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.
Algebraic geometry is an area of mathematics studying geometric objects defined by polynomial equations in the n-space (called affine varieties), as well as more general geometric objects obtained from affine varieties by a certain gluing process. It is a classical idea that an affine variety is determined by a purely algebraic object, namely, the algebra of polynomial functions on this variety. Passing to more general non-affine varieties led to an introduction of more complicated algebraic objects that are described in the framework of homological algebra. In this project the PI considered how some of the simplest non-affine varieties are described by A-infinity algebras. The varieties in question are projective curves, i.e., one-dimensional algebraic varieties given by equations in the projective n-space. The parameter spaces for projective curves with given discrete invariants called moduli spaces of curves play an important role in recent connections of algebraic geometry to physics. One of the major results achieved by PI's research is that a certain moduli space of curves is the same as a certain moduli space of A-infinity algebras. Other parts of the project are concerned with yet another powerful invariant ofalgebraic varieties, their derived categories of coherent sheaves. These categories feature prominently in Kontsevich's conjectural mathematical explanation of the mirror symmetry phenomenon in string theory. Jointly with Kuznetsov the PI found a construction of exceptional collections in the derived categories of certain classical varieties, whose existence was conjectured since late 80s. Exceptional collections in derived categories are like bases in linear spaces. They provide a simple description of the derived category in terms of modules over an associative algebra. Another project concerns developing an analogy between the derived categories arising from geometry and other categories defined in more algebraic way, such as the category of matrix factorizations of a polynomial W in several variables. Matrix factorizations are essentially ways of decomposing the scalar matrix associated with W into a product of matrices with polynomial entries. Jointly with Vaintrob the PI found an analog of the Hirzeburch-Riemann-Roch formula for categories of matrix factorizations (originally such a formula was proved in a geometric context). In this work we also developed a version of Gromov-Witten theory (a theory related to mirror symmetry) in the context of matrix factorizations. The PI has been advising two Ph.D. students and one postdoc who were working on problems related to the project. He also coorganized a workshop within Mathematics and was helping with organizing and leading a math circle for elementary school students.
