Bonn Symplectic Geometry Seminar

Abstract: In this talk, I will present some new results about skeleta of complete intersections inside (C*)^n. I will start by briefly reviewing the Batyrev-Borisov mirror construction, which uses combinatorial dualities between lattice polytopes to produce mirror pairs of Calabi-Yau complete intersections in Fano toric varieties. The main focus of the talk will be open Batyrev-Borisov complete intersections (BBCIs), which are Liouville manifolds obtained by removing the toric boundary in the Batyrev-Borisov construction. I will explain how one can use tropical geometry to compute Lagrangian skeleta of open BBCIs and decompose them into pieces mirror to certain toric varieties, which leads to a proof of homological mirror symmetry (generalising the work of Gammage-Shende and Zhou in the case of hypersurfaces). 

Abstract: Whitehead torsion originated in geometric topology as an invariant central to the s-cobordism theorem, detecting when a homotopy equivalence fails to be simple.  More recently, it has appeared in symplectic geometry through the work of Abouzaid - Kragh and Courte - Porcelli, in connection with closed Lagrangian submanifolds of cotangent bundles. In this talk, I will present a categorical framework for simple homotopy theory in Fukaya categories, based on the fundamental group of the ambient symplectic manifold. This framework provides a setting in which previous known results for cotangent bundles can be extended to general Weinstein manifolds. 

Abstract: The (small) quantum connection is one of the simplest objects built out of Gromov-Witten invariants, yet it gives rise to a repertoire of rich and important questions. This connection has a simple pole (well-behaved) at infinity and a quadratic pole ay 0. Despite its simple form, very little is understood about the latter. In this talk, I will discuss my result that for all Fano symplectic manifolds, this quadratic singularity is of unramified exponential type—-meaning it has a "spectral decomposition" where each piece is as simple as possible. Surprisingly, the proof follows a reduction mod p argument, and uses Katz's classical monodromy theorem and the more recent quantum Steenrod operations in symplectic Gromov-Witten theory. If time permits, I will discuss other developments in the field regarding (p-adic) analytic decompositions of the quantum connection.

Abstract: For a toric Fano surface, we study the counts of nodal curves having fixed intersection points with the boundary divisor, which may be interpreted as log Gromov-Witten invariants. We propose an approach using homological mirror symmetry. Specifically, following a Yau-Zaslow type argument, such counts should coincide with the Euler number of the moduli space of a certain family of Jacobians, which-by toric HMS-corresponds to moduli of certain constructible sheaves. We describe a ruling decomposition of this moduli space and thereby compute its Euler number. This is a joint work in progress with Tom Graber and Eric Zaslow.

Abstract: In joint work with Matessi-Ruddat-Zharkov, we prove the existence of Lagrangian torus fibrations on Calabi-Yau hypersurfaces in toric Fano manifolds given by a reflexive polytope. The result is motivated by the Strominger-Yau-Zaslow conjecture which predicts the existence of these fibrations on Calabi-Yau manifolds near large complex structure limits. In this talk I will outline the main set up of the construction. The idea is to replace the ordinary algebraic equation, with a new one involving "ironing coefficients" and a convex (but not strictly convex!) potential which has the effect of breaking the manifold in local models. Over these models we apply the Liouville flow technique in the style of Evans-Mauri. 

Abstract: I will discuss recent work with Soham Chanda on constructing flow categories and flow bimodules from moduli spaces in contact topology. I will outline the required constructions and show how this flow category recovers contact homology and chain maps between them.

Abstract: TBA