**Upcoming Talk:**

Speaker: Cezar Lupu

Affiliation: Beijing Institute of Mathematical Sciences and Applications

Title: Approximations by special values of the Riemann zeta function via cotangent integrals

Time: Thursday, April 18, 3:30-4:30 pm

Location: Shuangqing Complex, B725

Abstract: In this talk, we will prove some approximation results by using a surprising cotangent integral identity which involves the ratio ζ(2k+1)/π^{2k+1}. This cotangent integral is more flexible in controlling coefficients of zeta values compared to the one developed by Alkan (Proc. Amer. Math. Soc. 143 (9) 2015, 3743–3752.). Let A be a sufficiently dense subset of {ζ(3),ζ(5),ζ(7)…}. We show that real numbers can be approximated by certain linear combinations of elements in A, where the coefficients are values of the derivatives of rational polynomials. This is a joint work with Dongsheng Wu.

**Past Talks: **

Date: Thursday, March 28, 2024

Time: 3-4pm

Location: Shuangqing Complex, B725

Title: Geometric and arithmetic aspects of approximation vectors

Speaker: Barak Weiss (Tel Aviv University)

Abstract:

Let x in R^d be a vector and let (p_k, q_k) in Z^d \times N denote its sequence of best approximation vectors, with respect to some norm. In the case d=1, the properties of this sequence for a.e. x are understood via the continued fraction algorithm, and the ergodic theory of this algorithm can be used to obtain various limit laws such as the generic growth rate of the denominators, the distribution of the approximation, and more. In joint work with Uri Shapira, we extend many results in the one-dimensional setting, to d>1, and also study certain quantities associated with best approximations, that have no one-dimensional analogues. The main technique, inspired by work of Cheung and Chevalier, is the use of a certain well-adapted cross-section to a diagonal flow on the space of lattices.

Date: Thursday, March 28, 2024

Time: 1:30 - 2:30pm

Location: Shuangqing Complex, B725

Title: The translation geometry of Polya's shires

Speaker: Guillaume Tahar (BIMSA)

Abstract: In his shire theorem, Polya proves that the zeros of iterated derivatives of a rational function in the complex plane accumulate on the union of edges of the Voronoi diagram of the poles of this function. Recasting the local arguments of Polya into the language of translation surfaces, we prove a generalization describing the asymptotic distribution of the zeros of a meromorphic function on a compact Riemann surface under the iterations of a linear differential operator defined by meromorphic 1-form. The accumulation set of these zeros is the union of edges of a generalized Voronoi diagram defined jointly by the initial function and the singular flat metric on the Riemann surface induced by the differential. This process offers a completely novel approach to the practical problem of finding a flat geometric presentation (a polygon with identification of pairs of edges) of a translation surface defined in terms of algebraic or complex-analytic data. This is a joint work with Boris Shapiro and Sansan Warakkagun.

**Title: **Exploring isoperiodic foliations

**Speaker: **Guillaume Tahar, BIMSA

**Time: **Wed., 7:30-8:30 pm, Dec.21, 2022

**Venue：**Tencent Meeting ID：364-137-285

**Abstract:**

Strata of meromorphic $1$-forms are endowed with the atlas of period coordinates given by the periods of the differential on arcs joining the $n$ zeroes of the differential. Fixing the periods on absolute homology classes defines the isoperiodic foliation.

Isoperiodic leaves are complex manifolds of dimension $n-1$ endowed with a translation structure inherited from the period atlas of the stratum. In this talk, we give a description of some elementary examples.

In genus zero, isoperiodic leaves are compact and their translation structures have finite order singularities. On the other hand, for nonzero genus, In nonzero genus, even the simplest example displays wild behavior.

For strata H(1,1,-2) of meromorphic $1$-forms with one double pole and two simple zeroes on an elliptic curve, isoperiodic leaves are complex curves of infinite genus and their translation structures are given by differential forms with essential singularities.

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Exploring_isoperiodic_foliations.html

**Title:** Quantitative equidistribution of linear random walk on the torus.

**Speaker: **Weikun He (何伟鲲), Chinese Academy of Sciences

**Time: **Wed., 7:30-8:30 pm, Dec.14, 2022

**Venue: **Tencent Meeting ID：364-137-285

**Abstract: **

Consider the action of GL_d(Z) on the d-dimensional torus R^d/Z^d. Given a probability measure on GL_d(Z) and a point on the torus, a random walk is defined. In this talk, I will report some recent progress about the equidistribution of such random walks. More precisely, I will talk about quantitative results under the assumption that the acting group has semisimple Zariski closure. This is a generalisation of a theorem of Bourgain, Furman, Lindenstrauss and Mozes. This talk is based on a joint work with Nicolas de Saxcé.

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Quantitative_equidistribution_of_linear_random_walk_on_the_torus.html

**Title: **Towards a zero-one law for improvements to Dirichlet's approximation theorem

**Speaker: **Shucheng Yu (于树澄), USTC

**Time: **Wed.,7:30-8:30 pm,Nov.30,2022

**Venue: **Tencent Meeting ID：364-137-285

**Abstract:**

In this talk we discuss a notion of $\psi$-Dirichlet in Diophantine approximation which concerns improving Dirichlet’s approximation theorem to a general approximating function $\psi$. This notion was introduced by Kleinbock and Wadleigh in 2018 and generalizes the classical notion of a matrix being Dirichlet-improvable. In particular, we prove a partial zero-one law for the Lebesgue measure of the set of $\psi$-Dirichlet matrices. Joint with Dmitry Kleinbock and Andreas Strömbergsson.

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Towards_a_zero-one_law_for_improvements_to_Dirichlet_s_approximation_theorem.html

**Title: **Shrinking Target Problem for Matrix Transformations of Tori

**Speaker: **Lingmin Liao (廖灵敏), Wuhan University

**Time: **Wed.,7:30-8:30 pm,Nov.23,2022

**Venue: **Tencent Meeting ID：364-137-285

**Abstract: **

The shrinking target problem concerns the sizes of the sets of the points in a metric space whose orbits under a transformation fall into a family of shrinking subsets infinitely often. We study such problems for matrices with real coefficients which are transformations on the d-dimensional torus. We obtain a zero-one law for the Lebesgue measure of the corresponding shrinking target sets. A Hausdorff dimension formula is also given for the diagonal matrix transformations. This is a joint work with Bing Li, Sanju Velani and Evgeniy Zorin.

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Shrinking_Target_Problem_for_Matrix_Transformations_of_Tori.html

**Title: **Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner

**Speaker: **Davide Ravotti (University of Vienna)

**Time: **Wed., 7:30-8:30 pm, Nov.16,2022

**Venue: **(Online) Zoom Meeting ID: 399 587 2656 Password: 123456

**Abstract: **

In this talk I will present a simple method, inspired by the works of Ratner and Burger, to study ergodic integrals for the classical horocycle flow. The asymptotic expansion we can prove following this approach is a strengthening of the result by Flaminio and Forni in two ways: the coefficients in the expansion are shown to be Hölder continuous with respect to the base point and the term corresponding to the functions in the kernel of the Casimir operator is explicitly described.

Furthermore, we recover the spatial limit theorems by Bufetov and Forni and the temporal limit theorem by Dolgopyat and Sarig (this latter proof is by E. Corso).

Join Zoom Meeting: https://osu.zoom.us/j/3995872656?pwd=MGtibzdodXg3ZlR2RGZURnpXaTVyQT09

Video: http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Asymptotics_and_limit_theorems_for_horocycle_ergodic_integrals_a_la_Ratner.html

Tilte：Some ergodic optimization problems for expanding circle maps

Speaker: Rui Gao (高睿), Qufu normal university(曲阜师范大学)

Time: 2022.11.2, 7:30-8:30 pm, online

Tencent meeting：364-137-285

Abstract：In this talk, we consider some ergodic optimization problems for an expanding circle map. When the map is real analytic, we show that all optimizing measures of a real analytic potential function have zero entropy, unless the potential is cohomologous to constant. We also discuss some applications and related problems. This talk is based on a joint work with Weixiao Shen.

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Some_ergodic_optimization_problems_for_expanding_circle_maps.html

Title：Dynamics of some Fermi-Ulam acceleration models

Speaker: Jing Zhou (周晶), Penn state univeristy

Time: 2022.10.26, 10:00-11:00 am, online

Tencent meeting：364-137-285

Abstract: In this talk, I’ll brief introduce the Fermi-Ulam acceleration problem and the existing results on the subject. In particular, I’ll present my work on several variants of the Fermi acceleration models: the bouncing ball in gravity field and the billiard with moving platforms. We use techniques from elliptic as well as hyperbolic dynamics with singularities to study the ergodic ans statistical properties of these systems on infinite-volume phase.

报告人简介：周晶，于2020年在美国马里兰大学获得博士学位，博士导师为Dmitry Dolgopyat教授，现于 Pennsylvania State University进行博士后研究。

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Dynamics_of_Fermi-Ulam_acceleration_models.html

Title: Nondivergence of reductive group actions on homogeneous spaces

Speaker: Han Zhang （张涵）, Tsinghua University

Time: 2022.10.19, 7:30-8:30 pm, online

Tencent meeting：364-137-285

Abstract: Let $G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $G/\Gamma$. The question we are interested in is whether there is a compact set of $G/\Gamma$ that intersects every H-orbit. We show that the failure of this can be explained by a single algebraic reason, which generalizes several previous results towards this question. We also obtain a way to find this algebraic obstruction, if there is any. This talk is based on joint work with Runlin Zhang.

Title：Measure rigidity for a class of skew product systems

Speaker: Changguang Dong (董长光), Nankai University

Time: 2022.10.12, 7:30-8:30 pm

Online Tencent：364-137-285

Abstract: We will discuss old and new properties of skew product systems, we will focus on the rigidity phenomenon of the conditional measures on fibers. Based on joint work with Dolgopyat, Kanigowski, and Nandori.

Please join the seminar using the following link: https://meeting.tencent.com/dm/Qwu9hAyugbxj

Video：http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Measure_rigidity_for_a_class_of_skew_product_systems.html

Title: Pointwise convergence of multiple ergodic averages

Speaker: Song Shao（邵松） University of Science and Technology of China

Time: 2022.9.28 Wednesday 7:30pm-8:30pm (online).

#TencentMeeting：931-129-047

Abstract: Followed from Furstenberg's seminal work, problems concerning the convergence of multiple ergodic averages (in mean or almost surely) attract a lot of attention. In contrast to rich results on mean convergence, there are a few results on the problem of pointwise convergence for multiple averages. In this talk, we introduce our work on this topic. For example, we show the almost sure convergence of multiple ergodic averages along polynomials for distal systems, which answer a question by Derrien and Lesigne. The talk is based on the joint work with Wen Huang and Xiangdong Ye.

Click the link to join the webinar or to add it to your meeting list: https://meeting.tencent.com/dm/x9UaZQiGCtYb

Video: http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Pointwise_convergence_of_multiple_ergodic_averages.html

Title: Distinct distances on hyperbolic surfaces

Speaker:Xianchang Meng（孟宪昌） Shandong University

Time: 2022.9.21 Wednesday 7:30pm-8:30pm (online).

#TencentMeeting：378-821-659

Abstract: Erdős (1946) proposed the question of finding the minimal number of distinct distances among any N points in the plane. Guth-Katz (2015) gave almost sharp answer for this question using incidence geometry and polynomial partitioning. We consider this problem in hyperbolic surfaces associated with cofinite Fuchsian groups, i.e. the volume of the surface is finite. We prove a lower bound of the same strength as Guth-Katz. In particular, for any finite index subgroup of the modular group, we extract out the dependence of the implied constant on the index.

Click the link to join the webinar or to add it to your meeting list: https://meeting.tencent.com/dm/OzjM2fHijHqq

Video: http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Distinct_distances_on_hyperbolic_surfaces.html

Title: Effective joint equidistribution of primitive rational points on expanding horospheres

Speaker: Bingrong Huang(黄炳荣), Shandong University

Time: 2022.9.14 Wednesday 7:30pm-8:30pm (online)

Venue:Tencent Meeting：568-186-527,Meeting Password: 123456

Abstract: In this talk, we will discuss effective joint equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in any dimension. (For 1-dimentional case, this is joint equidistribtion of {a/q,b/q} where ab=1 mod q, as q goes to infinity.) The proof uses Fourier analysis and Weil’s bounds of Kloosterman sums. As an application, we provide a rate of convergence to the limiting distribution for the diameters of random circulant graphs. This is joint work with Daniel El-Baz and Min Lee.

Slides： Equidistribution-PRP-tsinghua.pdf