Lagrangian Cobordism and Categorification in Lagrangian Topology

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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.