Floer theory beyond Floer (FloerPlus35)

Floer 之外的 Floer 理论 (FloerPlus35)

• 批准号: EP/X030660/1
• 负责人: Ivan Smith
• 金额: $ 203.9万
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX030660%2F1
• 关键词: Floer; theory; beyond; FloerPlus35

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.

35年前，Gromov和Floer提出的全纯曲线理论彻底改变了辛拓扑的方方面面。Floer上同调现在包含了许多普通上同调中存在的代数结构：环结构，精确三角形，等变表亲，Steenrod运算。这些全纯曲线不变量的构造一般依赖于困难的虚拟摄动方法。在过去的几年里，从代数几何到镜像对称性，再到从稳定同伦理论到稳定同伦理论的新技术都进入了这门学科。这一提议的核心是在每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图的理论，这是一种绕过许多通常的技术细节的亏格零曲线理论的新方法。将这些与色同伦理论的输入相结合，产生了基本上新的辛不变量，适应了全纯曲线模空间的奥比福尔德性质。我们将使用它们来建立辛量子(Morava)K-理论，并将其应用于哈密顿的纤颤、乘积、爆破和奇异的辛结构。另外，PI与Sheridan一起发起了一个项目，通过镜面上代数圈的有理等价理论来研究拉格朗日共边。这表明，来自Chow群计算的对无障碍拉格朗日存在的限制，反映了“对Hodge猜想的反例”。在这两条链中，跟踪扭转信息的需要需要使用超越经典全纯曲线不变量的框架。