Mathematics Colloquia and Seminars

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One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings. We show that family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. Within this family, we provide some of the first examples of a Legendrian link that admits infinitely many planar exact Lagrangian fillings.

Weinstein domains are examples of symplectic manifolds with contact boundary that have a handle decomposition compatible with the symplectic structure of the manifold. Weinstein handlebody diagrams are given by projections of Legendrian submanifolds. We provide Weinstein handlebody diagrams of the $4$-dimensional Milnor fibers of $T_{p,q,r}$ singularities, which we then use to construct infinitely many Lagrangian tori and spheres in these spaces.