Analysis Seminars @ CUHK Math Dept

Graduate Seminars

This is an experimental page for the records of graduate seminars organized by De-Jun Feng. This series of seminars preferring the research interests of the group will normally be held in Friday afternoons at LSB 219. All the files and resources are and shall be purely for academic use. All are welcome.

Upcoming

2024

  • A series of seminars based on predetermined topics on Fridays at 3pm HKT in LSB 219.

Previous

2024

  • 03/29 Friday LSB 219
    Yuhao Xie 谢宇昊

  • 03/15 Friday LSB 219
    Tianhan Yi 易天涵

  • 03/01—03/08 Fridays LSB 219
    Zhou Feng 冯洲

  • 02/23 Friday LSB 219
    Lingmin Liao 廖灵敏

    • Affiliation: Wuhan University

    • Title: Hausdorff dimension of weighted singular vectors in the plane

    • Poster.

  • 02/02 Friday LSB 219
    Cai-Yun Ma 麻彩云

  • 01/29 Monday LSB 239
    Jiagang Yang

    • Affiliation: Universidade Federal Fluminense, Brazil

    • Titile: Maximal transverse measures of expanding foliations

    • Abstract
      For an unstable foliation of a diffeomorphism, we use a natural dynamical averaging to construct transverse measures, which we call maximal, describing the statistics of how the iterates of a given leaf intersect the cross-sections to the foliation. For a suitable class of diffeomorphisms, we prove that this averaging converges, even exponentially fast, and the limit measures have finite ergodic decompositions.

      This is a joint work with R. Ures, M. Viana, F. Yang.

  • 01/19 Friday LSB 239
    De-Jun Feng 丰德军

    • Some new results about the weak separation condition.

2023

  • 12/08 Friday LSB 222
    Ching Wei Ho

    • Affiliation: Academia Sinica, Taiwan

    • Titile: Heat flow on random polynomials

    • Abstract
      It is a classical problem to study the evolution of roots of polynomials under application of a differential operator. In this talk, I will discuss the heat evolution of random polynomials with a rotationally invariant root distribution on the complex plane. The limiting root distribution of the heat-evolved random polynomial can be completely determined in terms of its log potential. For example, when a Weyl polynomial, whose root distribution converges to the uniform distribution on the unit disk, undergoes heat flow, the limiting root distribution is uniform on some ellipse until time 1 at which it becomes exactly the semicircle law. This is joint work with Brian Hall, Jonas Jalowy, and Zakhar Kabluchko.

  • 11/10 Friday LSB 219
    Jian-Ci Xiao 萧建慈

    • Baker, Simon. Iterated function systems with super-exponentially close cylinders. Adv. Math. , Vol. 379 p. 107548, 13. 2021.

    • Summary
      I will talk about an interesting example given by Simon Baker in 2021, which indicates that there are homogeneous self-similar IFSs without exact overlaps but having super-exponentially close elementary copies.

  • 10/13–11/03 Fridays LSB 219
    Tianhan Yi 易天涵

  • 10/06 Friday LSB 219
    Lü Fan 吕凡

    • Lü, Fan & Wu, Jun. Diophantine analysis in beta-dynamical systems and Hausdorff dimensions. Adv. Math. , Vol. 290 p. 919-937. 2016.

    • Outline
      Let {xn}n≥1⊂[0,1] \{x_{n}\}_{n\ge 1}\subset [0,1] be a sequence of real numbers and let φ ⁣:N→(0,1] \varphi\colon \mathbb{N}\rightarrow (0,1] be a positive function. Using the mass transference principle established by Beresnevich and Velani [Ann. of Math., 164(3), 971–992], we prove that for any x∈(0,1] x\in(0,1] , the Hausdorff dimension of the set
      {β>1 ⁣:∣Tβnx−xn∣<φ(n) for infinitely many n∈N}\begin{aligned} \{\beta>1\colon |T^{n}_{\beta}x-x_{n}|<\varphi(n) \text{ for infinitely many } n\in \mathbb{N} \}\end{aligned}
      satisfies a so-called 00-11 law according to lim sup⁡n→∞log⁡φ(n)n=−∞ \limsup\limits_{n\rightarrow\infty}\frac{\log \varphi(n)}{n}=-\infty or not, where TβT_{\beta} is the β\beta-transformation.

    • Summary
      The mass transference principle established by Beresnevich and Velani allows us to transfer Lebesgue measure theoretic statements for limsup sets to Hausdorff measure theoretic statements, and has become a powerful tool in estimating the Hausdorff measures and the Hausdorff dimensions of limsup sets.

  • 09/22 — 09/29 Fridays LSB 219
    Edouard Daviaud

    • Mass transference principle and diophantine approximation

    • Summary for 09/22
      In this session, we present a mass transference principle which holds for any Borel probability measure and we apply this result to study natural dynamical coverings associated with overlapping self-similar IFS.

    • Summary for 09/29
      In this session we discuss a recent upper-bound theorem parallel to the mass transference principle from ball to open set from rams and Koivusalo.

  • 08/28 Monday 14:30 - 15:30 LSB 222
    Weikun He 何伟鲲

    • Affiliation: Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences

    • Titile: Dimension Theory of Groups of Circle Diffeomorphisms

    • Abstract
      In this talk, we consider the action of a finitely generated group on the circle by analytic diffeomorphisms. We will discuss some results concerning the dimensions of objects arising from this action. More precisely, we will present connections among the dimension of minimal subsets, that of stationary measures, entropy of random walks, Lyapunov exponents and critical exponents. These can be viewed as generalizations of well-known results in the situation of PSL(2,R) PSL(2,R) acting on the circle. This talk is based on a joint work with Yuxiang Jiao and Disheng Xu.

    • Poster

  • 08/17 Thursday LSB 222
    Wai Kit Lam 林偉傑

    • Affiliation: National Taiwan University

    • Titile: On the Geometry of the Growing Ball of First-Passage Percolation

    • Abstract
      In first-passage percolation, one puts i.i.d. nonnegative weights on the nearest-neighbor edges of the integer lattice Zd \mathbb{Z}^d , and studies the induced (pseudo)metric. It is well-known that a metric ball of radius tt, after rescaled by 1/t 1 / t , converges to a deterministic limit shape as t→∞ t \to \infty . However, not too much is known about the geometry of a ball for finite tt. In this talk, we will talk about the geometry of a ball of radius tt, by looking at its boundary. We will discuss the size of the boundary of a ball (based on an earlier joint work with M. Damron and J. Hanson), and then we will talk about holes in a ball (based on a recent joint work with M. Damron, J. Gold and X. Shen).

    • Poster

  • 08/07 16:00 - 17:00 Monday LSB 222
    Ying XIONG 熊瑛

    • Affiliation: South China University of Technology

    • Title: Equidistribution for measures defined by digit restrictions

    • Abstract
      In this talk, we discuss the pointwise equidistribution properties of measures μp \mu_p defined by digit restrictions on the b b -adic expansion, where b≥2 b \geq 2 is an integer. We prove that, if a sequence (αn)n≥1 (\alpha_n)_{n\geq 1} satisfies a certain b-adic diversity condition, then the sequence (αnx)n≥1 (\alpha_n x)_{n\geq 1} is uniformly distributed modulo one for μp \mu_p -a.e. x x . We also find some sufficient conditions to ensure the b b -adic diversity. Moreover, we apply these results to establish the b b -adic diversity for the sequences that can be written as a certain combination of polynomial and exponential functions.

    • Poster

  • 08/07 14:30 - 15:30 Monday LSB 222
    Xiong JIN 金雄

    • Affiliation: University of Manchester

    • Title: A Chung-Fuchs type theorem and degeneracy of critical Mandelbrot cascade measures

    • Abstract
      We will talk about the degeneracy of critical Mandelbrot cascades acting on ergodic measures of a symbolic space, that is the case when the entropy of the random weight equals the entropy of the ergodic measure. For the degeneracy we need to prove a Chung-Fuchs type theorem that, under mild assumptions, the summation of a random walk and a Brinkhoff sum, when their means are equal to zero, always has +∞ + \infty as limsup. The proof relies on Dekking’s generalised recurrence theorem (1982) and the filling scheme introduced by Chacon and Ornstein (1960).

    • Poster

  • 08/07 10:30 - 11:30 Monday LSB 222
    Qi ZHOU 周麒

    • Affiliation: Nankai University

    • Title: Fractal problems in Quasiperiodic Schrodinger operator

    • Abstract
      In this talk, I will talk about some fractal problems in Quasiperiodic Schrodinger operators, especially almost Mathieu operators. Topics will include the Hausdorff dimension of the spectrum, and the multifractal structure of absolutely continuous spectral measures.

    • Poster

  • 07/28 Friday LSB 222
    Chun-Kit LAI 赖俊杰

    • Affiliation: San Francisco State University

    • Title: On Measure and Topological Erdős Similarity Problems

    • Abstract
      A pattern is called universal in another collection of sets, when every set in the collection contains some linear and translated copy of the original pattern. Paul Erdős proposed a conjecture that no infinite set is universal in the collection of sets with positive measure.

      In this talk, we explore an analogous problem in the topological setting and consider the topologically universality and generically universality. We will show that Cantor sets on Rd \mathbb{R}^d are never topologically universal and Cantor sets with positive Newhouse thickness on R1 \mathbb{R}^1 are not generically universal. Moreover, we also obtain a higher dimensional generalization of the generic universality problem.

    • Poster

  • 07/10 Monday LSB 222
    Po Lam YUNG

    • Affiliation: Australian National University and The Chinese University of Hong Kong

    • Title: Some Recent Developments in Harmonic Analysis-

      Abstract
      I'll explain some work with my collaborators on Sobolev spaces and on Fourier decoupling. It will feature some new ways with which we can compute the Lp L^p norm of the derivative of a function, and some estimates that capture destructive interference between waves that propagate in different directions. Applications will be given in numerical approximations, PDEs, and analytic number theory. Efforts will be made to make this talk accessible to a wide audience.

      Joint work with Haim Brezis, Oscar Dominguez, Qingsong Gu, Shaoming Guo, Zane Kun Li, Andreas Seeger, Brian Street, Jean Van Schaftingen, and Pavel Zorin-Kranich.

    • Poster

  • 05/05 Friday LSB 222
    Ai-Hua FAN 范爱华

    • Affiliation: Huazhong Normal University and Université de Picardie

    • Title: A Topological Version of the Furstenberg-Kesten Theorem

    • Abstract
      The Lyapunov exponent of random products of non-negative matrices exists almost surely by the Furstenberg-Kesten theorem. But the Furstenberg-Kesten theorem tells us nothing about a given sample of matrices. We find conditions to ensure the existence of Lyapunov exponent for EVERY generic choice of matrices. If one of the matrices from which we select our matrices is of rank one, there is a closed formula for the Lyapunov exponent. Applications are given to the multifractal analysis of weighted Birkhoff averages. This is a joint work with Meng Wu.

    • Poster

  • 04/25 Tuesday LSB 219
    Tianhan Yi

    • Hochman, Michael. Lectures on dynamics, fractal geometry, and metric number theory. J. Mod. Dyn. , Vol. 8, No. 3-4 p. 437-497. 2014.

    • Outline
      Furstenburg ×2, ×3 \times 2,\, \times 3 projection problem.

    • Summary
      Mastrand Theorem tells us the dimension of a Borel set will not drop under the projection in almost every direction. Furstenberg (1969) conjectured for the product of a ×2 \times 2 -invariant set and a ×3 \times 3 -invariant set, the dimension will not drop under the projection of all directions except the axis directions. We discussed Hochman and Shmerkin’s proof of this conjecture. The main tools are CP chain and local entropy averages.

  • 03/28 Tuesday LSB 219
    Zhou Feng

  • 03/21 Tuesday LSB 219
    Yuhao Xie

  • 03/14 Tuesday LSB 219
    Yuhao Xie

  • 03/14 Tuesday LSB 219
    Caiyun Ma

    • Walters, Peter. An introduction to ergodic theory. Vol. 79, Graduate Texts in Mathematics Springer-Verlag, New York-Berlin p. ix+250. 1982.

    • Outline
      We show that P(σ,f)=sup⁡{hμ(σ)+∫fdμ:μ∈Mσ(Σ)}P(\sigma, f)=\sup \left\{h_\mu(\sigma)+\int f d \mu: \quad \mu \in M_\sigma(\Sigma)\right\}.

    • Note.

2022

2021

  • 11/12 — 12/3 Fridays LSB 222
    Tianhan Yi

  • 10/15 — 11/5 Fridays LSB 222
    Caiyun Ma

    • Shmerkin, P. Ergodic Geometric Measure Theory. Lecture notes (Please contact members to get the unpublished notes).

    • Outline
      Firstly, we introduce some general notation and definitions for measures. In Section 3 we recall the notions of entropy and dimension and some of their properties. In Section 4 we study measures on trees and obtain bounds on the image of such a measure under a tree morphism.

  • 9/17 — 9/24 Fridays LSB 222
    Yuhao Xie

    • Feng, D.-J.; Huang, W. & Rao, H. Affine embeddings and intersections of Cantor sets. J. Math. Pures Appl. (9), 2014, 102, 1062-1079

    • 9/17 Outline
      Let EE be a self-similar set satisfying (OSC). FF is also a self-similar set satisfying dim⁡HF=dim⁡SF\dim_{H}F=\dim_{S}F. Then FF can be C1C^{1}-embedded into EE if and only if FF can be affinely embedded into EE.

    • 9/24 Outline
      We will talk about some special circumstances under which if a self-similar set FF can be affinely embedded into another self-similar set EE, then the contraction ratios of EE and FF are logarithmic commensurable.

  • 8/27 Friday LSB 222
    Yuhao Xie

    • Bary, N. K. A treatise on trigonometric series. Vols. I, II. Chp1. The Macmillan Co., New York, 1964, Vol. I: xxiii+553 pp. Vol. II: xix+508.

    • Outline
      We will continue the proof of one part of Salem-Zygmund Theorem (To construct the sets of multiplicities), and talk about Riemann's classic results about trigonometric series.

    • Summary
      We use some useful tools, including Riemann's First Theorem and Riemann's Localization Principle to construct the sets of multiplicities.

  • 8/20 Friday LSB 222
    Yuhao Xie

    • Feng, D.-J.; Rao, H. & Wang, Y. Self-similar subsets of the Cantor set. Adv. Math., 2015, 281, 857-885

    • Salem, R. Algebraic numbers and Fourier analysis. D. C. Heath and Co., Boston, Mass., 1963, x+68

    • Outline
      The contraction factor of the self-similar subsets of Cantor set and the uniqueness and multiplicity problem.

    • Summary
      We apply Salem-Zygmund theorem to prove that the contraction factors of the IFS generating self-similar subsets of Cantor sets must be ±3−n\pm 3^{-n}, where nn is a positive integer. Then we use Riemann’s trigonometric series theory to prove one part of the Salem-Zygmund theorem.

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