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A fundamental and remarkably subtle question in symplectic geometry is “when does one symplectic manifold embed in another?”. There are two paths to approaching such problems: constructing embeddings, and obstructing embeddings; I will focus on the latter. Connections with algebraic geometry emerged from work of Biran and McDuff-Polterovich relating embeddings of disjoint unions of balls (i.e. ball packing problems) and the algebraic geometry of blowups of $\mathbb{P}^2$, and this talk will describe work over the last few years continuing in the vein of employing algebraic techniques to study symplectic embedding problems. We describe a sequence of invariants of a polarised algebraic surface that obstruct symplectic embeddings, in many interesting cases sharply. Using this perspective we prove a combinatorial bound on the Gromov width of toric surfaces conjectured by Averkov-Nill-Hofscheier, and discuss related phenomena in algebraic positivity inspired by these symplectic findings.

Pre-talk at 10:30am with same Zoom information.