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Noncommutative toric geometry and multilinear series

非交换环面几何和多线性级数

基本信息

批准号：
EP/G004048/1
负责人：
Alastair Craw
金额：
$ 38.34万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG004048%2F1
关键词：
Noncommutative toric geometry multilinear series

项目摘要

When faced with a new question in any walk of life, a natural first step is to test potential answers against one's current understanding. Given that mathematics aims to answer new questions, one of the key tools in a mathematician's toolbox is to have an appropriate collection of examples to hand against which to test new questions or theories. One such collection of examples that arise naturally in the field of algebraic geometry, which studies questions concerning the geometry of solutions to polynomial equations, is provided by toric varieties. While polynomials are regarded as rather simple, the polynomials that describe toric varieties are especially simple. In fact, these curved geometric objects can be encoded by very elementary combinatorial data involving collections of cones with straight sides. Despite this, the study of toric varieties has proved to be a remarkable testing ground for questions and conjectures in algebraic geometry. The primary goal of this proposal generalises the construction of toric varieties from the classical field of algebraic geometry to the emerging field of `noncommutative' algebraic geometry. The new field of study will provide a simple testing ground for new questions and, moreover, it will have provide a new method with which to answer a number of older questions that arise in both algebraic geometry and theoretical physics. These applications aim to shed new light on certain rather complicated algebraic structures, called derived categories, that are associated naturally to geometric objects in algebraic geometry. While toric varieties are simple geometric objects, we are nevertheless unable to answer certain questions about the much more complicated algebraic structure that is encoded in the corresponding derived categories. In particular, we are unable to answer questions about derived categories of certain toric varieties that arise naturally, though perhaps surprisingly, in the study of string theory in theoretical physics. The noncommutative approach described here provides a new, examples-based approach to the study of these complicated structures that provides a concrete approach to certain questions arising in theoretical physics.

在任何行业中，当面对一个新问题时，第一步自然是对照自己现有的理解来测试可能的答案。考虑到数学的目标是回答新问题，数学家工具箱中的关键工具之一就是有一个适当的例子集合来测试新的问题或理论。在研究多项式方程解的几何问题的代数几何领域中自然出现的一个这样的例子集合是由环面变分提供的。虽然多项式被认为是相当简单的，但描述环面变异的多项式尤其简单。事实上，这些弯曲的几何对象可以通过非常基本的组合数据进行编码，这些组合数据涉及具有直边的锥体集合。尽管如此，环面变体的研究已被证明是代数几何问题和猜想的一个显著的试验场。本建议的主要目标是将环变的构造从代数几何的经典领域推广到“非交换”代数几何的新兴领域。新的研究领域将为新问题提供一个简单的试验场，而且，它将提供一种新的方法来回答代数几何和理论物理中出现的一些老问题。这些应用旨在揭示某些相当复杂的代数结构，称为派生范畴，自然地与代数几何中的几何对象相关联。虽然环面变体是简单的几何对象，但我们仍然无法回答有关编码在相应派生范畴中的复杂得多的代数结构的某些问题。特别是，我们无法回答关于某些环面变体的衍生类别的问题，这些问题在理论物理弦理论的研究中自然出现，尽管可能令人惊讶。这里描述的非交换方法为研究这些复杂结构提供了一种新的、基于实例的方法，为理论物理中出现的某些问题提供了一种具体的方法。