Loop and double loop geometry

该项目的长期目标之一是更好地理解库仑规范理论的库仑分支，这导致了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发布了一个预印本，标题为“Kac-Moody仿射格拉斯曼切片的基本基本运算符和嵌入”。在本文中，我们构造嵌入的Kac-Moody仿射格拉斯曼切片到另一个使用基本单极算子。通过这种方式，我们回答了Finkelberg在2018年ICM（国际数学家大会）演讲中提出的一个问题。特别地，我们能够以这种方式构造许多Kac-Moody仿射格拉斯曼切片的Poisson子簇。我们希望基本单极算子将是进一步研究Kac-Moody仿射格拉斯曼切片的重要工具。此外，我在与Auguste Hebert的项目中取得了进一步的进展，了解Kac-Moody仿射Hecke代数的完成。我也取得了进展的项目与安娜Puskas的T-基础上的卡茨穆迪仿射Hecke代数及其关系的长度功能和Bruhat秩序。

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.