学术科研-【短课】Continued fractions, Diophantine approximations and elements of Dynamics

学术科研

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

演讲者：Carlos Gustavo Moreira（巴西国家理论数学与应用数学研究所）
时间：2022-09-20 14:00-15:30
地点：台州楼240A，同步腾讯会议 ID 952-8738-9500

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.