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.
 
Bio-Sketch:

Carlos Gustavo Moreira  holds a degree in Mathematics from the Federal University of Rio de Janeiro (UFRJ), a Master's degree in Mathematics from the Institute of Pure and Applied Mathematics (IMPA) and a PhD in Mathematics from IMPA in 1993, under the supervision of Jacob Palis.  He is currently a Full Professor of IMPA and a Distinguished Visiting Professor of SUSTech. He has around 100 published papers in international mathematical journals. His main research areas are Dynamical Systems, Combinatorics and Diophantine Approximations. He has been a member of the Mathematical Olympiads Committee of the Brazilian Mathematical Society (SBM) since 1992, and is its General Coordinator since 2012. He has been a member of the SBM Director Board since 2009. He is a Full Member of the Brazilian Academy of Sciences  since 2008 and a Full Member of TWAS since 2012. He won gold medals at the Ibero-American Mathematical Olympiad in 1989 and at the International Mathematical Olympiad (IMO) in 1990.  He won the UMALCA Prize in 2009 and the TWAS Prize in Mathematics in 2010. He gave an invited lecture at the International Congress of Mathematicians in 2014, in Seoul, South Korea and a plenary lecture at the International Congress of Mathematicians in 2018, in Rio de Janeiro, Brazil. In 2023, he received from the President of Brazil the National Medal of Scientific Merit - National Order of Scientific Merit.