Floer theory beyond Floer (FloerPlus35)

Gromov和Floer在35年前提出的全纯曲线理论彻底改变了辛拓扑的各个方面。花上同调现在包含了普通上同调中存在的许多代数结构：环结构、精确三角形、等变表兄妹、Steenrod操作。这些全纯曲线不变量的构造一般依赖于困难的虚摄动方法。在过去的几年里，来自代数几何的新技术，通过镜像对称，并且与稳定同伦理论分开，已经进入了这个主题。这个建议的核心是在每一个新主题中发展一个中心思想。与Abouzaid和McLean一起，PI正在开发一种全局Kuranishi图理论，这是一种绕过许多通常技术细节的属零曲线理论的新方法。将这些与来自色同伦理论的输入相结合，产生了新的辛不变量，适应于全纯曲线模空间的轨道性质。我们将使用这些来建立辛量子（Morava） k理论，并将其应用于哈密顿纤维，产物，膨胀和奇异辛结构。另外，与Sheridan一起，PI发起了一个项目，通过镜面上代数循环的理性等价理论来研究拉格朗日协同性。这表明了来自Chow群计算的无阻碍拉格朗日存在的限制，反映了“霍奇猜想的反例”。在这两种情况下，需要跟踪扭转信息就需要使用超越经典全纯曲线不变量的框架。

The holomorphic curve theory of Gromov and Floer, introduced 35 years ago, revolutionised all aspects of symplectic topology. Floer cohomology now incorporates many of the algebraic structures present in ordinary cohomology: ring structures, exact triangles, equivariant cousins, Steenrod operations. The construction of these holomorphic curve invariants in general relies on difficult virtual perturbation methods. In the last years, new techniques from algebraic geometry, via mirror symmetry, and separately from stable homotopy theory, have entered the subject. This proposal is centred on developing one central idea in each of these new themes.With Abouzaid and McLean, the PI is developing a theory of global Kuranishi charts, a new approach to genus zero curve theory bypassing many of the usual technicalities. Combining these with input from chromatic homotopy theory yields fundamentally new symplectic invariants, adapted to the orbifold nature of holomorphic curve moduli spaces. We will use these to build symplectic quantum (Morava) K-theory, with applications to Hamiltonian fibrations, products, blow-ups and exotic symplectic structures. Separately, with Sheridan, the PI initiated a program to study Lagrangian cobordism via the theory of rational equivalence of algebraic cycles on the mirror. This suggests constraints on the existence of unobstructed Lagrangians coming from Chow group computations, mirroring `counterexamples to the Hodge conjecture'. In both strands, the need to keep track of torsion information necessitates using frameworks going beyond classical holomorphic curve invariants.