Interactions between representation theory, algebraic geometry, and physics

表征理论是研究对称性的数学领域，是许多科学研究领域的基础。例如，氢原子的电子轨道部分是通过与球体对称性相关的表示理论来分类的。表征理论在数学本身也有不同的理论应用。这一领域的一些最现代的发展都属于高等表征理论的范畴。在这里，问题被重新定义为几何或抽象术语，允许新技术的应用。这通常发生在代数几何的框架中：研究多项式方程组的解。高等表示理论和代数几何之间的相互作用是非常丰富的，并导致解决具有挑战性的问题，如Kazhdan-Lusztig猜想。人们对高等表征理论和理论物理学之间的关系了解较少，但可能同样丰富。许多作为高级表征理论中心的空间和概念也出现在量子场论中，物理学的视角为新的证明、技术和直觉打开了大门。一个典型的例子是Kapustin和Witten用超级Yang-Mills理论对几何朗兰兹规划的物理解释，这启发了许多数学家。提出的研究重点是高等表征理论、几何和物理之间的相互作用。它将解决这些领域交叉出现的自然问题，例如：-我们可以使用物理学的思想来理解高等表征理论中的哪些问题和结构，反之亦然？-在高等表征理论中产生的空间的特殊代数几何性质是什么？它们与物理结构有什么关系？-如何进一步拓展高等表征理论与物理学的联系，促进跨学科合作？这个提议的主要焦点是库仑分支，这是量子场论中产生的空间。最近，布雷弗曼、芬克尔伯格和中岛给出了一个严格的数学定义。随后的工作建立了库仑分支与数学的几个领域之间的联系，包括表示理论、代数几何、数论、量子群和可积系统。提出的研究旨在发展和理解这些联系，并应用库仑分支技术来解决这些领域的重要数学问题。这项研究的结果将引起数学家和物理学家的兴趣，并为发展中的研究领域贡献基础成果。这个研究项目为各级培训提供了机会。它将使学生和博士后接触到一个非常活跃的研究领域，并促进萨斯喀彻温省新研究专业知识的发展。

Representation theory is an area of mathematics which deals with the study of symmetry, and is fundamental to many areas of scientific study. For example, the electron orbitals of a hydrogen atom are classified in part through the representation theory associated to the symmetries of a sphere. Representation theory also has diverse theoretical applications within mathematics itself. Some of the most modern developments in this field fall under the umbrella of higher representation theory. Here, problems are recast in geometric or abstract terms, allowing for the application of novel techniques. This often takes place in the framework of algebraic geometry: the study of solutions of systems of polynomial equations. The interplay between higher representation theory and algebraic geometry is extremely rich, and has led to the resolution of challenging problems such as the Kazhdan-Lusztig Conjectures. Less understood, but perhaps equally as rich, are emerging relations between higher representation theory and theoretical physics. Many of the spaces and concepts that are central to higher representation theory also appear in quantum field theory, and the perspective of physics opens the door to new proofs, techniques, and intuition. A prime example is Kapustin and Witten's physical interpretation of the Geometric Langlands program in terms of super Yang-Mills theory, which has inspired many mathematicians. The proposed research focuses on the interactions between higher representation theory, geometry, and physics. It will address natural questions which arise at the intersection of these fields, such as: - What problems and structures in higher representation theory can we understand using ideas from physics, and vice versa? - What are the special algebro-geometric properties of spaces arising in higher representation theory, and how do they relate to physical constructions? - How can we further expand the connections between higher representation theory and physics, and foster interdisciplinary collaboration? A primary focus of this proposal is Coulomb branches, which are spaces arising in quantum field theory. They were very recently given a rigourous mathematical definition by Braverman, Finkelberg, and Nakajima. Subsequent work has established ties between Coulomb branches and several areas of mathematics, including representation theory, algebraic geometry, number theory, quantum groups, and integrable systems. The proposed research aims to develop and understand these connections, and apply Coulomb branch techniques to resolve important mathematical problems in these fields. The results of this research will be of interest to mathematicians and to physicists, and contribute foundational results to a developing area of research. This research program offers opportunities for training at all levels. It will expose students and postdocs to a very active field of study, and foster the development of new research expertise in Saskatchewan.