A-infinity structures and derived categories in algebraic geometry

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This research project is in the field of algebraic geometry with some connections to string theory and noncommutative geometry. Algebraic geometry is a branch of mathematics studying geometric objects defined by polynomial equations and related mathematical structures. In classical algebraic geometry one associates with such geometric objects (called algebraic varieties) a space of functions that is a commutative ring. In this research project, more sophisticated algebraic structures associated with algebraic varieties, such as A-infinity algebras and derived categories of sheaves, will be studied.The research project will focus on the following topics:1) A-infinity structures associated with curves and their relation to the moduli spaces of curves,2) Semiorthogonal decomposition of the derived categories of equivariant sheaves for finite group actions,3) Cohomological field theories associated with quasihomogeneous polynomials,4) Sheaves on NC-thickenings and a characterization of Jacobians.The first project is about some A-infinity algebras associated with curves with marked points. The research will study normal forms of these A-infinity algebras up to homotopy and to relate them to the moduli spaces of curves. In the second project a construction of a canonical semiorthogonal decomposition of the derived category of equivariant coherent sheaves for some actions of finite reflection groups is outlined and will be studied. The third project is concerned with applications of categories of matrix factorizations with computation in the cohomological field theories attached to quasihomogeneous polynomials with isolated singularities. The fourth project focuses on which coherent sheaves on an abelian variety can be extended to a noncommutative thickening, which is a quantization of the Poisson envelope of the sheaf of regular functions. This may lead to a new characterization of Jacobians of curves.