Lagrangian Cobordism and Categorification in Lagrangian Topology

拉格朗日子流形在起源于物理学的问题中扮演着重要的角色：它们表现为描述物理现象的各种数学对象的边界。这个项目致力于从全球的角度来理解这种类型的子流形。简而言之，其目的是使用弗拉基米尔·阿诺德于1980年首次提出的“拉格朗日协边”的概念来寻求对拉格朗日的系统理解，类似于从托姆的工作开始，在20世纪50年代和60年代使用限制较少的光滑协边理论来理解一般流形的方式。这种方法之所以及时，主要有两个原因：第一，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好的认识，并通过一种名为Floer理论的工具在它们的研究中取得了重大进展，第二，最近发现(在作者和来自苏黎世高等理工学院的Paul Biran的联合工作中)拉格朗日子流形非常适合Floer技巧。此外，与此同时，其他作者，特别是伯克利的纳德勒和西北大学的田中，也研究了这种Cobordism，因此这个话题最近受到了相当大的兴趣，它的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，这是几何学中一种非常强大的现代范式，但尚未完全应用于拉格朗日子流形的研究。

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.