Applications of Topos Theory & Shape Theory

在我的研究生研究中，有两个（相互关联的）数学主题我打算继续研究-拓扑理论和形状理论。回顾：（1）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）;（2）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，那么它们共同的分类拓扑可以作为它们之间传递信息的“桥梁”：粗略地说，我们选择一个拓扑不变量I（即，I是在范畴等价下稳定的topose上的性质/构造）并分析了I在两个（不同的）理论T和T '中的表达。正如Caramello所说，这种拓扑理论框架形式化了“从不同的角度看同一事物”的直觉，并为我们提供了一种严格检查在不同数学背景下出现的模糊或暗示性类比的方法。在我的研究中，我想概括Caramello的“桥”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）如何启动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。特别是，史蒂夫·维克斯博士一直致力于研究几何逻辑在这项研究计划中的应用。所有这一切都表明，Caramello的“桥梁”技术可能是一个强大的武器选择在这种情况下。这两种研究方法不仅有着相似的数学元素（特别是，它们都与拓扑和几何逻辑有关），现代数学和量子理论之间以及量子理论本身之间也有许多明确的关系。（例如S-对偶），这反过来又可以通过Caramello的桥接技术来理解（因为这些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，注意，一般数学兴趣的一个重要主题是分析局部数据的结构与全局不变量之间的关系。功能.特别是在几何学中，已经开发了许多工具来通过理解它们的局部/简单性质来分析全局/复杂对象，并且这些工具在朗兰兹纲领和量子理论中找到了令人兴奋的应用。但如果局部结构是病态的呢？那么，关于全局对象，我们能说些什么呢？这就是形状理论的用武之地，我感兴趣的是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。特别是，我很好奇的应用形状理论：（i）p-adic和perfectoid几何（特别是，彼得·肖尔茨（Peter Scholze）对重量单值猜想（Weight Monodromy Conjecture）的重新表述让我觉得它在精神上特别符合形状理论）;（ii）量子结构;以及（iii）拓扑理论（特别地，形状理论提出了一个有趣的方式推广Caramello的桥1一个有趣的例子是几何朗兰兹对应和S-对偶之间的关系，通过工作的Kapustin和维滕。另一个例子是费曼积分和代数簇的动机之间的关系，正如马科利的“费曼动机”中所写的那样。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问题方面的潜力。

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.