Mathematical Sciences: The Geometry of Langlands Duality

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Abstract Mirkovic The main project: geometry of the loop Grassmannian and applications to representation theory: (i) geometric Langlands conjectures and (ii) modular representations. Here (i) is based on our minor role in the completion of Drinfelds project of constructing the dual reductive group geometricaly, and (ii) on our observation that the loop Grassmannian is the geometric setting for the localization of algebraic representations of split reductive groups over arbitrary commutative rings. Related projects: (a) study of intersection homology sheaves and characteristic cycles via the p-transversal stratifications and (b) the role of the hyperkahler structure in representation theory. A long term goal of the above work is to relate certain highly developed parts of abstract mathematics (geometry and number theory) with the ideas developed recently by theoretical physicist (particle theory). Both of these developments share common mathematical language of representation theory and certain parts of algebraic geometry. The exdpected unification should dramaticaly increase our understanding of both fields.