The Homological Minimal Model Program

20世纪数学最令人震惊的发现之一是认识到，与曲面理论相反，与许多结果和流行的智慧相比，三维几何极小模型可能是奇异的。因此，“最好的”答案，实际上是我们所瞄准的空间，本身可能有奇点。该项目旨在从同源的观点理解这些结构，并将提供新的信息，即使在充分研究的情况下。当前的派生范畴技术主要集中在极小模型光滑的情况下，而这个项目将推动我们的理论理解和计算能力深入到奇异集。本项目的第一部分发展了围绕重构代数和极大修改代数的同调代数，加强了它们，并在许多新的方向上扩展了它们。然后将这些新技术应用于代数几何，并将作为特例恢复三维几何极小模型的存在。然而，这些技术将提供更多，在这项工作的主体中，我们使用这些额外的数据来获得关于曲线和翻转收缩的信息；这随后允许我们以算法方式运行最小模型程序的各个方面。

One of the more startling discoveries of twentieth century mathematics was the realization that, in contrast to the theory of surfaces, and in contrast to many results and the prevailing wisdom, in dimension three geometric minimal models may be singular. Thus the `best' answers, indeed the spaces we are aiming for, may themselves have singularities. This project aims to understand these structures from a homological viewpoint, and will give new information even in the well-studied cases. Current derived category techniques focus on situations where the minimal models are smooth, whereas this project will push both our theoretical understanding and our computational ability deep into the singular setting.The first part of this project develops the homological algebra surrounding reconstruction algebras and maximal modification algebras, strengthening them and extending them in many new directions. These new techniques will then be applied to algebraic geometry, and will recover the existence of geometric minimal models in dimension three as a special case. The techniques, however, will give much more, and in the main body of this work we use this extra data to obtain information regarding contraction of curves and flops; this then allows us to run aspects of the minimal model program in an algorithmic manner.