In the paper L. Katzarkov, M. Kontsevich, T. Pantev "Hodge theoretic aspects of mirror symmetry", the authors suggested existence of a functorial "noncommutative Hodge sctructure" on the periodic cyclic homology of a smooth proper dg-category. In particular, it was conjectured that periodic homology possess a functorial rational structure, analogous to the Betti one in the commutative case. Recently Anthony Blanc proposed a possible candidate, so-called topological K-theory of dg-categories. In the talk, I will remind the definition of this invariant, discuss some of its properties and tell about some cases when it can be shown that topological K-theory indeed provides us with a rational structure on HP.
We construct natural virtual fundamental classes for nested Hilbert schemes on a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as Poincare invariants of Duerr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In the case of the nested Hilbert scheme of points, we can express these invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. The virtual fundamental classes of the nested Hilbert schemes play a crucial role in the local Donaldson-Thomas theory of threefolds that I will talk about, in talk 2. This talk is based on arXiv:1701.08899.
We discuss the 'moduli of objects' MD in a dg category D and construct a map from cyclic homology of D to functions on the moduli space MD. When D is a smooth, oriented dg category ('Calabi-Yau'), the cyclic homology HC(D) is endowed with a shifted Lie bracket ('algebraic string bracket') and the functions on M_D are endowed with a shifted Poisson bracket. We show that the map from cyclic homology to functions entwines the brackets. Examples include the Goldmann bracket of free loops on a surface, the string bracket of Chas-Sullivan, and the Hitchen system for Higgs bundles. This is joint work very much in progress with Nick Rozenblyum.
In this talk we will discuss how to compute the Stokes matrices for some semisimple Frobenius manifolds by using the so-called monodromy identity. In addition, we want to discuss the case when we get integral matrices and their relations with mirror symmetry. This is part of an ongoing project with Maxim Smirnov which extends previous work with Marius van der Put for the case of quantum cohomology of projective and weighted projective spaces to other Frobenius manifolds not necessarily of quantum cohomology type.
We will show that the bounded derived category of a generic curve of genus g at least two can be embedded as a semiorthogonal component into the bounded derived category of a smooth Fano variety. Namely, the moduli space of stable rank 2 vector bundles with fixed odd determinant. This is joint work with A. Kuznetsov.