Project descriptionThe concept of an oriented cohomology theory is well known in algebraic topology. In algebraic geometry it was introduced and systematically studied by Levine, Morel, Panin and Smirnov. Moreover, there exist an equivariant version of oriented cohomology theories. Similar to Grothendieck's construction of Chow motives, one can define the category of motives with respect to any oriented cohomology theory (ordinary or equivariant). Motives play a central role in understanding of the cohomologies of schemes and in the algebraic geometry by itself. There exists a broad literature devoted to classical Chow motives, but so far there are very few results about the structure of motives with respect to arbitrary oriented cohomology theories. To make progress on this is one of the goals of the present project.
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, arXiv: 1805.09059 (2018)
accepted in Ann. Sci. Ec. Norm. Sup.
V. Petrov, N. Semenov, Hopf-theoretic approach to motives twisted flag varieties, abs/arXiv: 1908.08770 (2019).