Current NSERC Project:   Algebraic cycles on projective homogeneous varieties   (2010 - 2014)

I. The Grothendieck gamma-filtration and the topological filtration on PHVs

Slides of recent talks

• Boston, 2012
• Alberta, 2011
• Vancouver, 2011

Subprojects:

• Grothendieck's gamma-filtration, motives and the Rost invariant (see the recent preprint with S.Garibaldi)
  
• Bounds for the torsion in Chow groups of homogeneous varieties (see the recent preprint with S.Garibaldi and S.Baek - E.Neher)
• The gamma filtration, the J-invariant and orthogonal involutions (see the recent preprint with A.Quiguiner and N.Semenov)
• Equivariant cohomology and invariants of torsors (see the recent preprint with S.Gille)
• T-equivariant Chern class map, exponents of W-actions and the Dynkin indices of Lie algebras (see the recent preprint with S.Baek and E.Neher)

II. Schubert calculus for oriented cohomology theories, e.g. for algebraic cobordism

Subprojects:

• Characteristic map for general oriented cohomology theories (see the recent preprint with Calmes and Petrov)
• T-equivariant Chern class map for oriented theories (work in progress with Malagon-Lopez)

III. Motivic decompositions of PHVs

Objectives:

• Given an anisotropic linear algebraic group G and a projective G-homogeneous variety X to describe all motivic decompositions of X
• Knowing motivic decompositions of X to compute various cohomological invaraints of G, for instance, canonical and essential dimensions.

Maple package to work with algebraic cycles

Subprojects, works in progress:

• Vishik's J-invariant and GDI for projective homogeneous varieties
  See our paper on J-invariant which has appeared in [Ann.Sci.ENS.] There we provided complete description of motivic decompositions of an arbitrary generically split projective homogeneous variety.
• Algebraic cycles via algebraic cobordisms
  To apply the techniques of symmetric operations in algebraic cobordism in order to construct "new" algebraic cycles on PHV. See the paper in [Crelle's Journal].
• Motives with integer coefficients. Liftings of motivic decompositions.
• Degree formula for connective K-theory
  This is a further development of ideas of Rost and Merkurjev. See the paper in [Invent.Math.].
• Essential, canonical dimensions of linear algebraic groups and motives
  See the paper in [Math.Ann.].

Old project: The Grothendieck-Serre conjecture, the Purity and the Gersten conjecture

Directions:

• Gersten resolutions, Geometric presentation lemmas, General position arguments.
  See the recent joint paper with I.Panin published in Manuscripta Math.
• Grothendieck-Serre's conjecture on G-torsors.  (short overview on what was done)