Derived categories techniques in algebraic geometry

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The proposed research will focus on three topics, all applying the techniques of derived categories in algebraic geometry:1) cohomological field theories associated with isolated hypersurface singularities;2) real multiplication functors and stability spaces for derived categories related to abelian varieties;3) exceptional collections on Lagrangian Grassmannians.The first project is to construct a cohomological field theory associated with an isolated quasi-homogeneous hypersurface singularity.Essentially, this amounts to constructing a collection of cohomology classes on the moduli spaces of stable curves with marked points that satisfy certain factorization rules over the boundary of the moduli spaces, as in the theory of Gromov-Witten invariants.The second project is to study certain functors between derived categories of abelian varieties that can be used in Manin's real multiplication program for noncommutative tori. It is also proposed to define and study the action of these functors on the Bridgeland's stability spaces.The third project is to construct full exceptional collections of vector bundles in the derived category of coherent sheaves on the Grassmannian of Lagrangian subspaces in a symplectic vector space.Such collections, defined by certain cohomological conditions, facilitate the study of coherent sheaves on an algebraic variety by transferring geometric questions into linear algebra problems.The proposed research is in the field of algebraic geometry with some connections to string theory and noncommutative geometry. Algebraic geometry is a classical branch of mathematics studying geometric objects defined by polynomial equations and related mathematical concepts. Many recent advances in some parts of algebraic geometry involving moduli spaces (parameter spaces classifying various geometric structures) were motivated by their use in string theory. Derived categories arose from studying categories of chain complexes and form a part of a vast algebra machinery needed for modern algebraic geometry.