Dualities in the representation theory and geometry of loop groups

表象理论是对出现在数学、物理或任何其他自然科学中的对称性的研究。它的主要目标是对可能出现在具有给定对称性的系统中的数据进行先验描述。对于“经典”对称性(有限群、紧致李群等)，如今已有大量这样的数据。近年来，在无限维环群的背景下出现了各种新的对称类型。令人惊讶的是，这些结构与其他数学领域之间发现了显著的相互关系。“对偶”的反复出现属于当今数学中最深层次的谜团。该项目旨在通过结合环群表示理论和相关结构(如模表示或量子表示)的两种方法来研究这些对偶。

Representation theory is the study of symmetries appearing in mathematics, physics or in any other natural science. Its main objective is to give an a priori description of data that can appear in a system that exhibits a given symmetry. For „classical" symmetries (finite groups, compact Lie groups, etc.) a large amount of such data is known today. In recent years various new classes of symmetries appeared in the context of infinite dimensional loop groups. Quite surprisingly, significant interrelations between these structures and other areas of mathematics were found. The recurrent appearance of “dualities” belongs to the deepest mysteries in today's mathematics. The project intends to study these dualities by combining two approaches towards the representation theory of loop groups and related structures such as modular or quantum representations.