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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570988
• 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.

拉格朗日子流形在物理问题中起着重要的作用：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来理解这种类型的子流形。简而言之，目的是使用最初由弗拉基米尔阿诺德在1980年引入的“拉格朗日配边”概念来系统地理解拉格朗日，类似于限制较少的光滑配边概念在50年代和60年代被使用的方式，从Thom的工作开始，以理解一般流形。这种做法之所以及时，主要有两个原因：首先，Lagaragian子流形的重要性和作用近年来在几何学、数学物理学等领域得到了越来越好的认识，并且借助于Floer理论的工具在其研究中取得了重大进展，其次，最近发现了（在作者和来自苏黎世ETH的Paul Biran的联合工作中），拉格朗日配边法非常适合Floer技术。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边性，因此这个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，这是一个非常强大的现代几何范式，但尚未完全应用于拉格朗日子流形的研究。