Georgia Institute of Technology College of Sciences

Given a Legendrian link L in the contact 3-sphere, one can hope to classify the set of exact Lagrangian fillings of L, i.e. exact Lagrangian surfaces in the symplectic 4-ball with boundary equal to L. Much of the recent progress towards this classification relies on establishing a connection between sheaf-theoretic invariants of Legendrians and cluster algebras. In this talk, I will describe this connection and how these invariants behave with respect to certain symmetries of Legendrian links and their fillings. Parts of this are joint work with Agniva Roy.

In this talk, I will present recent advancements in the study of smooth mapping class groups of 4-manifolds. Our work focuses on diffeomorphisms arising from Dehn twists along embedded 3-manifolds and their interaction with Seiberg-Witten theory. These investigations have led to intriguing applications across several areas, including symplectic geometry (related to Torelli symplectomorphisms), algebraic geometry (concerning the monodromy of singularities), and low-dimensional topology (involving exotic diffeomorphisms). This is collaborative work with Hokuto Konno, Jianfeng Lin, and Juan Munoz-Echaniz.