Moduli of A-Infinity Structures and Related Topics

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The project is in the field of algebraic geometry with some connections to string theory. Algebraic geometry is a branch of mathematics studying geometric objects defined by polynomial equations and related mathematical structures. Classically one associates with such geometric objects (called algebraic varieties) the set of algebraic functions on them which forms a commutative ring (i.e., functions can be added and multiplied). The PI studies much more sophisticated algebraic structures associated with algebraic varieties, such as A-infinity algebras and derived categories of sheaves. In an A-infinity algebra, in addition to taking product of two elements, one has higher operations of multiplying together n-tuples of elements. These structures appeared independently in a very different way in works of Fukaya. The research is partly motivated by the desire to match different constructions of A-infinity algebras. The PI also intends to apply A-infinity structures to solve some problems in algebraic geometry.The project will focus mostly on the following two topics: moduli spaces of A-infinity structures and moduli spaces related to algebraic curves. In the first part of the project the goal is to construct moduli spaces (i.e., parameter spaces) of A-infinity structures related to various moduli spaces of geometric objects (sometimes involving non-commutative geometry). In particular, the PI plans to construct moduli spaces of A-infinity algebras related to algebraic surfaces, double covers of non-commutative projective lines, noncommutative orders over curves, as well as moduli spaces of A-infinity modules related to birational transformations. The second part of the project is devoted to studying toric GIT picture for moduli spaces of curves with nonspecial divisors, as well as moduli spaces of curves with vector bundles on them and their non-commutative analogs. Also, the PI plans to study natural vector fields on the moduli spaces of curves with nonspecial divisors and use them to produce explicit rational functions on the moduli spaces of curves with no marked points.