Homotopical Approaches to Algebraic Structures

在这一建议中，PI试图从几个方面扩展前人对同伦理论的同伦理论的研究。一个令人感兴趣的主要议题是继续进行关于这些结构的更高范畴概括的工作。虽然已经引入了几个这样做的模型，部分原因是代数几何和拓扑量子场论的工作，但重要的是要确定它们都是彼此等价的，类似于PI之前的工作。在相关工作中，这些模型在歌剧世界中的类似正在开发中，再次重要的是看看哪些已知结果可以继续以及如何进行比较。在另一个方向，PI寻求使用同伦论工具来进一步探索霍尔代数和量子群之间的关系。表象理论中的一个长期悬而未决的问题是找到霍尔代数的一个派生版本，从它可以恢复所有相应的量子群。预计同伦理论中的新技术将为解决这一猜想提供新的见解。

In this proposal, the PI seeks to extend previous research on the homotopy theory of homotopy theories in several ways. A major topic of interest is the continuation of work in progress on higher-categorical generalizations of these structures. While a few models have been introduced for doing so, in part motivated by work in algebraic geometry and topological quantum field theory, it is important to establish that they are all equivalent to one another analogous to the PI's previous work. In related work, analogues of these models in the world of operads are being developed, and again it will be important to see which known results can be carried over and how comparisons can be made.In another direction, the PI seeks to use homotopy-theoretic tools to explore further the relationship between Hall algebras and quantum groups. A longtime open question in representation theory is to find a derived version of the Hall algebra from which all of the corresponding quantum group can be recovered. It is anticipated that newly-developed techniques in homotopy theory will provide new insights into solving this conjecture.