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【短课】Continued fractions, Diophantine approximations and elements of Dynamics

Time: Tue 20 2-3:30 pm, Fri 23 2-3:30 pm, Mon 26 4-5:30 pm, Tue 27 2-3:30 pm, Thu 29 2-3:30 pm



Program

Approximations of real numbers by rational numbers and representation in continued fractions - basic theory: definitions; examples; characterization of the convergents from the continued fractions representation as the best rational approximations of a real number; Hurwitz-Markov theorem; characterization of (eventually) periodic continued fractions: Lagrange's theorem.


An application: Pell's equation
Metric theory of Diophantine approximations: Khintchine's theorem
Simultaneous Diophantine approximations
Inhomogeneous approximations: Kronecker's theorem
Liouville numbers
Dynamics of the Gauss map g(x)={1/x} and continued fractions
Comments on theMarkov and Lagrange spectra (and their relations with regular Cantor sets and the Gauss map).



Biography

Carlos Gustavo T. de A. Moreira is a Brazilian mathematician working on dynamical systems, ergodic theory, number theory and combinatorics. Moreira is currently a researcher at the National Institute of Pure and Applied Mathematics (IMPA, Brazil). He is member of the Brazilian Academy of Science and of the Third World Academy of Sciences. He was awarded with the UMACA award (2009) and the TWAS Prize (2010). He was invited speaker at the International Congress of Mathematicians of 2014 and plenary speaker in 2018.