权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Applications of homotopy theory to algebraic geometry

同伦理论在代数几何中的应用

基本信息

批准号：
EP/I004130/2
负责人：
Jonathan Pridham
金额：
$ 41.11万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI004130%2F2
关键词：
Applications homotopy theory algebraic geometry

项目摘要

In homotopy theory, topological spaces (i.e. shapes) are regarded as being the same if we can deform continuously from one to the other. Algebraic varieties are spaces defined by polynomial equations, often over the complex numbers; studying their homotopy theory means trying to tell which topological spaces can be deformed continuously to get algebraic varieties, or when a continuous map between algebraic varieties can be continuously deformed to a map defined by polynomials.If the polynomials defining a variety are rational numbers (i.e. fractions), this automatically gives the complex variety a group of symmetries, called the Galois group. Although these symmetries are not continuous (i.e. nearby points can be sent far apart), they preserve something called the etale topology. This is an abstract concept which looks somewhat unnatural, butbehaves well enough to preserve many of the topological features of the variety. Part of my project will involve investigating how the Galois group interacts with the etale topology. I also study algebraic varieties in finite and mixed characteristics. Finite characteristics are universes in which the rules of arithmetic are modified by choosing a prime number p, and setting it to zero. For instance, in characteristic 3 the equation 1+1+1=0 holds. In mixed characteristic, p need not be 0, but the sequence 1,p, p^2, p^3 ... converges to 0.Although classical geometry of varieties does not make sense in finite and mixed characteristics, the etale topology provides a suitable alternative, allowing us to gain much valuable insight into the behaviour of the Galois group. This is an area which I find fascinating, as much topological intuition still works in contexts far removed from real and complex geometry. Indeed, many results in complex geometry have been motivated by phenomena observed in finite characteristic.Moduli spaces parametrise classes of geometric objects, and can themselves often be given geometric structures, similar to those of algebraic varieties. This structure tends to misbehave at points parametrising objects with a lot of symmetry. To obviate this difficulty, algebraic geometers work with moduli stacks, which parametrise the symmetries as well as the objects. Sometimes the symmetries can themselves have symmetries and so on, giving rise to infinity stacks.Usually, the dimension of a moduli stack can be calculated by naively counting the degrees of freedom in defining the geometric object it parametrises. However, the space usually contains singularities (points where the space is not smooth), and regions of different dimensions. Partially inspired by ideas from theoretical physics, it has been conjectured that every moduli stack can be extended to a derived moduli stack, which would have the expected dimension, but with some of the dimensions only virtual. Extending to these virtual dimensions also removes the singularities, a phenomenon known as hidden smoothness . Different classification problems can give rise to the same moduli stack, but different derived moduli stacks. Much of my work will be to try to construct derived moduli stacks for a large class of problems. This has important applications in algebraic geometry, as there are many problems for which the moduli stacks are unmanageable, but which should become accessible using derived moduli stacks. I will also seek to investigate the geometry and behaviour of derived stacks themselves.A common thread through the various aspects of my project will be to find ways of applying powerful ideas and techniques from a branch of topology, namely homotopy theory, in contexts where they would not, at first sight, appear to be relevant.

在同伦理论中，如果拓扑空间（即形状）可以连续地从一个变形到另一个，则认为拓扑空间是相同的。代数变量是由多项式方程定义的空间，通常在复数上；研究它们的同伦理论意味着试图判断哪些拓扑空间可以连续变形以得到代数变体，或者代数变体之间的连续映射何时可以连续变形为多项式定义的映射。如果定义一个变量的多项式是有理数（即分数），这将自动给这个复数变量一组对称，称为伽罗瓦群。尽管这些对称不是连续的（即附近的点可以相距很远），但它们保留了一种被称为etale拓扑的东西。这是一个抽象的概念，看起来有些不自然，但表现得很好，足以保留该品种的许多拓扑特征。我项目的一部分将涉及调查伽罗瓦组如何与以太拓扑相互作用。我也研究有限和混合特征的代数变异。有限特征是这样一种宇宙，在这种宇宙中，通过选择一个素数p并将其设置为零来修改算术规则。例如，在特征3中，等式1+1+1=0成立。在混合特征中，p不一定是0，但序列1，p, p^2, p^3…收敛于0。尽管变种的经典几何在有限和混合特征中没有意义，但ettale拓扑提供了一个合适的替代方案，使我们能够获得对伽罗瓦群行为的许多有价值的见解。这是一个我觉得很吸引人的领域，因为很多拓扑直觉仍然在远离真实和复杂几何的环境中起作用。事实上，复杂几何中的许多结果都是由在有限特征中观察到的现象引起的。模空间参数化几何对象的类别，并且通常可以给出类似于代数变体的几何结构。这种结构在具有大量对称性的参数化对象的点上往往表现不佳。为了避免这个困难，代数几何学家使用模堆，它参数化了对称和物体。有时对称本身也有对称，以此类推，产生无限堆叠。通常，模堆栈的维数可以通过简单地计算其参数化的几何对象的自由度来计算。然而，空间通常包含奇异点（空间不光滑的点）和不同维数的区域。部分受到理论物理思想的启发，人们推测每个模栈都可以扩展为派生模栈，它将具有预期的维度，但其中一些维度只有虚的。扩展到这些虚拟维度也消除了奇点，一种被称为隐藏平滑的现象。不同的分类问题可以得到相同的模堆，但可以得到不同的派生模堆。我的大部分工作将是尝试为一大类问题构造派生的模栈。这在代数几何中有重要的应用，因为有许多问题的模栈是无法管理的，但应该可以使用派生的模栈来访问。我还将试图研究衍生堆栈本身的几何形状和行为。贯穿我项目各个方面的一个共同主线将是找到一种方法，将来自拓扑学分支的强大思想和技术，即同伦理论，应用于乍一看似乎不相关的环境中。