Toric methods in homotopy theory

From a historical perspective, algebraic topology came into being as combinatorial topology. Indeed, some of the oldest geometric objects considered by mathematicians, Platonic solids and more general polytopes, are inherently combinatorial in nature. An important feature is that polytopes can be constructed by gluing lower-dimensional, and hence simpler, polytopes.This idea of describing a complicated object by gluing simpler pieces is ubiquitous in modern mathematics. Topological spaces equipped with a triangulation or a more general kind of cell decomposition, sheaves, and localisation results in algebra may serve as examples. The main idea is that the properties of the global object are entirely determined by the properties of the building blocks, and that one can pass from (known) local properties to (unknown) global properties by virtue of sophisticated gluing arguments.Around the mid-70s, algebraic geometers developed the theory of toric varieties, providing a formalism for describing algebraic varieties by rather simple-looking combinatorial objects (collections of cones in a finite-dimensional real vector space). This is in fact another example of the method described before to obtain interesting and complicated global objects by gluing well-understood simple small pieces, and has the added benefit that the abstract gluing process is readily visualised by gluing cones along common faces. A modification of the construction led to the notion of toric manifolds developed by Davis and Januskiewicz, initiating a new branch in topology nowadays known as toric topology . Both the algebraic and the topological side are active research areas today, and provide deep links between such diverse subjects as convex geometry, algebraic geometry, algebra, topology, polytope theory and combinatorics.The proposed research is about the less obvious relation between toric methods and homotopy theory. The connection works both ways. One can use toric methods to attack finiteness issues in algebra, topology and combinatorics. Going the other way, modern homotopy theory provides the means to give new descriptions of objects from toric geometry, or to extend the scope of toric geometry into new and exciting contexts, eg, algebraic geometry over the sphere spectrum , or over the field with one element .

The proposed research is on the borderline of various areas of pure mathematics, and brings together aspects of topology, geometry, algebra and combinatorics. Beneficiaries are, in the first instance, mathematicians working in any of these areas. The results of my research will be presented at international conferences and colloquia in the UK and abroad, and will be published in peer reviewed journals of international standing. Thus the research will reach a broad academic audience in the UK and worldwide. As is the case with all research in pure mathematics, this project is about generating knowledge. Economic impact is indirect and will be achieved largely by two means: firstly, by enhancing the reputation of Queen's University Belfast as a research-active environment, and hence enhancing the reputation of the UK as a leading research nation; secondly, by the influence of research on student education via research led teaching , thus contributing to high quality teaching at Queen's University Belfast.