Details of Grant 

EPSRC Reference:	EP/Z534705/1
Title:	The periodic homotopy type of spaces of embeddings
Principal Investigator:	Taggart, Dr N
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Sch of Mathematics and Physics
Organisation:	Queen's University of Belfast
Scheme:	EPSRC Fellowship TFS
Starts:	01 September 2025	Ends:	31 August 2028	Value (£):	309,131
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
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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Organisation Website:	http://www.qub.ac.uk