Noncommutative toric geometry and multilinear series

在任何一个行业中，当面对一个新问题时，第一步自然是根据自己目前的理解来测试潜在的答案。鉴于数学的目的是回答新的问题，数学家工具箱中的关键工具之一是拥有适当的例子集合，以测试新的问题或理论。一个这样的收集自然出现在代数几何领域的例子，其中研究问题的几何解决方案多项式方程，是由复曲面品种。虽然多项式被认为是相当简单的，但描述复曲面簇的多项式特别简单。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圆锥体的集合。尽管如此，研究环面品种已被证明是一个显着的试验场的问题和代数几何。这一建议的主要目标概括了从经典领域的代数几何的复曲面品种的建设新兴领域的“非交换”代数几何。新的研究领域将提供一个简单的试验场，新的问题，而且，它将提供一种新的方法，以回答一些老的问题，出现在代数几何和理论物理。这些应用的目的是揭示新的光某些相当复杂的代数结构，称为派生类别，自然地关联到代数几何中的几何对象。虽然环面簇是简单的几何对象，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导出范畴的问题，这些环面变体在理论物理学弦理论的研究中自然出现，尽管可能令人惊讶。这里描述的非对易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。

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.