Modern Dynamics Seminar

Organizer:Yitwah Cheung / Jinxin Xue
Time:3:20-4:55
Venue:Conference Room 3, Jinchun Yuan West Building

Upcoming talks:

Time:April 20th, 3:20-4:55pm

Title: Khintchine’ problem in metric Diophantine Approximation and approximation by reduced fractions

Speaker: Yuming Wei (Tsinghua University)

Slot: The 3rd conference room, Jinchunyuan west building

Abstract: It has long been know that if x is any real number there exists an infinitude of rational numbers p/q which satisfy |x-p/q|<1/q^2. There are two natural questions to ask: one is can the function 1/q^2 on the right be repalced by a smaller function to obtain a sharper inequality? The other is what where happen if we order the reduced fractions, i.e.(p,q)=1. In this talk, I will introduce a few results to answer this two questions.

Welcome all of you to attend!


History talks


2022 Spring Semester



Time: April 13th, 3:20-4:55pm

Title: Hausdorff dimension of weighted singular vectors

Speaker: Bohan Yang, Tsinghua University

Abstract: In this talk, I'll show Lingmin Liao, Ronggang Shi, Omri N. Solan and Nattalie Tamam’s work in computing the Hausdorff dimension of w-weighted singular vectors in \mathbb{R}^2. This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the unweighted singular vectors. The proof of upper bound uses best approximation, and the proof of the lower bound uses self-affine cover. I’ll show the proof of upper bound and sketch proof of lower bound.



Time: April 6th, 3:20-4:55pm

Title: Itinerary function of BCZ map

Speaker: Yiming Li, Tsinghua University

Abstract: In this talk, I'll first share some new results of itinerary function, then explain the relationship between return function and itinerary function and why it perhaps brings a new route to RH. If time allows, I’ll share some more results about cylinder sets and distribution of gaps in a circle.



Time: March 30th, 3:20-4:55pm

Title: On the Moebius function randomness and dynamics

Speaker: Fei Wei (魏菲), Tsinghua University

Abstract: In this talk, I will introduce Sarnak's lecture notes on the Mobius Disjointness Conjecture. This includes the investigation of the Mobius flow and the square-free flow, and the entropy of these flows. This also includes the connections between the dynamics of these flows and Chowla's conjecture, which impies the Mobius Disjointness Conjecture.



Time: March 23rd, 3:20-4:55pm

Tittle: Gap Distributions for Saddle Connection Directions

Speaker: Sixu Liu (刘思序) Tsinghua University

Abstract: I will talk about the paper written by J. S. Athreya and J. Chaika (2012, GAFA). It proved that for almost every holomorphic differential ω on a Riemann surface of genus g 2, the smallest gap between saddle connection directions of length at most a fixed length decays faster than quadratically in the length. Besides, the decay rate is not faster than quadratic if and only if ω is a lattice surface.



Time: March 16th, 3:20-4:55pm

Title: Divergent trajectories in Sp4R and 2-convergents

Speaker: Zhijing Wang (王芝菁), Tsinghua University

Abstract: In this talk we consider the signature 2,2 diagonal flow diag(e^t,e^t,e^(-t),e^(-t)) on SL4R/SL4Z and Sp4R/Sp4Z. I’ll introduce the 2d-convergents, which are “best approximates”of 2-planes in R^4, and use it to characterize the orbits of the diagonal flow. As an application, I’ll construct non-degenerate divergent orbits in Sp4R/Sp4Z.



Time: March 9th, 3:20-4:55pm

Title: Diophantine exponents for 2D-convergents

Speaker: Yitwah Cheung (张翼华), Tsinghua University

Abstract: The Diophantine exponent of a real number has a dynamical interpretation as a measure of the degree of unboundedness of a geodesic ray in the modular surface. It can readily be expressed in terms of the sequence of denominators of the convergents of the associated continued fraction. In this talk, I will begin by recalling a symbolic characterization of bounded trajectories of a diagonal flow that is analog to the characterization of badly approximable numbers in terms of continued fractions. Then I will recall the notion of a higher dimensional convergent and explain how an analogous notion of Diophantine exponent for 2D-convergents could be used as an approach to proving Littlewood Conjecture.



Time: March 2nd, 3:20-4:55pm

Tittle: Orbit closures on homogeneous spaces

Speaker: Han Zhang (张涵) Tsinghua University

Abstract: Studying orbit closures of different group action is a classical problem in dynamical systems. The celebrated Theorem of Ratner asserts that any orbit closure for unipotent group action is homogeneous, that is, the orbit closure of unipotent subgroup is equal to the closed orbit of some larger subgroup. Margulis conjectured that the same phenomenon also holds for the diagonalizable subgroup action. More precisely, let A be the group of all diagonal matrices in SL_n(R), he conjectured that except those A orbit arising from rank 1 torus action, all other orbit closure of A are homogeneous. I will discuss this conjecutre a little bit and then a result of Lindenstrauss and Weiss showing that if the orbit closure of A containing a compact A orbit, then this orbit closure is homogeneous.



2021 Fall Semester



Time: December 29th, 3:20-4:55pm

Tittle: Hyperbolic surfaces and Teichmueller space

Speaker: Yi Huang(黄意) Tsinghua University

Abstract: The term "moduli space" is a somewhat popular term in modern mathematics, and roughly describes a space whose individual points represent/parametrise the collection of all spaces of a specific type. For example: Grassmannians may be regarded as a moduli space of subspaces of a fixed vector space; Riemann's moduli space is the space of all Riemann surfaces; and Teichmueller spaces are moduli spaces of marked hyperbolic surfaces. "Nice" moduli spaces often inherit properties from the objects such as algebraicity, complex and/or symplectic structure, and the Teichmueller space is particularly nice in this regard. We explore this theme from the Thurston-school hyperbolic-surfaces perspective of Teichmueller space, and try to emphasis some of the dynamical aspects of this picture.



Time: December 22, 3:20-4:55pm

Tittle: An introduction to joinings in ergodic theory and their applications to statistical inference

Speaker: Sixu Liu (刘思序)Tsinghua University

Abstract: Since their introduction by Furstenberg in 1967, joinings have proved a very powerful tool in ergodic theory. We discuss some aspects of the use of joinings in the study of measurable dynamical systems, including how joinings can be employed to provide elegant proofs of classical results. Moreover, we explain how joinings connect to the analysis of empirical risk based inference for dynamical models.



Time: December 8th, 3:20-4:55pm

Tittle: Minkowski conjecture and stable lattices

Speaker: Zhang Run Lin(张润林) Peking University

Abstract: Let N be the function defined on R^n by taking products of the absolute value of its coordinates. From this function one can build another function defined on the space of unimodular lattices. It is a conjecture, attributed to Minkowski, that this function is bounded by 2^{-n}. As this function is invariant under the action of diagonal matrices of determinant 1 (call this group A), one may wish to find good representatives within an A-orbit. This leads to the concept of well-rounded lattices and stable lattices. In this talk we will follow a paper of Solan (Israel Journal of Math, 2019) to show the existence of stable lattices within an A-orbit. You will see some geometry of numbers and differential topology in the talk.



Time: December 1st, 3:20-4:55pm

Tittle: Semitoric systems and their combinatorial classification

Speaker: Xiudi Tang(唐修棣) Beijing Institute of Technology

Abstract: A four dimensional integrable system is semitoric if one of the components of the momentum map is proper and generates a circle action. We would explain the combinatorial invariants that classify all semitoric systems, which is analogous to the Delzant polytopes for toric systems and the five invariants for simple semitoric systems in the sense that each fiber of the momentum map of the circle action contains at most one singular point of focus-focus type, invented by Pelayo & Vu Ngoc about 10 years ago. This talk is based on joint work with J. Palmer and A. Pelayo, see arXiv:1909.03501.



Time: November 24th, 3:20-4:55pm

Tittle: Closed Geodesics And Class Numbers

Speaker: Yang Bohan (杨博寒) Tsinghua University

Abstract: In this talk I will introduce the relation between closed geodesics and class numbers, and show the distribution of closed geodesics. Then I will show Bourgain and Kontorovich's work in low-lying geodesics and give some idea of cubic field.



Time: November 18th, 21:00-22:30 (Beijing time)

Zoom Meeting ID: 839 9242 9123

Passcode: 609565

Tittle: Polynomial decay of correlations of geodesic flows on some nonpositivity curved surfaces

Speaker: Yuri Lima, Universidade Federal do Ceara, Brazil

Abstract: We consider a class of nonpositively curved surfaces and show that their geodesic flows have polynomial decay of correlations. This is a joint work with Carlos

Matheus and Ian Melbourne.



Time: November 11th, 21:00-22:30 pm (Beijing time

Zoom Meeting ID: 843 0140 5760

Passcode: 342344

Tittle: Symbolic dynamics for maps with singularities in high dimension

Speaker: Yuri LimaUniversidade Federal do Ceara, Brazil

Abstract: We construct Markov partitions for non-invertible and/or singular nonuniformy hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, as well as includes all the recent results of the literature. As an application, we prove exponential growth rate on the number of periodic orbits. Joint work with Ermerson Araujo and Mauricio Poletti.



Time: November 3rd, 3:20-4:55pm

Tittle: Invariants For Diagonal Actions On Moduli Spaces

Speaker: Yuming Wei (魏瑜铭)Tsinghua University

Abstract: In this talk, I will try to answer the question does it exist a 3-parameter family of divergent A-orbits that have dual pivots which do not contain any pivots. Before that, I will first introduce some definitions like pivot, dual pivot, and A-invariant coordinates. Then I will give our construction about the 3-parameter family, and, if time permits, a brief proof why it satisfies.



Time: October 27, 3:20-4:55pm

Tittle: From Horocycle Flow To Riemann Hypothesis

Speaker: Yiming Li (李一鸣)Tsinghua University

Abstract: In this talk, I will first introduce the horocycle flow, then construct a transversal to the horocycle flow so that the first return map is the BCZ map. Some interesting facts about horocycle flow and BCZ map will be shown. An equivalence established by Franel made it possible for us to view Riemann hypothesis from the BCZ map perspective. I will give some detailed proof if time allows.



Time: October 20, 3:20-4:55

Tittle: Introduction To Staircases

Speaker: Yitwah CheungTsinghua University

Abstract: The notion of a staircase is a rectilinear analog of that of a Kleinian sail, with a slight advantage from the dynamics point of view of being more easily renormalizable. It serves as a backdrop for both higher genus as well as higher dimensional generalizations of the rich interplay between continued fractions and the geodesic flow on the modular surface. In particular, there is an extremely natural generalization of the notion of a convergent to higher dimensions that serves to symbolically encode the excursions of A-orbits in SL(n,R)/SL(n,Z). Elementary properties of staircases and related notions where explored in a series of masters theses projects (see http://math.sfsu.edu/cheung/ ), e.g. Knitter showed the complexity of domains of approximations associated to a rational point in the plane is O(log q) where q is the lowest common denominator. In this talk, I will give a brief overview of these results as well as discuss some interesting open questions for which this technology is likely relevant.



Time: October 6th, 3:20-4:55

Tittle: Proof of Mostow Rigidity Theorem Using Ergodic Theory

Speaker: Lan Qing (蓝青) Tsinghua University

Abstract: In this talk I will present the original proof of Mostow rigidity, which essentially states that the geometry of a closed hyperbolic manifold of dimension greater than two is determined by the fundamental group. I will also sketch another proof given by Gromov using Gromov norm.



Time: September 29th, 3:20-4:55

Tittle: Introduction to a dynamical generalization of the Prime Number Theorem

Speaker: Wang biao (王标) Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstract: In this talk, we mainly introduce a dynamical generalization of the classical Prime Number Theorem by Bergelson and Richter (i.e., Theorem A in arXiv:2002.03498v1). Their result generalizes a theorem of Pillai-Selberg and a theorem of Erdős-Delange on the distribution of the number of prime factors as well. The detailed proof will also be introduced in the talk.



Time: September 22th, 3:20-4:55

Tittle: Oppenheim Conjecture and Its Quantatitive version

Speaker: Han Zhang (张涵) Tsinghua University

Abstract: I will give a brief introduction to Oppenheim conjecture and its quantitative version obtained by Eskin, Margulis and Mozes. Oppenheim conjecture was formulated in 1929 and was originally attacked by analytic number theory method. Only some partial results were known until it was proved by Margulis in full generality in 1980s. Later, quantatitive version of Oppenheim conjecture was obtained using Ratner's theorem on unipotent flows for certain unbounded continuous functions on homogeneous spaces. If time allows, I will also discuss the difficulty of quantitative Oppenheim conjecture for signature (2,2) form.



Time: September 15th, 3:20-4:55

Tittle: Ergodicity of complex structures of hyperkahler manifolds

Speaker: Zhijing Wang (王芝菁) Tsinghua University

Abstract: In this talk, I will introduce hyperkahler manifolds and their complex moduli spaces, especially Verbitsky's work on the highly non-Hausdorff property of the complex structures. More precisely, he applied Ratner's theorem on the moduli space of complex structures of hyperkahler manifolds or complex tori to show that every "irrational" complex structure is ergodic in the moduli space.



Time: September 8th, 3:20-4:55

Tittle: Intruduction to Sarnak's Moebius Disjointness Conjecture

Speaker: Fei Wei (魏菲), Tsinghua University

Abstract: In this talk, I will mainly introduce Sarnak's conjecture and some recent developments. Meanwhile, I will also show some related research topics to Sarnak's conjecture.