Stability, derived categories, and mirror symmetry

本项目以理论物理为灵感，通过同调代数来理解代数几何中的问题。代数几何是数学中最古老和最丰富的研究领域之一，可以追溯到公元前五世纪。它涉及到对几何对象的理解，称为代数变异，由多项式函数的消失来定义，如圆或双曲线。几个世纪以来，出现了大量用于研究代数变量的技术工具。最近，物理学中的同调代数和镜像对称提供了令人印象深刻的见解。即便如此，紧迫的问题依然存在。一个特别的问题是：当使用同调代数或物理时，可以检测到的两个变体之间的差异的精确程度是多少？本项目旨在阐明这一问题及相关问题。该项目关注于比较派生类别的新工具：窗口。窗口是建立衍生等价和半正交分解的强大框架，但由于非常新，它们仍在发展中。该项目试图通过推广框架和重新解释紧化和核的几何形状产生的函子来推动窗口理论。该项目旨在通过镜像对称将这种新的理解应用于辛拓扑。此外，该项目建议通过核语言将代数几何与非交换代数联系起来，并使用霍尔代数来研究三角化范畴的生成式不变量。

This project strives to understand problems in algebraic geometry through homological algebra, with inspiration from theoretical physics. Algebraic geometry is one of the oldest and richest fields of study in mathematics, dating back to the fifth century B.C. It is concerned with understanding geometric objects, called algebraic varieties, defined by the vanishing of polynomial functions, like circles or hyperbolas. Over the centuries, a vast array of technical tools have arisen for studying algebraic varieties. Relatively recently, homological algebra and mirror symmetry from physics have provided impressive insight. Even so, pressing questions persist. One in particular: What is the precise extent of the differences that can be detected between two varieties when using homological algebra or physics? This project seeks to elucidate this and related questions.The project focuses on a new tool for comparing derived categories: windows. Windows are a robust framework for establishing derived equivalences and semi-orthogonal decompositions, but, being very new, they are still developing. The project seeks to push the theory of windows by generalizing the framework and by reinterpreting the functors arising in terms of the geometry of compactifications and kernels. The project seeks to apply this new understanding to symplectic topology through mirror symmetry. Additionally, the project proposes to relate algebraic geometry to noncommutative algebra through the language of kernels and to use Hall algebras to study generation-style invariants of triangulated categories.