Mathematicians are taking apart the abstract spaces where one shape sits inside another, using tools that break these spaces into simpler pieces like a prism splits light into colours. This matters because manifolds—high-dimensional analogues of the curved space we live in—are fundamental to topology and appear across mathematics. Understanding how one manifold can be embedded inside another has been studied since the mid-twentieth century and is currently seeing a revival. The gap this project fills is a lack of methods to fully compute the homotopy type of spaces of embeddings, which describes how embeddings behave up to continuous deformation. The research is purely curiosity-driven fundamental science. It combines two existing techniques—functor calculus, which breaks mathematical objects into manageable pieces, and chromatic homotopy theory, which splits spaces into frequency-like components—to weave them together in a novel way. Success would give mathematicians a complete description of the homotopy type of spaces of embeddings, enabling concrete computations in manifold topology. While no immediate practical application exists, similar fundamental work in homotopy theory has historically underpinned advances in fields from quantum field theory to data analysis.
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Manifolds are, in many ways, the building blocks of topology and appear in virtually every area of mathematics. They are high-dimensional analogues of the space in which we live: to the inhabitants, manifolds look flat just as the earth appears flat to us, but in reality, they can have a much wilder global shape. One approach to the study of manifolds is to study their embeddings, that is, how a copy of one manifold can sit inside another. This way of studying manifolds has been employed since the mid-twentieth century and is currently seeing a great revival. The primary aim of this project is to use and develop modern techniques in homotopy theory to compute the homotopy type of spaces of embeddings, or in other words, calculate how embeddings behave up to homotopy. The project will use two main tools, the first of which is functor calculus. Functor calculus is an abstraction of the ideas of differential calculus to a categorical setting where functions on the real line are replaced by the notion of a functor: i.e., a "function" which sends one mathematical object to another mathematical object, i.e., a topological space being sent to its fundamental group. The calculus of functors provides a method for breaking a functor into smaller pieces that are easier to handle and gives a prescribed way to build the original functor from these pieces. There are three well-known versions of functor calculus employed in homotopy theory: Goodwillie calculus, orthogonal calculus, and embedding calculus, and this project will utilise all three and their interactions to dismantle spaces of embeddings into smaller pieces which are easier to manipulate. The second primary tool is chromatic homotopy theory. At its heart, the purpose of chromatic homotopy theory is similar to that of functor calculus: it splits spaces of objects into "frequencies", much like how a prism separates white light into colours. From these "frequencies," one may reconstruct the original space. Typically, these "frequencies" capture the periodic homotopy type of the space of embeddings. At its core, the project will examine the interaction between these two theories. Both functor calculus and chromatic homotopy theory provide filtrations on a space by easier-to-manipulate pieces, but how do these filtrations interact? Much of this project will explore yoga relating to various calculi and chromatic homotopy theory. The project aims to use the interactions of these two methods to completely decompose the space of embeddings into pieces that we understand and can compute and then reassemble our computations to give a complete description of the homotopy type of spaces of embeddings. Although some interactions between chromatic homotopy and functor calculi are known, we will weave a novel web between all of the theories so that by travelling along one of the threads, one can play these theories against each other to make concrete computations in manifold topology. Start date
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