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This survey article, in honor of G. Tians 60th birthday, is inspired by R. Pandharipandes 2002 note highlighting research directions central to Gromov-Witten theory in algebraic geometry and by G. Tians complex-geometric perspective on pseudoholomorphic curves that lies behind many important developments in symplectic topology since the early 1990s.
Quivers, gauge theories and singular geometries are of great interest in both mathematics and physics. In this note, we collect a few open questions which have arisen in various recent works at the intersection between gauge theories, representation
We construct families of quilted surfaces parametrized by the multiplihedra, and define moduli spaces of pseudoholomorphic quilted disks using the theory of pseudoholomorphic quilts of Wehrheim and Woodward. We prove a gluing theorem for regular, iso
We construct orientations on moduli spaces of pseudoholomorphic quilts with seam conditions in Lagrangian correspondences equipped with relative spin structures and determine the effect of various gluing operations on the orientations. We also invest
We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstruc
Given a symplectic three-fold $(M,omega)$ we show that for a generic almost complex structure $J$ which is compatible with $omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $ggeq 0$ representing a homology class $beta$ in $H