Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

David White (North Carolina State University):  Symplectic Instanton Homology of Knots and Links in 3-manifolds

Powerful homology invariants of knots in 3-manifolds have emerged from both the gauge-theoretic and the symplectic kinds of Floer theory: on the gauge-theoretic side is the instanton knot homology of Kronheimer-Mrowka, and on the symplectic the (Heegaard) knot Floer homology developed independently by Ozsváth-Szabó and by Rasmussen. These theories are conjecturally equivalent, but a precise connection between the gauge-theoretic and symplectic sides here remains to be understood. We describe a construction designed to translate singular instanton knot homology more directly into the symplectic domain, a so-called symplectic instanton knot homology: We define a Lagrangian Floer homology invariant of knots and links which extends a 3-manifold invariant developed by H. Horton. The construction proceeds by using specialized Heegaard diagrams to parametrize an intersection of traceless $SU(2)$ character varieties. The latter is in fact an intersection of Lagrangians in a symplectic manifold, giving rise to a Lagrangian Floer homology. We discuss its relation to singular instanton knot homology, as well as the formal properties which this suggests and methods to prove these properties.

Kai Hugtenburg (University of Edinburgh): Open Gromov-Witten Invariants from the Fukaya Category

Enumerative mirror symmetry is a correspondence between closed Gromov-Witten invariants of a space X, and period integrals of a family Y. One of the predictions of Homological Mirror Symmetry is that the closed Gromov-Witten invariants can be obtained from the Fukaya category. For Calabi-Yau varieties this has been demonstrated by Ganatra-Perutz-Sheridan. Recently, enumerative mirror symmetry has been extended, by including open Gromov-Witten invariants and extended period integrals. It is natural to expect that open Gromov-Witten invariants can be obtained from the Fukaya category. In this talk I will outline a construction which will demonstrate this for certain open Gromov-Witten invariants.

Patricia Dietzsch (ETH Zürich) : Lagrangian Hofer Metric and Barcodes

Filtered Lagrangian Floer homology gives rise to a barcode associated to a pair of Lagrangians.  It is well-known that the lengths of the finite bars and the spectral distance are lower bounds of the Lagrangian Hofer metric. In this talk we are interested in a reverse inequality.  I will explain an upper bound of the Lagrangian Hofer distance between equators in the cylinder in terms of a weighted sum of the lengths of the finite bars and the spectral distance.