Applications of Topos Theory & Shape Theory

在我的研究生研究中，我打算追求两个(相互关联的)数学主题--拓朴理论和形状理论。让我们从拓朴理论开始。回想一下：(I)任何一阶几何理论都可以唯一地与分类拓扑点相关联(直到等价)和(Ii)如果两个数学理论具有相同的分类拓扑点(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，我们选取了定义在共同分类Topos上的一些拓扑论不变量I(即i是在范畴等价下稳定的拓扑论上的性质/结构)，并分析了I在两个(不同的)理论T和T’中是如何表达的。正如卡拉梅罗所说，这一拓扑论框架形式化了“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在不同数学背景下出现的模糊或暗示类比的方法。在我的研究中，我想概括卡拉梅罗的“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(例如Chris Isham)如何发起了一项雄心勃勃的计划，利用TOPOS理论重新阐述量子物理的基础。特别是，史蒂夫·维克斯博士致力于研究几何逻辑在这一研究计划中的应用。所有这些都表明，在这种情况下，卡拉梅洛的“桥牌”技术可能是一个强大的选择武器。这两种研究方法不仅共享相似的数学元素(特别是，它们都与拓扑学和几何逻辑有关)，而且在现代数学和量子理论之间以及量子理论本身之间也存在许多猜想关系(例如S对偶性)，这些猜想关系反过来可以用卡拉梅罗的桥技术进行有益的理解(因为这些猜想关系本质上是发生在不同数学背景下的提示性类比)。要启发形状理论，请注意，一般数学兴趣的一个重要主题是分析局部数据的结构和全局不变特征之间的关系。特别是，在几何学中，已经开发了许多工具来通过了解全局/复杂对象的局部/简单性质来分析它们，这些工具在朗兰兹计划和量子理论中得到了令人兴奋的应用。但如果当地的结构是病态的呢？那么，关于全球物体，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一个有价值的框架。特别是，我对形状理论在以下方面的应用感到好奇：(I)p-进几何和完美体几何(特别是彼得·肖尔茨对重量单列猜想的重新表述，我觉得这在精神上特别是形状理论的东西)；(Ii)量子结构；以及(Iii)拓扑论(特别是形状理论提出了一种推广Caramello桥的有趣方法1一个有趣的例子是几何朗兰兹对应和S-二元性之间的关系，通过Kapustin和Witten的工作。另一个例子是费曼积分和代数簇的动机之间的关系，就像马尔科利在《费曼动机》中所写的那样。最终，这个项目的希望是：(I)在量子理论及其与现代数学的相互作用方面获得一些有趣的新结果，(Ii)发展某些形状理论和拓扑论工具，并探索它们在解决某些类型问题方面的潜力。

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.