Loop and double loop geometry

这个项目的长期目标之一是更好地理解箭图规范理论的库仑分支，它引起了Kac-Moody仿射Grassman切片的定义。具体地说，目标是明确了解它们的几何图形。2022年11月，我和我的合作者Alex Weekes发布了一本预印本，标题是《基本单极子算子和Kac-Moody仿射Grassmanian切片的嵌入》。本文利用基本单极算子构造了Kac-Moody仿射Grassman切片的相互嵌入。通过这种方式，我们回答了芬克尔伯格在2018年国际数学家大会(ICM)上发表演讲时提出的一个问题。特别地，我们可以用这种方法构造Kac-Moody仿射Grassman切片的许多Poisson亚簇。我们希望基本单极子算子将成为进一步研究Kac-Moody Affine Grassmanian切片的重要工具。此外，我与Auguste Hebert在理解Kac-Moody Affine Hecke代数的完备化方面取得了进一步的进展。我还与Anna Puskas在Kac-Moody Affine Hecke代数的T-基及其与长度函数和Bruhat序的关系方面取得了进展。

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.