Tensor Categories and Geometric Representation Theory

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One of the basic ideas in the science and art is the idea of {\em symmetry}. For exampleone expects that the best solutions of technical problems are symmetric (an archetypicalexample here is the wheel -- highly symmetric shape that is so useful in our everydaylife). On a deeper level it is expected that all the physical laws should be symmetric withrespect to a change of point of view and this appears to be the guiding principle ofmany modern physical theories (starting with Einstein's Relativity Theory). It is quite anontrivial problem to express the idea of symmetry in mathematical terms; but wehave a great progress in this direction. Actually there are at least three levels of abstractionin the solution of this problem: groups, Hopf algebras, tensor categories. The first partof this project is related with highest level of abstraction in this list, that is with tensorcategories. 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 phasetransitions and surface phenomena (and this is important if we want to produce moreefficient computers); on the other hand the tensor categories are used in computerscience as a working tool for the constructing of the "quantum computer", a newconceptual approach to the computer architecture. So one expects that the study oftensor categories would be useful in the development of modern technologies.Thus it is suggested to study the tensor categoriesfocusing on the construction of new interesting examplesand the solution of classification problems.The second part of the project is devoted to the representation theory. One can thinkabout representation theory as about the study how a symmetry can be realized inmathematical equations. Thus for a given symmetry (which is described by group,Hopf algebra etc) representation theory aims to describe all the ways this symmetrycan show up in mathematical models. In the second part of the proposal it is suggested tostudy certain questions of representation theory of groups using insightsfrom the theory of tensor categories (thus we would like to employ symmetryfor 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 examplesof fusion (= semisimple tensor) categories by studying fusion categories graded byfinite groups and by constructing new examples of module categories (then dualcategories will provide new examples of tensor categories). The second partdeals with geometric representation theory. An extremely important class of objects ofinterest for the representation theory is formed by character sheaves that wereintroduced and classified by G.~Lusztig.