Algebraic and Symplectic Geometry Seminar

For a holomorphic function (“superpotential”)  W: X —> C on a complex manifold X, one defines the (2-periodic) matrix factorisation category MF(X;W), which is supported on the critical locus Crit(W) of W. At a Morse singularity, MF(X;W) is equivalent to the category of modules over the Clifford algebra on the tangent space TX. It had been suggested by Kapustin and Rozansky that, for Morse-Bott W, MF(X;W) should be equivalent to the (2-periodicised) derived category of Crit(W), twisted by the Clifford algebra of the normal bundle. I will discuss why this holds when the first neighbourhood of Crit(W) splits, why it fails in general, and will explain the correct general statement.

I will present a definition of symplectic homology groups for pairs of Liouville cobordisms with fillings, and explain how these fit into a formalism of homology theory similar to that of Eilenberg and Steenrod. This construction allows to understand form a unified point of view many structural results involving Floer homology groups, and yields new applications. Joint work with Kai Cieliebak.

Given some class of "geometric spaces", we can make a ring as follows. Additive structure: when U is an open subset a space X,  [X] = [U] + [X - U]. Multiplicative structure:  [X][Y] = [XxY]. In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable
statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural).  A motivating example will be polynomials in one variable. This is joint work with Melanie Matchett Wood.

The topological Fukaya category is a combinatorial model of the Fukaya category of exact symplectic manifolds which was first proposed by Kontsevich. In this talk I will explain work in progress (joint with J. Pascaleff and S. Scherotzke) on gluing techniques for the topological Fukaya category that are closely related to Viterbo functoriality. I will emphasize applications to homological mirror symmetry for three-dimensional CY LG models, and to Bondal's and Fang-Liu-Treumann-Zaslow's coherent constructible correspondence for toric varieties.  

I will describe 5 definitions of Euler characteristic for a space with perfect obstruction theory (i.e. a well-behaved moduli space), and their inter-relations. This is joint work with Yunfeng Jiang. Then I will describe work of Yuuji Tanaka on how to this can be used to give two possible definitions of Vafa-Witten invariants of projective surfaces in the stable=semistable case.