珠海数学杰出学者讲座第44讲-Fractal geometry of the Markov and Lagrange spectra and their set difference

讲座题目：Fractal geometry of the Markov and Lagrange spectra and their set difference
讲座时间 Datetime: 2025年 3月7日下午4:30-6:00
地点 Venue: 海琴2号楼455
主持人 Host：Sergio Augusto Romana Ibarra 副教授
报告人 Speaker: Carlos Gustavo Tamm de Araujo Moreira 院士
单位 Affiliation: National Institute of Pure and Applied Mathematics IMPA-Brazil
报告摘要 Abstract:
We will discuss some recent results on the fractal geometry of the Markov and Lagrange spectra, M and L which are classical objects from the theory of Diophantine approximations, and their set difference. In particular, we discuss recent results in collaboration with Erazo, Gutiérrez-Romo and Romaña, which give precise asymptotic estimates for the fractal dimensions of the Markov and Lagrange spectra near 3 (their smaller accumulation point) and other recent collaborations with Jeffreys, Matheus, Pollicott and Vytnova, in which we prove that the Hausdorff dimension of the complement of the Lagrange spectrum in the Markov spectrum has Hausdorff dimension between 0.594561 and 0.796445. Finally we will discuss a recent work in collaboration with H. Erazo, D. Lima, C. Matheus and S. Vieira in which we prove that inf(M\L)=3.
We will relate these results to symbolic dynamics, continued fractions and to the study of the fractal geometry of arithmetic sums of regular Cantor sets, a subject also important for the study of homoclinic bifurcations in Dynamical Systems.
报告人简介：
Carlos Gustavo Moreira，巴西科学院（BAC）院士，第三世界科学院（TWAS）院士, 巴西数学奥林匹克委员会（BMOC）成员。Moreira 教授主要研究动力系统，遍历理论，数论以及组合数学。在丢番图逼近，分型几何学等具体研究领域做出了重要贡献。他曾于1990年获得了世界数学奥林匹克（IMO）金牌，2009年获得了UMALCA奖，2010年获得了TWAS奖。他是2014年世界数学家大会（ICM）的45分钟报告人以及2018年世界数学家大会（ICM）60分钟报告人。他曾在《数学年刊》、《数学学报》、《欧洲数学学会杂志》等期刊上发表过论文。