Marcelo Viana

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Marcelo Viana

Marcelo Viana (born March 4, 1962 in Rio de Janeiro ) is a Brazilian mathematician who deals with dynamic systems and chaos theory.

Viana is the son of Portuguese immigrants. He grew up in Portugal, where he studied at the University of Porto (graduated in 1984). In 1990 he did his doctorate at the mathematics institute IMPA ( Instituto de Matemática Pura e Aplicada ) in Rio de Janeiro ( Strange Attractors in Manifolds of Any Dimension) with Jacob Palis . He then worked as a post-doc at the University of California, Los Angeles and Princeton University . Today he is a professor at IMPA in Rio de Janeiro. From 2004 to 2007 he was its Deputy Director. Viana was visiting professor at the ETH Zurich , the University of Paris-South , at the Institut des Hautes Études Scientifiques (IHES) and the University of Dijon .

Viana is concerned with chaotic dynamic systems and in particular with the existence of strange attractors. After Lennart Carleson and Michael Benedicks proved the existence of strange attractors in the Henon map, he and Leonardo Mora showed their frequency in a more general class of maps (with homoclinic bifurcation ), proving a conjecture by Jacob Palis. He also demonstrated the existence of strange attractors for images with bifurcations via saddle point cycles. Viana also found new types of the Lorenz attractor in more than three dimensions (with any dimension of the expansion directions)

With Palis he also generalized a theorem by Newhouse from the 1970s to higher dimensions. The theorem claims that in the vicinity of a diffeomorphism with a homoclinic tangent there exist many diffeomorphisms which have an infinite number of attractive periodic orbits.

In 2001 he and Michael Benedicks solved a problem posed by David Ruelle and Jakow Sinai in the 1970s for attractors of the Hénon type (to prove that its catchment area, the Basin of Attraction , has no "holes" ).

In 2005 he and Artur Avila proved a conjecture by Maxim Kontsevich and Anton Zorich about the Lyapunov exponents of the Teichmüller river on the modular space of Abelian differentials on compact Riemann surfaces (namely that the non-trivial Lyapunov exponents are all different).

In 1994 he was invited speaker at the International Congress of Mathematicians (ICM) in Zurich (Homoclinic bifurcations and persistance of non uniformly hyperbolic attractors) and in 1998 he gave a plenary lecture at the ICM in Berlin (Dynamics: a probabilistic and geometric perspective). In 1994 he gave a plenary lecture at the International Congress of Mathematical Physicists in Paris (Chaotic dynamical behavior). He is head of the organizing committee for the ICM 2018 in Rio de Janeiro .

In 1993/4 he was a Guggenheim Fellow. Since 1995 he has been a member of the Council of the Brazilian Mathematical Society (and from 2009 its Vice-President) and since 1997 a member of the Brazilian Academy of Sciences. In 2000 he received the Grand Cross of the National Order of Merit for Science in Brazil. In 2005 he received the ICTP Ramanujan Prize in Trieste . Since 2009 he has been a member of the Chilean Academy of Sciences. In 1998 he received the Mathematics Prize of the Third World Academy of Sciences.

Fonts

  • with Christian Bonatti, Lorenzo Diaz: Dynamics beyond uniform hyperbolicity , Springer 2004 (Encyclopedia of Mathematical Sciences)
  • What's new on Lorenz Attractors? , Mathematical Intelligencer 2000, volume 3
  • Dynamical systems - moving into the next century in Björn Engquist, Wilfried Schmid (editor) Mathematics Unlimited - 2001 and beyond , Springer 2001
  • with Mora : Abundance of strange attractors. Acta Math. 171 (1993) no. 1, 1-71
  • with Palis : High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann. of Math. (2) 140 (1994) no. 1, 207-250.
  • Homoclinic bifurcations and persistence of nonuniformly hyperbolic attractors. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 1221-1229, Birkhäuser, Basel, 1995.
  • Dynamics: a probabilistic and geometric perspective. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 557-578
  • mit Alves , Bonatti : SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), no. 2, 351-398.
  • with Bochi : The Lyapunov exponents of generic volume-preserving and symplectic maps. Ann. of Math. (2) 161 (2005), no. 3, 1423-1485.
  • with Ávila : Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. Acta Math. 198 (2007), no. 1, 1-56.
  • with Ávila: Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181 (2010), no. 1, 115-189.
  • with Liao , Yang : The entropy conjecture for diffeomorphisms away from tangencies. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 6, 2043-2060.

Web links

Individual evidence

  1. ^ Viana, Mora Abundance of Strange Attractors , Acta Mathematica, Vol. 171, 1993, pp. 1-71. Expanded from two to any number of dimensions in Viana Strange Attractors in higher dimensions , Bol. Braz.Math.Soc., Vol. 24, 1993, pp. 13-62, at the same time as his doctoral thesis
  2. ↑ Diaz, Rocha, Viana Strange attractors with saddle-node cycles: prevalence and globality , Inventiones Mathematicae, Vol. 125, 1996, p. 34
  3. ↑ Bonatti, Pumarino Viana Lorenz attractors with arbitrary expanding dimensions , Compte Rendu Acad. Paris, Vol. 325, 1997, p. 883
  4. ^ Palis, Viana: High dimension diffeomorphisms displaying infinitely many periodic attractors, Annals of Mathematics. Vol. 140, 1994, pp. 207-250
  5. ^ Benedicks, Viana Solution of the basin problem for Hénon attractors , Invent. Math. 143: 375-434 (2001)
  6. ↑ Avila, Viana Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture , Acta Mathematica, Volume 198, 2007, pp. 1-56 (PDF; 441 kB)
personal data
SURNAME Viana, Marcelo
BRIEF DESCRIPTION Brazilian mathematician
DATE OF BIRTH March 4th 1962
PLACE OF BIRTH Rio de Janeiro