Date/Time : 15 Jan 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Han Zhang, Soochow University

In 1984, Mahler proposed the following question on Diophantine approximation : How close can irrational numbers in the middle-thirds Cantor set be approximated by rational numbers? One way to reformulate Mahler’s question is to ask if Khintchine’s theorem extends to the middle-thirds Cantor set.

It turns out that random walks on homogenous sapces and slicing theorems in fractal geometry play crucial roles in answering Mahler’s question. In this talk, I will survey works regarding Khintchine’s theorem on fractals, discuss the connection between homogeneous dynamics and Diophantine approximation, and sketch the proof of Khintchine’s theorem on the middle-thirds Cantor set using tools from homogeneous dynamics and certain slicing theorem from fractal geometry.

This is based on a joint work with Timothée Bénard and Weikun He.

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Date/Time : 26 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sang-hyun Kim, Korea Institute for Advanced Study

Given a topological (C0) action of a group on a manifold, one can ask the best possible regularity (Cr) with which the group may act smoothly. Estimation of such regularities involve interplays among group theory (nilpotence, solvability and so forth), analysis (Hölder regularity) and dynamics (tracing orbits). In this talk, We prove that if a countable group does not contain a finitely generated subgroup of exponential growth, then every topological action of the group on a compact connected one-manifold can be blown-up to a C1 action. The proof is based on a functional characterisation of such groups. Joint work with Mikael de la Salle, Nicolas Matte Bon and Michele Triestino.

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Date/Time : 10 Dec 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Gianluca Faraco, University of Milano Bicocca

A translation surface is the datum of an abelian differential on a Riemann surface. Every such pair determines a representation called absolute period representation or period character. In this talk we discuss the realization of a given representation as the period character of some translation surface.

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Date/Time : 09 Dec 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Daniel Mathews, Monash University

Work of Penrose and Rindler in the 1980s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors as flags in Minkowski space. We present some extensions of this work involving hyperbolic geometry. In particular, we give a correspondence between between spinors and horospheres with certain decorations, and show how spinors can be used to calculate a complex-valued, 3-dimensional generalisation of Penner’s lambda lengths. Time permitting, we may discuss some further applications, including inter-cusp distances in knot complements, higher-dimensional generalisations, and circle packings.

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Date/Time : 03 Dec 2024 15:30
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Federico Rodriguez Hertz, The Pennsylvania State University

In a joint work in progress with Pablo Carrasco we show that to any quasi-morphism for a hyperbolic group can be associated a function with the Bowen property for some hyperbolic dynamics. In this talk I plan to describe the construction and how to prove the CLT for a quasi-morphism w.r.t. some length function.

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Date/Time : 02 Dec 2024 16:00
Venue : S17 #04-04 (SR3)
Speaker : Changguang Dong, Nankai University

We will discuss a dynamically defined zeta function for partially hyperbolic systems. This is a generalization of the classical Ruelle zeta function for Anosov (more generally Axiom A) diffeomorphisms/flows, which may reflect dynamical invariants such as entropy and pressure, as well as the geometry/topology of the manifolds. Based on an ongoing project with Qiujie Qiao.

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Date/Time : 27 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Marco Vergamini, Scuola Normale Superiore

Let f be a complex Hénon map and µ its unique measure of maximal entropy. Recently, Bianchi-Dinh proved that µ is exponentially mixing of all orders for all Hölder observables, and that all such observables satisfy the central limit theorem with respect to µ. De Thélin-Vigny generalized these results for a certain class of bounded plurisubharmonic observables. We prove that these properties hold for all, not necessarily bounded, plurisubharmonic observables. This is a joint work with Hao Wu.

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Date/Time : 13 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Duc-Bao Nguyen, National University of Singapore

We generalize previous diameter estimates and local non-vanishing of volumes for Kähler metrics to the case of big cohomology classes. In our proof, among other things, we will prove a uniform diameter estimate for a family of smooth Kähler metrics only involving an integrability condition. We also have to use fine stability properties of complex Monge-Ampere equations with prescribed singularities. This is joint work with Duc-Viet Vu

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Date/Time : 06 Nov 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Lei Yang, National University of Singapore

In this talk, I will give a new (and shorter) proof of Strombergsson’s Theorem on effective equidistribution of unipotent orbits in ASL(2, R)/ASL(2,Z). The proof is based on a general framework developed in recent works by Lindenstrauss-Mohammadi-Wang and myself, which relates the problem to incidence geometry. It relies on the study of a Kakeya-type model. This is joint with Elon Lindenstrauss.

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