Proposer plans to investigate several topics in motivic cohomology theory. The first project is to compare two constructions of the motivic spectral sequence, the first due to Bloch, Lichtenbaum, Friedlander, Suslin and Levine and the second one due to Grayson and Suslin. Since the first construction was shown by M. Levine to coincide with the Voevodsky construction via the slice filtration this will prove that all the known approaches to the construction of the spectral sequence give the same answer. The second project concerns the motives of non split reductive algebraic groups like GL_n,D and SL_n,D, where D is a central division algebra over a field F. The third project is an attempt to compute the group H^n-1,n(F) for a field F. Finally the last project concerns the comparison between two constructions of the algebraic cobordism theory.
The main objective of mathematics is to provide an accurate picture to the physical world or at least an appropriate approximation of that picture. From this point of view algebraic varieties are of principal importance, first they are relatively easy to understand since they are just defined by polynomial equations, next they usually give a rather accurate approximation to other shapes, most importantly they do appear naturally in quite a lot of subjects from theoretical physics to coding theory. That is why algebraic geometry - the theory of algebraic varieties is so important for the development and applications of mathematics. This project is devoted to the study of certain fundamental problems of motivic cohomology theory - a relatively new and very quickly developing branch of algebraic geometry. Geometry is blended with algebra and topology in this part of mathematics, ideas and methods to be used come equally from all these directions.
