Oriented cohomology theories and equivariant motives
Principal Investigator
Prof. Dr. N. Semenov (LMU München)Scientific staff
Andrei Lavrenov (01.07.2015-30.06.2018, LMU München) a.lavrenov@lmu.de
Project description
This project will investigate the following areas:• Equivariant cohomology theory and equivariant Chow motives
We are interested in the structure of the equivariant motives of the variety of Borel subgroups of a split semisimple group G.
• The A-motives for an ordinary oriented cohomology theory A We would like to investigate the structure of the A-motives for an arbitrary oriented cohomology theory A and in particular, consider the case of the Morava K-theory. The existing methods to compute Chow motives cannot be directly carried over to arbitrary cohomology theories. One main goal here is the development of new methods to attack this problem.
• J-invariant for arbitrary oriented cohomology theories The J-invariant is a discrete motivic invariant of a semisimple algebraic group G over a field F, which measures the behaviour of the variety of Borel subgroups of G in the category of Chow motives (see Section 1). It would be very interesting to give an appropriate definition and compute the J-invariant with respect to an arbitrary oriented cohomology theory, e.g. for the Morava K-theory.
• Operations in oriented cohomology theories
In the present project we propose to further the research on cohomological operations, and, in particular, extend the investigations to arbitrary (oriented) cohomology theories with applications to classical problems in algebra.
Related publications
V. Petrov, N. Semenov, Rost motives, affine varieties, and classifying spaces, Journal of London Math. Soc. 95 (2017), issue 3, 895-918.
A. Neshitov, V. Petrov, N. Semenov, K. Zainoulline, Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras, Advances in Math. 340 (2018), 791-818.
M. Borovoi, N. Semenov, M. Zhykhovich, Hasse principle for Rost motives, Int. Math. Res. Not., https://doi.org/10.1093/imrn/rny300
P. Sechin, N. Semenov, Applications of the Morava K-theory to algebraic groups, Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), no. 4, 945–990.
V. Petrov, N. Semenov, Hopf-theoretic approach to motives twisted flag varieties, Compos. Math. 157 (2021), no. 5, 963–996.
A. Lavrenov, V. Petrov, P. Sechin, N. Semenov, Morava K-theory of orthogonal groups and motives of projective quadrics, arxiv: 2011.14720 (2020).