Mathematical Sciences: Operads, Representation Theory and Algebraic Geometry

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Kapranov proposes to extend the prior study of moduli spaces of stable curves by means of operads and their duality to more sophisticated moduli spaces of stable maps and Fulton-McPherson compactified configuration spaces. He proposes to relate two branches of representation theory which previously were not connected: theory of automorphic representations and theory of affine quantum groups, by looking at Eisenstein series and their functional equations from a new point of view. He also proposes to study Hecke operators on vector bundles on algebraic surfaces over finite fields, in order to establish an analog of the geometric Langlands correspondence, a fundamental principle in the theory of vector bundles on curves. Working with V. Ginzburg, Kapranov proposes to study natural analogs of Hecke algebras for matrix groups over 2-dimensional local fields like the field of power series in two variables over a finite field. In addition, Kapranov proposes to generalize the theory of Drinfeld modules to algebraic varieties of arbitrary dimension, again with the aim of finding the correct analog of the Langlands correspondence. He also proposes to look for a generalization of the theory of character sheaves to the case of $p$-adic fields, which would involve stacks. This research is in the fields of algebraic geometry and representation theory. Algebraic geometry is one of the oldest branches of mathematics which initially studied plane geometric images defined by simple equations but in last few decades has developed very extensively, finding applications not only inside mathematics but in such diverse fields as particle physics (geometry of microscopic degrees of freedom), car design (use of algebraic surfaces to develop new aesthetically appealing shapes), robotics, theoretical computer science and others. In particular, algebraic geometry over finite fields (to which a large part of this proposal is devoted) has found applications in constructing error-correcting codes, optimal net works and in several other important problems related to communications and information transmission.