The Homological Minimal Model Programme (Extension)

自然界中惊人的大部分，其令人难以置信的丰富结构，系统和复杂的生命形式，可以使用科学原理来理解，这些原理要么是支撑，控制，要么可以近似，数学对象称为多项式。这些基本对象建立在坚实的、长期存在的数学基础上，并且具有能够精确而优雅地描述关系的优势。然而，它们看似简单的形式往往掩盖了一个深刻而复杂的基本几何，而这又往往表现出非常违反直觉的行为。这个提议位于多项式及其附属几何变体的框架内。它试图回答一系列开放的问题，在双理性手术，他们的分类，并枚举问题，通过使用新构建的非交换不变量，并使用额外的结构，这些编码区分几何对象到一个更精细的程度。该建议的第一部分寻求通过非交换技术的几何结构的分类，并在过程中提出了某些Jacobi代数的ADE分类。这一点，以及以前的工作，强烈建议在其他领域的结果，第二部分的项目涉及这些，从生成集的纯编织群，新的组合平铺的平面这calculally控制许多结构在代数和几何，与强大的联系考克斯特群。第三部分统一和概括成一个更广泛的框架，这涉及到理解两个单一的触发器和翻转枚举和结构的问题。

A surprisingly large proportion of the natural world, with its incredibly rich structure, systems and complex lifeforms, can be understood using scientific principles that are either underpinned, controlled, or can be approximated by, mathematical objects called polynomials. These fundamental objects are built on solid, long-standing mathematical foundations, and have the advantage of being able to describe relationships with both precision and with grace. Their deceptively simple form, however, often masks a deep and complicated underlying geometry, which in turn often exhibits very counter-intuitive behaviour.This proposal lies within the framework of polynomials, and their attached geometric varieties. It seeks to answer a series of open questions in birational surgeries, their classification, and enumerative questions by using newly constructed noncommutative invariants, and using the additional structure that these encode to distinguish geometric objects to a much finer degree. The first part of this proposal seeks classification of geometric structures via noncommutative techniques, and in process proposes an ADE classification of certain Jacobi algebras. This, and previous work, strongly suggests results in other areas, and the second part of the project involves these, from generating sets of the pure braid groups, to new combinatorial tilings of the plane which conjecturally control many structures in both algebra and geometry, with strong links to Coxeter groups. The third part unifies and generalises into a wider framework, which involves understanding enumerative and structural questions for both singular flops, and for flips.