The Homological Minimal Model Program

20世纪世纪数学的一个更令人吃惊的发现是认识到，与曲面理论相反，与许多结果和流行的智慧相反，三维几何最小模型可能是奇异的。因此，"最佳"答案，实际上我们所追求的空间，本身可能具有奇点。这个项目的目的是从同源的角度来理解这些结构，并将提供新的信息，即使在充分研究的情况下。当前的导出范畴技术集中在最小模型是光滑的情况下，而这个项目将推动我们的理论理解和我们的计算能力深入到奇异设置。这个项目的第一部分开发了围绕重构代数和最大修改代数的同调代数，加强它们并将它们扩展到许多新的方向。这些新的技术将被应用到代数几何，并将恢复存在的几何最小模型在三维作为一个特殊情况。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触发器收缩的信息;这使我们能够以算法的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.