Yves Benoist

Yves Benoist is a French mathematician, known for his work on group dynamics on homogeneous spaces. He is currently a Senior Researcher (Directeur de Recherche) of CNRS at the University of Paris-Sud.

In 1990 Benoist proved a longstanding open conjecture with Patrick Foulon and François Labourie about Anosov flows on compact, negatively curved manifolds.[1] In the 2000s he wrote a series of papers on the divisible convex sets in projective space and periodic tilings by such sets.

In 2011 he was awarded the Clay Research Award along with his former doctoral student, Jean-François Quint.[2] The prize citation highlighted their work on stationary measures and closed orbits for non-abelian group actions on homogeneous spaces, solving a long-standing conjecture of Hillel Furstenberg. They showed that in homogeneous spaces of finite volume, orbits of a Zariski dense subgroup of a semisimple group equidistribute towards algebraic measures.

He gave the 2012 Takagi Lectures[3] in Kyoto at the Research Institute for Mathematical Sciences (RIMS). In 2014 he was an Invited speaker at the International Congress of Mathematicians in Seoul.

Selected works
  • With J.-F. Quint: Mesures stationaires et fermés invariants of homogeneous espaces. Annals of Mathematics, vol 174, 2011, pp 1111–1162.
  • With J.-F. Quint: Random walks on finite volume homogeneous spaces. Inventiones Mathematicae, vol 187, 2012, pp 37–59.
  • Convexes divisibles I: Algebraic groups and arithmetic, 339–374, Tata Inst Fund.. Res., Mumbai in 2004.
  • Convexes divisibles II. Duke Math J. 120 (2003), No 1, 97–120.
  • Convexes divisibles III: Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 5, 793–832.
  • Convexes divisibles IV: Invent. Math. 164 (2006), no. 2, 249–278.
  • Propriétés asymptotiques of groupes linéaires. Geom. Funct. Anal. 7 (1997), no. 1, 1–47.
  • Actions propres sur les espaces homogeneous réductifs. Ann. of Math. (2) 144 (1996), no. 2, 315–347.
  • With P. Foulon, F. Labourie: Flots d'Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5 (1992), no. 1, 33–74.

Lecture notes on Yves Benoist's work
  • Jean-François Quint: Convexes divisibles, d'après Yves Benoist. Séminaire Bourbaki, June 2008.
  • François Ledrappier: Mesures sur les espaces station aires homogeneous, d'après Yves Benoist et Jean-François Quint Séminaire Bourbaki, 2012.
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References
  1. Y. Benoist, P. Foulon, F. Labourie: Flots d'Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5 (1992), no. 1, 33–74.
  2. Clay Research Award
  3. 10th Takagi Lectures
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