Morava J-invariant(joint with Nikita Geldhauser, Andrei Lavrenov and Victor Petrov)DOI[arXiv]
We compute the co-multiplication of the algebraic Morava K-theory for split orthogonal groups.
This allows us to compute the decomposition of the Morava motives of generic maximal orthogonal Grassmannians
and to compute a Morava K-theory analogue of the J-invariant in terms of the ordinary (Chow) J-invariant.
Forum of Mathematics, Sigma, Volume 13, 2025
Morava K-theory of orthogonal groups and motives of projective quadrics(j.w. Nikita Geldhauser, Andrei Lavrenov and Victor Petrov)DOI [arXiv]
We compute the algebraic Morava K-theory ring for all split orthogonal groups.
As an application we describe the Morava motive of a generic quadric and of its maximal orthogonal Grassmannian.
Advances in Mathematics, Volume 446, June 2024, 109657
Applications of the Morava K-theory to algebraic groups (joint with Nikita Geldhauser)DOI[arXiv]
We study relations between vanishing of cohomological invariants of algebraic groups and splitting of Morava K-theory motives.
For quadrics the n-th Morava K-theory detects vanishing of all invariants of degree less than n+2.
The second Morava K-theory detects vanishing of Rost invariants, the fourth Morava K-theory detects splitting of groups of type E8.
We provide new estimates on torsion in codimensions up to 2n of quadrics as above as well as provide a general method for estimation.
Annales scientifiques de l'ENS, Tome 54, Fascicule 4, Pages 945-990, 2021
On the Structure of Algebraic Cobordism DOI[arXiv]
We investigate the structure of algebraic cobordism of Levine-Morel as a module
over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it.
We also prove the Syzygies Conjecture of Vishik on the existence of certain free resolutions of algebraic cobordism,
and show that algebraic cobordism of a smooth surface can be described in terms of K-theory together with a topological filtration.
Advances in Mathematics, Volume 333, 31 July 2018, Pages 314–349
Chern Classes from Algebraic Morava K-theories to Chow Groups DOI[arXiv]
We calculate the ring of unstable (possibly nonadditive) operations from Morava K-theory
to Chow groups with p-local coefficients.
More precisely, we prove that it is a formal power series ring on generators which satisfy a Cartan-type formula.
International Mathematics Research Notice, August 2018, Vol. 2018, No. 15, pp. 4675–4721
The Category of Flat Hodge–Tate Structures DOI
We describe the full subcategory
of the category of Hodge-Tate structures on which
the (essentially unique) arithmetic Gauss-Manin connection (constructed by M. Rovinsky) is flat.
Mathematical Notes, 2016
Invertible Morava motives in quadrics[arXiv](joint with Andrei Lavrenov)
We construct invertible motives associated to elements in Milnor K-theory modulo 2
and investigate the Morava motives of quadrics.
We conjecture that Morava motives yield a new approach to cohomological invariants.
Chern classes from Morava K-theories to pn-typical oriented theories [arXiv]
We generalize the notion of p-typical formal group laws to pn-typical and study operations
from n-th Morava K-theory K(n) to an arbitrary pn-typical oriented theory.
If the ring of coefficients of the pn-typical theory is torsion-free,
we show that all operations from K(n) are generated by 'Chern classes'.
In particular, this allows us to introduce the gamma filtration on Morava K-theories.