The proposed research 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). Modern research involves more sophisticated algebraic structures associated with algebraic varieties, such as the category of coherent sheaves (the notion of a category is a generalization of that of an associative ring). One part of the project is to establish some cases of the homological mirror symmetry conjecture which identifies categories appearing in geometry in two seemingly unrelated contexts. Another part of the project aims to give a rigorous mathematical foundation to some aspects of the use of super Riemann surfaces (a generalization of the usual surfaces) in string theory. This project provides research training opportunities for undergraduate and graduate students.
More specifically, the first part of the project is on homological mirror symmetry for symmetric powers of punctured spheres. The goal is to identify categorical resolutions of derived categories of coherent sheaves on certain algebraic varieties with partially wrapped Fukaya categories of the symmetric powers of punctured spheres. This may help to find a new construction of Ozsvath-Szabo's categorical knot invariant. The second part is to work out a generalization of the Hirzebruch-Riemann-Roch formula to the categories of matrix factorizations over non-affine varieties and stacks. The PI also would like to use categories of matrix factorizations to find a Landau-Ginzburg counterpart of the G-equivariant Gromov-Witten theory. The third part of the project is to realize trigonometric solutions of the associative Yang-Baxter equation in terms of noncommutative orders over nodal cubics. The fourth part is related to the geometry of stable supercurves. The PI proposes to understand the poles of the analog of Mumford's isomorphism for the Berezinian of the moduli of supercurves near the boundary of the compactification by stable supercurves and to study some problems arising in integration over the moduli space of supercurves.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
