Current NSERC Project: Algebraic cycles on projective homogeneous varieties (2010 - 2014)
Active members of my research group: Sanghoon Baek, Caroline Junkins, Jose Malagon-Lopez, Changlong Zhong
I. The Grothendieck gamma-filtration and the topological filtration on PHVs
with applications to invariant theory of finite reflection groups,
motivic decompositions and algebraic cycles on twisted flag varieties,
invariants of torsors - Tits algebras and the Rost invariant,
torsion in Chow groups of twisted flag varieties.
Slides of recent talks
- 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
- 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
- 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
See the paper in [Math.Ann.].
Old project: The Grothendieck-Serre conjecture, the Purity and the Gersten conjecture
- 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)