Lagrangian Cobordism and Categorification in Lagrangian Topology

拉格朗日拓扑中的拉格朗日配边和分类

• 批准号: 261277-2013
• 负责人: Cornea, Octavian
• 金额: $ 3.21万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=594518
• 关键词: Lagrangian; Cobordism; Categorification; Topology

Lagrangian submanifolds play an important role in problems originating in physics: they appear as boundaries for a variety of mathematical objects that describe physical phenomena. This project is concerned with understanding this type of submanifold from a global point of view. In short, the purpose is to use a notion of ``Lagrangian cobordism'' initially introduced by Vladimir Arnold in 1980 to pursue a systematic understanding of Lagrangians similar to the way the less restrictive notion of smooth cobordism has been used in the '50s and '60s, starting with the work of Thom, to understand general manifolds. There are two main reasons why this approach is timely: first, the importance and the role of Lagaragian submanifolds has been better and better recognized in recent years in geometry, mathematical physics and beyond, and significant progress was made in their study by means of a tool called Floer theory, secondly, it was recently discovered (in joint work of the author and Paul Biran from ETH, Zürich) that Lagrangian cobordism fits very well with the Floer technique. Moreover, simultaneously, other authors, in particular Nadler from Berkeley together with Tanaka from Northwestern, have also looked at this type of cobordism so that this topic has recently received considerable interest and its exploration promises a high impact. The precise approach proposed here is through the prism of categorification, a very powerful modern paradigm in geometry, but one not yet fully implemented to the study of Lagrangian submanifolds.

拉格朗日子流形在源自物理学的问题中扮演着重要的角色：它们作为描述物理现象的各种数学对象的边界出现。这个项目关注的是从全局的角度理解这种类型的子流形。简而言之，我们的目的是使用“拉格朗日协同”这个概念，这个概念最初是由Vladimir Arnold在1980年引入的，目的是为了系统地理解拉格朗日量，类似于50年代和60年代，从Thom的工作开始，人们使用较少限制的光滑协同概念来理解一般流形。这种方法之所以及时，主要有两个原因：首先，拉格朗日子流形的重要性和作用近年来在几何、数学物理和其他领域得到了越来越多的认识，并且通过一种叫做Floer理论的工具在他们的研究中取得了重大进展，其次，最近发现（在作者和来自苏黎世联邦理工学院的Paul Biran的共同工作中）拉格朗日协数论与Floer技术非常适合。此外，与此同时，其他作者，特别是来自伯克利的Nadler和来自西北大学的Tanaka，也研究了这种类型的协同，因此该主题最近受到了相当大的兴趣，其探索有望产生很大的影响。这里提出的精确方法是通过分类的棱镜，这是几何中一个非常强大的现代范例，但尚未完全应用于拉格朗日子流形的研究。