Topic: Fractal Geometry of the Markov and Lagrange Spectra and Their Set Difference

Speaker: Carlos Gustavo Moreira, Member of the Brazilian Academy of Sciences

Time: Monday, June 16, 2025, 16:00

Venue: Room 309, Building 2, Xueyuan Road Campus

Host: School of Mathematical Sciences

Registration: All university members (faculty, students, alumni) and scholars from partner institutions are cordially invited to attend. Please scan the QR code below to complete the registration form.

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.

About the Speaker:

Carlos Gustavo Moreira is a full professor at Brazil’s National Institute for Pure and Applied Mathematics (IMPA). He is a member of the Brazilian Academy of Sciences (ABC) and The World Academy of Sciences (TWAS), and serves on the Brazilian Mathematical Olympiad Committee (BMOC). His research spans dynamical systems, ergodic theory, number theory, and combinatorics, with notable contributions to Diophantine approximation and fractal geometry. He won a gold medal at the International Mathematical Olympiad (IMO) in 1990, the UMALCA Award in 2009, the TWAS Award in 2010, and the Paul Erdős Award from the World Federation of National Mathematics Competitions (WFNMC) in 2018. He was an invited speaker at the Dynamical Systems session of the ICM 2014 in South Korea and a plenary speaker at the ICM 2018 in Brazil.