Hong-kwon Jo - pre-course on symplectic homology

We review a basic construction of symplectic homology.

Jungsoo Kang - Algebraic structures in symplectic homology 

Symplectic homology is a powerful invariant of Liouville domains, defined via the Hamiltonian Floer theory of periodic orbits. Beyond its fundamental structure as a graded vector space, it carries rich algebraic operations induced by counting pseudoholomorphic curves on Riemann surfaces with multiple punctures. We will explore the geometric constructions and properties of these operations, including TQFT structures, the BV operator, the secondary coproduct, and DGA structures. We will also examine S1-equivariant symplectic homology and the associated Gysin exact sequence.

Yoosik Kim - Almost toric fibrations and exotic Lagrangian tori

In this lecture, we discuss techniques for visualizing symplectic manifolds. When a symplectic manifold admits an effective Hamiltonian torus action, its geometry can be understood through the image of the moment map. There have been several attempts to extend this viewpoint to more general symplectic manifolds and their fibrations. After reviewing symplectic toric manifolds, we introduce almost toric fibrations as one such generalization. Topics include completely integrable systems, action–angle coordinates, and integral affine structures. We then discuss almost toric fibrations and their associated operations introduced by Symington. Using these tools, we explain how to construct exotic monotone Lagrangian tori, following the work of Vianna. If time permits, we also discuss constructions of such tori via deformation methods.

Sangjin Lee - Weinstein manifold and Lefschetz fibration

Abstract: Weinstein manifolds are symplectic manifolds, which admit "nice" symplectic structures. For example, every Weinstein manifold admits a Weinstein handle decomposition that can help us to understand its symplectic topology—according to Bourgeois–Ekholm–Eliashberg / Ganatra–Pardon–Shende, one can compute the symplectic homology / wrapped Fukaya category from a Weinstein handle decomposition. In this series of talks, after introducing basic notions such as Weinstein handles and Lefschetz fibrations, we will see that Lefschet fibrations provide a combinatorial/topological tool for studying Weinstein handle decompositions, together with examples of applications. The specific topics of each lecture are as follows: 

• First: Basic definitions (Lefschetz fibration, Weinstein handle, etc)

• Second: Relations between Weinstein handle decompositions and Lefschetz fibrations. 

• Third: An application — A construction of diffeomorphic, but symplectically distinct Weinstein manifolds via Lefschetz fibrations.