits when d=m+1. However, there are many more unsolved problems along this direction which might be of interest not only to homogeneous dynamicists, but also to number theorists and analysts as well. In this talk, I will explain the problem, our result, an overview of methods and further directions of research in a user-friendly way.
Date: 26 October 2023
Speaker: Noy Soffer Aranov
Title: Covering Radii in Positive Characteristic
Abstract: A fascinating question in geometry of number pertains to the covering radius of lattice with respect to an interesting function. For example, given a convex body C and a lattice L in R^d, it is interesting to ask what is the infimal r ≥ 0 such that L + rC = R^d. Another interesting covering radius is the multiplicative covering radius, which connects to dynamics due to its invariance under the diagonal group. It was conjectured by Minkowski that the multiplicative covering radius is bounded above by 2^{-d} and that this upper bound is obtained only on AZ^d. In this talk I will discuss surprising results pertaining to covering radii in the positive characteristic setting and discover several surprising results. Some of my results include explicitly connecting between the covering radii with respect to convex bodies and successive minima and proving a positive characteristic analogue of Minkowski’s function.
Date: 2 November 2023
Speaker: Alon Agin
Title: Constructing best approximation vectors
Abstract: For v in R^d and arbitrary norm, we define the best approximation sequence of v and the displacement vectors sequence of v. We will discuss classical and recent works in Diophantine approximations in the language of these objects – focusing on their length, direction and congruence class.
Date: 9 November 2023
Speaker: Sam Chow
Title: Dispersion and Littlewood’s conjecture
Abstract: I’ll discuss some problems related to Littlewood’s conjecture in diophantine approximation, and the role hitherto played by discrepancy theory. I’ll explain why our new dispersion-theoretic approach should, and does, deliver stronger results. Our dispersion estimate is proved using Poisson summation and diophantine inequalities. Joint with Niclas Technau.
Date: 16 November 2023
Speaker: Mikey Chow
Title: Jordan and Cartan spectra in higher rank with applications to correlations
Abstract: The celebrated prime geodesic theorem for a closed hyperbolic surface says that the number of closed geodesics of length at most t is asymptotically e^t/t. For a closed surface equipped with two different hyperbolic structures, Schwartz and Sharp (’93) showed that the number of free homotopy classes of length about t in both hyperbolic structures is asymptotically a constant multiple of e^{ct} /t^{3/2} for some 0<c<1.
We will discuss the asymptotic correlations of the length spectra of convex cocompact manifolds, generalizing Schwartz-Sharp’s results. Surprisingly, it is helpful for us to relate this problem with understanding the Jordan spectrum of a discrete subgroup in higher rank. In particular, we will explain the source of the exponential and polynomial factors in Schwartz-Sharp’s asymptotics from a higher rank viewpoint.
We will also discuss the asymptotic correlations of the displacement spectra and the ratio law between the asymptotic correlations of the length and displacement spectra.
This is joint work with Hee Oh.
Date: 30 November 2023
Speaker: Daniel El-Baz
Title: Primitive rational points on expanding horospheres: effective joint equidistribution
Abstract: I will present joint work with Min Lee and Andreas Strömbergsson. Using techniques from analytic number theory, spectral theory, geometry of numbers as well as a healthy dose of linear algebra and building on a previous work by Bingrong Huang, Min Lee and myself, we furnish a new proof of a 2016 theorem by Einsiedler, Mozes, Shah and Shapira. That theorem concerns the equidistribution of primitive rational points on certain homogeneous spaces and our proof has the added benefit of yielding a rate of convergence. It turns out to have several (perhaps surprising) applications to number theory and combinatorics, which I shall also discuss.
Date: 7 December 2023
Speaker: Florian Richter
Title: Finding Infinite arithmetic structures in sets of positive density
Abstract: In the 1970’s Erdos asked several questions about what kind of infinite arithmetic structures can be found in every set of natural numbers with positive density. In recent joint work with Bryna Kra, Joel Moreira, and Donald Robertson we use ergodic methods to resolve some of these long-standing conjectures. This talk will provide a gentle introduction into this topic, with an overview of our results and the dynamical structures that are used to prove them.
Date: 14 December 2023
Speaker: Oleksiy Klurman
Title: Partition regularity of Pythagorean pairs
Abstract: Is there a partition of the natural numbers into finitely many pieces, none of which contains a Pythagorean triple (i.e. a solution to the equation x2+y2=z2)? This is one of the simplest (to state!) questions in arithmetic Ramsey theory which is still widely open. I will talk about a recent partial result, showing that “Pythagorean pairs” are partition regular, that is in any finite partition of the natural numbers there are two numbers x,y in the same cell of the partition, such that x2+y2=z2 for some integer z (which may be coloured differently). The proof is a blend of ideas from ergodic theory and multiplicative number theory. Based on a joint work with N. Frantzikinakis and J. Moreira.
