
Arithmetic homogeneous spaces
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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 19551965, 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, Requivalence 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,
 Requivalence 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 Requivalence 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:
 [1] V. Chernousov, P. Gille, A. Pianzola, Whitehead
groups of loop group schemes of nullity one, Journal of the Ramanujan Mathematical Society 29 (2014), 126.
 [2] P. Gille, Sur la classification des schémas en groupes semisimples,
``Autour des schémas en groupes, III'', Panoramas et Synthèses 47 (2015), 39110.
 [3] V. Chernousov, P. Gille, A. Pianzola,
Threepoint Lie algebras and Grothendieck's dessins d'enfants,
Mathematical Research Letters 23 (2016), 81104.
Preprints:
Book: [8] P. Gille, ''Cohomologie galoisienne des groupes semisimples 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: MargauxVinberg'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.
