Loop and double loop geometry

该项目的长期目标之一是更好地理解颤振规范理论的库仑分支，它产生了Kac-Moody仿射格拉斯曼切片的定义。具体来说，目标是明确地理解它们的几何形状。2022年11月，我和我的合作者Alex Weekes发表了一篇题为“Kac-Moody仿射格拉斯曼切片的基本单极算子和嵌入”的预印本。本文利用基本单极子算子构造了Kac-Moody仿射格拉斯曼切片的嵌入。这样，我们就回答了芬克尔伯格在2018年国际数学家大会（ICM）上的演讲中提出的一个问题。特别地，我们能够用这种方法构造许多Kac-Moody仿射格拉斯曼切片的泊松亚种。我们的希望是，基本单极子算子将成为进一步研究Kac-Moody仿射格拉斯曼切片的关键工具。此外，我与Auguste Hebert在理解Kac-Moody仿射Hecke代数的补全方面取得了进一步的进展。我和Anna Puskas在Kac-Moody仿射Hecke代数的t基及其与长度函数和Bruhat阶的关系的项目中也取得了进展。

One of the long term goals of this project is to better understand the Coulomb branches of quiver gauge theories, which give rise to a definition of Kac-Moody affine Grassmannian slices. Specifically the goal is to understand their geometry explicitly. In November 2022, my collaborator, Alex Weekes, and I posted a preprint titled "Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices". In this paper, we construct embeddings of Kac-Moody affine Grassmannian slices into one another using Fundamental Monopole Operators. In this way, we answer a question posed by Finkelberg in his 2018 address at the ICM (International Congress of Mathematicians). In particular, we are able to construct many Poisson subvarieties of Kac-Moody Affine Grassmannian slices in this way. Our hope is that the Fundamental Monopole Operators will be a crucial tool in further investigations of Kac-Moody Affine Grassmannian slices.Additionally, I have made further progress in the project with Auguste Hebert on understanding completions of Kac-Moody Affine Hecke algebras. I have also made progress in the project with Anna Puskas on the T-basis of Kac-Moody Affine Hecke algebras and its relationship with the length function and Bruhat order.