Obstructions to the existence of monotone Lagrangian embeddings into cotangent bundles of manifolds fibered over the circle

Bourgeois contact structures: tightness, fillability and applications

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of w…

Local systems on the free loop space and finiteness of the Hofer-Zehnder capacity

In this article we examine under which conditions symplectic homology with local coefficients of a unit disk bundle $D^*M$ vanishes. For instance this is the case if the Hurewicz map $π_2(M)\to H_2(M;\mathbb{Z})$ is nonzero. As an application we prove finiteness of the $π_1$-sensitive Hofer-Zehnder capacity of unit disk bundles in these cases. We also prove uniruledness for such cotangent bundles. Moreover, we find an obstruction to the existence of $H$-space structures on general topological spaces, formulated in terms of local systems.

奇异辛流形上的可积系统：从局部到全局

Integrable systems on singular symplectic manifolds: From local to global

本文研究一阶奇异辛流形上的可积系统，证明了折叠辛可积系统的作用-角定理，给出两类奇异辛结构下可积系统的全局构造并发现源于临界集拓扑的全局作用-角坐标存在障碍。

Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory

A local normal form theorem for smooth equivariant maps between Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence of this equivariant normal form theorem, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi spac…

Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory

We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplect…

二维球面与双曲空间的超凯勒结合

Hyperkaehler Marriage of the two sphere with the hyperbolic space

本文研究二维球面超凯勒延拓在双曲圆盘上继承的复结构，证明其嵌入超凯勒流形并非全纯映射。