Mathematics Colloquium: Dax invariants, light bulbs, and isotopies of symplectic structures

Speaker: Boyu Zhang, University of Maryland

Title: Dax invariants, light bulbs, and isotopies of symplectic structures

Abstract:  In this talk, I will present several results about isotopy problems in dimension 4.  First, we give a classification of the isotopy classes of embeddings of $\Sigma$ in $\Sigma\times S^2$ that are geometrically dual to $\{pt\}\times S^2$, where $\Sigma$ is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic. This answers a question of Gabai. Second, we show that the space of symplectic forms on an irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components. This gives the first such example among closed 4-manifolds and answers a question of McDuff-Salamon.  We also show that symplectic forms on a closed 4-manifold with a fixed cohomology class do not admit the h-principle, which answers a question of Cieliebak-Eliashberg-Mishachev.  The proofs are based on a generalization of the Dax invariant to embedded closed surfaces.  This is joint work with Jianfeng Lin, Weiwei Wu, and Yi Xie. 

Time:  Friday, March 6, 3:30pm – 4:30pm

Place:  Exploratory Hall, room 4106 or Zoom