Applications of Topos Theory & Shape Theory

拓扑斯理论的应用

• 批准号: 2041753
• 金额: --
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2041753
• 关键词: Applications; Topos; Theory; Shape

There are two (inter-related) mathematical themes I intend to pursue in my postgraduate research - topos theory and shape theory.Let's start with topos theory. Recall that: (i) Any 1st-order geometric theory can be uniquely associated with a classifying topos (up to equivalence) and (ii) Two mathematical theories are said to be Morita-equivalent if they have the same classifying topos (up to equivalence). In her research, Olivia Caramello proposed that if T and T' are Morita-equivalent 1st-order geometric theories, then their common classifying topos can be used as a "bridge" for transferring information between them: roughly speaking, we pick some topos-theoretic invariant I(i.e. I is a property/construction on toposes that is stable under categorical equivalence) that is defined on the common classifying topos and analyze how I is expressed in the two (different) theories T and T'. As Caramello remarked, this topos-theoretic framework formalizes the intuition of "looking at the same thing from different perspectives", and gives us a way of rigorouslyexamining vague or suggestive analogies that occur in different mathematical contexts. In my research I would like to generalise Caramello's "bridge" technique as well as apply it to interesting mathematical contexts. One exciting example is how theoretical physicists (e.g. Chris Isham) have initiated an ambitious project to reformulate the foundations of quantum physics using topos theory. In particular, Dr Steve Vickers has worked on investigating the applications of geometric logic to this research programme. All this suggests that Caramello's "bridge" technique might be a powerful weapon of choice in this context. Not only do both research approaches share similar mathematical elements (in particular, they are both concerned with toposes and geometric logic), there are also many conjectured relationships between modern mathematics and quantum theory1 as well as within quantum theory itself (e.g. S-duality), which in turn may be profitably understood using Caramello's bridge technique (since these conjectured relationships are essentially suggestive analogies occurring in different mathematical contexts).To motivate shape theory, note that an important theme of general mathematical interest is to analyse the relationship between the structure of local data and global invariant features. In particular, in geometry, many tools have been developed to analyse global/complicated objects through understanding their local/simple nature, and such tools have found exciting applications in the Langlands programme and quantum theory. But what if the local structure is pathological? What can we say about the global object then? This is where shape theory comes in, and I am interested in how shape-theoretic thinking might provide a valuable framework for investigating various mathematical questions. In particular, I am curious about the applications of shape theory to: (i) p-adic and perfectoid geometry (in particular, Peter Scholze's reformulation of the Weight Monodromy Conjecture strikes me as something that is particularly shape-theoretic in spirit); (ii) quantum structures; and (iii) topos theory (in particular, shape theory suggests an interesting way of generalizing Caramello's bridge 1 One interesting example would be the relationship between geometric Langlands correspondence and S-duality via the work of Kapustin and Witten. Another example would be the relationship between Feynman integrals and the motives of algebraic varieties, as written about in Marcolli's "Feynman Motives". technique).Ultimately, the hope of this project will be to: (i) get some interesting new results about quantum theory and its interactions with modern mathematics, and (ii) develop certain shape theoretic and topos-theoretic tools, and investigate their potential in tackling certain kinds of questions.

在我的研究生研究中，有两个（相互关联的）数学主题——拓扑理论和形状理论。让我们从拓扑理论开始。回想一下：(i)任何一阶几何理论都可以唯一地与一个分类拓扑相关联（直到等价）；（ii）如果两个数学理论具有相同的分类拓扑（直到等价），则称它们是森田等价的。Olivia Caramello在她的研究中提出，如果T和T′是morita等价的一阶几何理论，那么它们的共同分类拓扑可以作为它们之间传递信息的“桥梁”：粗略地说，我们选择一些拓扑理论不变量I(即。I是在常见分类拓扑上定义的在范畴等价下稳定的拓扑属性/结构，并分析I在两种（不同的）理论T和T'中是如何表达的。正如Caramello所说，这种拓扑理论框架形式化了“从不同角度看同一事物”的直觉，并为我们提供了一种严格检查不同数学环境中出现的模糊或暗示性类比的方法。在我的研究中，我想推广Caramello的“桥梁”技术，并将其应用于有趣的数学背景。一个令人兴奋的例子是，理论物理学家（如克里斯·伊沙姆）发起了一个雄心勃勃的项目，利用拓扑理论重新制定量子物理学的基础。特别值得一提的是，Steve Vickers博士致力于研究几何逻辑在该研究项目中的应用。所有这些都表明，在这种情况下，Caramello的“桥梁”技术可能是一种强大的武器。这两种研究方法不仅具有相似的数学元素（特别是，它们都与拓扑和几何逻辑有关），而且在现代数学和量子理论之间以及量子理论本身（例如s -对偶性）中也有许多推测关系，这些关系反过来可以利用Caramello的桥技术来理解（因为这些推测关系本质上是发生在不同数学背景下的暗示性类比）。为了激发形状理论，请注意，一般数学兴趣的一个重要主题是分析局部数据结构与全局不变特征之间的关系。特别是在几何学中，人们开发了许多工具，通过理解局部/简单的性质来分析全局/复杂的物体，这些工具在朗兰兹程序和量子理论中得到了令人兴奋的应用。但如果局部结构是病态的呢？那么我们对全局对象能说些什么呢？这就是形状理论出现的地方，我对形状理论思维如何为研究各种数学问题提供有价值的框架感兴趣。特别是，我对形状理论在以下方面的应用感到好奇：(I) p进几何和完美几何（特别是，彼得·肖尔兹（Peter Scholze）对权重单性猜想的重新表述给我的印象是，它在精神上特别具有形状理论的性质）；（ii）量子结构；（iii）拓扑理论（特别是形状理论）提出了一种有趣的推广Caramello桥的方法1一个有趣的例子是通过Kapustin和Witten的工作，几何朗兰兹对应和s -对偶之间的关系。另一个例子是费曼积分和代数变量动机之间的关系，正如Marcolli在《费曼动机》中所写的那样。技术)。最终，这个项目的希望将是：(i)获得一些关于量子理论及其与现代数学相互作用的有趣的新结果，（ii）开发某些形状理论和拓扑理论工具，并研究它们在解决某些类型问题方面的潜力。