Noncommutative toric geometry and multilinear series

在各行各业面对新问题时，第一步自然是对照自己目前的理解来测试可能的答案。鉴于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组适当的例子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由Toric变体提供。虽然多项式被认为是相当简单的，但描述环状簇的多项式特别简单。事实上，这些弯曲的几何对象可以由非常基本的组合数据编码，这些组合数据涉及具有直边的圆锥体集合。尽管如此，环面簇的研究已被证明是代数几何中问题和猜想的一个非凡的试验场。这一建议的主要目的是将环簇的构造从经典的代数几何领域推广到新兴的“非对易”代数几何领域。新的研究领域将为新问题提供一个简单的试验场，此外，它还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旧问题。这些应用旨在阐明某些相当复杂的代数结构，称为派生范畴，它们与代数几何中的几何对象自然相关。虽然圆环簇是简单的几何对象，但我们无法回答某些关于更复杂的代数结构的问题，这些代数结构编码在相应的派生范畴中。特别是，我们无法回答在理论物理中弦理论的研究中自然产生的关于某些环状变体的派生范畴的问题，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的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.