Applications of Topos Theory & Shape Theory

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 rigorously
examining 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.