Interactions between representation theory, algebraic geometry, and physics

表象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，氢原子的电子轨道在一定程度上是通过与球的对称性相关的表示理论来分类的。表示理论在数学本身也有不同的理论应用。这个领域的一些最现代的发展属于更高代表性理论的保护伞。在这里，问题被重新塑造成几何或抽象的术语，允许应用新的技术。这通常发生在代数几何的框架内：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非常丰富，并导致了一些具有挑战性的问题的解决，如Kazhdan-Lusztig猜想。人们不太了解，但可能同样丰富的是，更高表示理论和理论物理之间正在形成的关系。许多对更高表示理论至关重要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。一个最好的例子是Kapustin和Witten用超级杨-米尔斯理论对几何朗兰兹程序的物理解释，这启发了许多数学家。建议的研究集中在高等表示理论、几何和物理之间的相互作用。它将解决在这些领域的交叉点上出现的自然问题，例如：-利用物理学的思想，我们可以理解高等表示理论中的哪些问题和结构，反之亦然？-高等表示理论中出现的空间的特殊代数几何性质是什么，它们与物理构造是如何联系的？-我们如何进一步扩大高等表示理论和物理之间的联系，并促进跨学科合作？这一提议的一个主要焦点是库仑分支，它是量子场论中出现的空间。最近，Braverman、Finkelberg和Nakajima给它们下了严格的数学定义。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，包括表示论、代数几何、数论、量子群和可积系统。拟议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中的重要数学问题。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性成果。这项研究计划为所有级别的培训提供机会。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.