层量子化与有理拉格朗日浸入的相交

通过层量子化研究余切丛中的紧致正合拉格朗日相交（附Tomohiro Asano的附录）

Compact exact Lagrangian intersections in cotangent bundles via sheaf quantization (with an appendix by Tomohiro Asano)

该研究用层论方法证明余切丛中紧致正合拉格朗日子流形横截相交基数的下界，且方法可推广至清洁与退化拉格朗日相交情形。

Exact Lagrangian immersions with a single double point

Let k > 2. We show that if a closed orientable 2k-manifold K, with Euler characteristic not equal to -2, admits an exact Lagrangian immersion into complex Euclidean 2k-space with one transverse double point and no other self-intersections, then K is diffeomorphic to the standard sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with boundary in a monotone Lagrangian submanifold obtained by Lagrange surgery on K.

The classification of Lagrangians nearby the Whitney immersion

The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number $+1.$ This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal cubic curve inside the projective plane, and the latter Weinstein manifold is thus the `standard' neighbourhood of Lagrangian immersions of this type. We classify the Lagrangians inside such a neighbourhood which are homologically essential, and which either are embedded or immersed with a single double point; they are shown to be …

通过微局部层理论研究清洁拉格朗日交集的拓扑约束

Topological constraints on clean Lagrangian intersections via microlocal sheaf theory

本文利用微局部层理论和3维流形理论，证明了清洁拉格朗日交集的结类型受原结类型严格约束，并建立了一个保持Seifert框架的群同态。

Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections

Let Y,Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf O_X to a sheaf of noncommutative algebras and of the sheaves O_Y and O_Z to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf Tor^{O_X}_*(O_Y, O_Z). The induced Gerstenhaber bracket on the Tor-sheaf turns out to be canonically defined; it is independent of the choices of deformations involved. There are similar results for Ext-sheaves as…

Constructing exact Lagrangian immersions with few double points

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding $S^1\times S^2\to\R^6_\st$ with vanishing Maslov class.