GRANT CNCS IDEI 2012

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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 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:

[1] V. Chernousov, P. Gille, A. Pianzola, Whitehead groups of loop group schemes of nullity one, Journal of the Ramanujan Mathematical Society 29 (2014), 1-26.

[2] P. Gille, Sur la classification des schémas en groupes semi-simples, ``Autour des schémas en groupes, III'', Panoramas et Synthèses 47 (2015), 39-110.

[3] V. Chernousov, P. Gille, A. Pianzola, Three-point Lie algebras and Grothendieck's dessins d'enfants, Mathematical Research Letters 23 (2016), 81-104.

[4] C. Beli, P. Gille, T.-Y. Lee, Maximal tori of algebraic groups of type G2, Pacific Journal of Mathematics 279 (2015), 101-134

[5] C. Beli, P. Gille, T.-Y. Lee, Examples of algebraic groups of type G2 having the same maximal tori, arXiv:1503.07046, Proceedings of the Steklov Institute of Mathematics 292 (2016), 10-19.

[6] V. Chernousov, P. Gille, A. Pianzola, A classification of torsors over Laurent polynomial rings, Commentarii Mathematici Helvetici 92 (2017), 37-55.

Preprints:

[7] C. Beli, Common slot lemmas in characteristic 2, arXiv:1510.08330.

Book: [8] P. Gille, ''Groupes algébriques semi-simples en dimension cohomologique <=2'', volume 2238 (2019) of the Lecture Notes in Mathematics series (to appear).

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, Multiloop Lie algebras of exceptional type, Montréal, CRM, March 12 of 2014.

P. Gille, Topological properties of torsors and homogeneous spaces over valued fields, Ann Arbour (University of Michigan), March 24 of 2014.

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

P. Gille, Maximal tori of algebraic groups of type G2, Séminaire Variétés Rationnelles, Paris, October 3 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, New homogeneous spaces with a 0-cycle of degree one and no rational point, Banff, December 9 of 2014.

P. Gille, New homogeneous spaces with a 0-cycle of degree one and no rational point, Charlottesville (University of Virginia), February 4 of 2015.

P. Gille, Topological properties of torsors and homogeneous spaces over valued fields, IPAM (Los Angeles), February 12 of 2015.

P. Gille, Serre's conjecture II : beyond the de Jong-He-Starr's theorem, CRG Geometry and Physics Seminar (Vancouver, UBC), February 24 of 2015.

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

P. Gille, Parabolic subgroups of reductive group schemes, Conference ``From Lie algebras to group schemes" Ottawa, May 4 of 2015.

P. Gille, On maximal tori of algebraic groups of type G2, Minisymposium on Algebras and Algebraic Groups, Uppsala, May 22 of 2015.

P. Gille, Octonion algebras and torsors, Rencontres arithmétiques de Caen, June 26 of 2015.

P. Gille, Unramified non-abelian cohomology, Cohomological Methods in the Theory of Algebraic Groups, Luminy, September 8 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.

P. Gille, Some geometric aspects of the quaternions and the octonions (Notes), IMAR monthly's conference, Bucharest, October 14 of 2015.

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

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