Shenzhen University School of Mathematical Sciences

Liyuan Distinguished Scholars Lecture Series Session 37

Lecture Title: Fractal geometry of the Markov and Lagrange spectra and their set difference

Speaker: Professor Carlos Gustavo Moreira (Southern University of Science and Technology)

Time: 15:30-16:30, November 6, 2025 (Thursday)

Location: Classroom 2, Huixing Building, Yuehai Campus, Shenzhen University

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). We will also discuss 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.

Speaker Profile:

Carlos Gustavo Moreira is a Distinguished Visiting Professor at the Southern University of Science and Technology, a Member of the Brazilian Academy of Sciences (BAC), a Fellow of The World Academy of Sciences (TWAS), and a member of the Brazilian Mathematical Olympiad Committee (BMOC). Professor Moreira's research primarily focuses on dynamical systems, ergodic theory, number theory, and combinatorics. He won a gold medal at the International Mathematical Olympiad (IMO) in 1990, received the UMALCA Prize in 2009, and the TWAS Prize in 2010. He was an invited 45-minute speaker at the International Congress of Mathematicians (ICM) in 2014 and a 60-minute plenary speaker at the ICM in 2018.

All faculty and students are welcome!

School of Mathematical Sciences

November 3, 2025