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荔园杰出学者讲座第三十七期

讲座题目: Fractal geometry of the Markov and Lagrange spectra and their set difference

讲座人：Carlos Gustavo Moreira 教授（IMPA）

讲座时间：2025年 11月 6日 下午15:30-16:30

讲座地点：深圳大学粤海校区汇星楼二号教室

讲座摘要：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.

讲座人简介：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。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分钟讲座人以及2018年世界数学家大会（ICM）60分钟讲座人。他曾在《数学年刊》、《数学年报》、《欧洲数学学会杂志》等期刊上发表过论文。

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2025年11月3日