One of the basic ideas in the science and art is the idea of {em symmetry}. For example one expects that the best solutions of technical problems are symmetric (an archetypical example here is the wheel -- highly symmetric shape that is so useful in our everyday life). On a deeper level it is expected that all the physical laws should be symmetric with respect to a change of point of view and this appears to be the guiding principle of many modern physical theories (starting with Einstein's Relativity Theory). It is quite a nontrivial problem to express the idea of symmetry in mathematical terms; but we have a great progress in this direction. Actually there are at least three levels of abstraction in the solution of this problem: groups, Hopf algebras, tensor categories. The first part of this project is related with highest level of abstraction in this list, that is with tensor categories. Despite of this high level of abstraction the theory is useful in the applications: in physics the language of tensor categories is used in theoretical study of phase transitions and surface phenomena (and this is important if we want to produce more efficient computers); on the other hand the tensor categories are used in computer science as a working tool for the constructing of the "quantum computer", a new conceptual approach to the computer architecture. So one expects that the study of tensor categories would be useful in the development of modern technologies. Thus it is suggested to study the tensor categories focusing on the construction of new interesting examples and the solution of classification problems. The second part of the project is devoted to the representation theory. One can think about representation theory as about the study how a symmetry can be realized in mathematical equations. Thus for a given symmetry (which is described by group, Hopf algebra etc) representation theory aims to describe all the ways this symmetry can show up in mathematical models. In the second part of the proposal it is suggested to study certain questions of representation theory of groups using insights from the theory of tensor categories (thus we would like to employ symmetry for studying symmetry and this is indeed one of the guiding principles of the theory).
More technically, in the first part of the projectit is suggested to find new examples of fusion (= semisimple tensor) categories by studying fusion categories graded by finite groups and by constructing new examples of module categories (then dual categories will provide new examples of tensor categories). The second part deals with geometric representation theory. An extremely important class of objects of interest for the representation theory is formed by character sheaves that were introduced and classified by G.~Lusztig.
