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    Grant CNCS IDEI Nr PN-II-ID-PCE-2012-4-0364 (cod proiect)
                            (November 2013- October 2015)

                                           financed by

UNITATEA EXECUTIVA PENTRU FINANTAREA INVATAMANTULUI SUPERIOR, A CERCETARII DEZVOLTARII SI INOVARII

                                           Contract n0 89/24/09/2013


             Arithmetic homogeneous spaces





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Contact

Annual budget (RON):    2013: 50830,    2014: 316480,    2015: 248400.


Abstract : Homogeneous spaces under the action of a Lie group are a classical notion in differential geometry. From Grothendieck and Serre in the years 1955-1965, these notions passed to the algebraic and arithmetic geometry setting. The groups are then algebraic groups (namely defined by polynomial equations as the orthogonal groups) and more generally group schemes. The work plan deals with three exploratory research tasks : arithmetic homogeneous spaces, R-equivalence on homogeneous spaces and applications of torsors to infinite dimensional Lie theory.



Expected results of the proposal: There are three main directions of work:

  • Arithmetical homogeneous spaces,
  • R-equivalence and homogeneous spaces,
  • Application of torsors to infinite dimension Lie theory.

The first one includes the study of maximal tori of algebraic groups over arithmetic fields. The second one is about the study of R-equivalence of homogeneous spaces G/H over nice fields around Serre's vanishing conjecture II in Galois cohomology. The last one continues the collaboration with V. Chernousov and A. Pianzola and focuses mainly on the abstract group properties of group of points of loop group schemes and relevant generalizations of loop torsors.


Activity reports


Publications produced as a result of this research

Published/to appear:





Preprints:


Book: [8] P. Gille, ''Cohomologie galoisienne des groupes semi-simples sur un corps de dimension cohomologique séparable <=2''.



Talks

    P. Gille, Attractors for the action of G_m on schemes (after V. Drinfeld), seminar of Algebraic Geometry, Imar, December 12 of 2013.

    P. Gille, Gabber's compactifications of algebraic groups, seminar of Algebraic Geometry, Imar, April 10 of 2014.

    P. Gille, Group schemes I: introduction, seminar of Algebraic Geometry, Imar, October 9 of 2014.

    P. Gille, Group schemes II: Grothendieck's deformation theorem for subtori, seminar of Algebraic Geometry, Imar, October 23 of 2014.

    P. Gille, Group schemes III: Margaux-Vinberg's rigidity for homomorphisms of linearly reductive algebraic groups, seminar of Algebraic Geometry, Imar, October 30 of 2014.

    P. Gille, Group schemes IV: Parabolic subgroups of reductive group schemes, seminar of Algebraic Geometry, Imar, November 13 of 2014.

    P. Gille, The cohomological Brauer group, seminar of Algebraic Geometry, Imar, March 26, April 2 and 9 of 2015.

    C. Beli, Octonion algebras having the same quaternion subalgebras, ICTAMI, Alba Iulia, September 17 of 2015.

    P. Gille, Some geometric aspects of the quaternions and the octonions, ICTAMI, Alba Iulia, September 18 of 2015.