# EP/H018743/1 - Toric methods in homotopy theory

Research Perspectives grant details from EPSRC portfolio

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Research Perspectives grant details from EPSRC portfolio

Principal Investigator - School of Mathematics and Physics, Queen's University of Belfast

Start Date

10/2010

End Date

09/2012

Value

£101,318

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Summary and Description of the grant

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 .

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Name: Toric methods in homotopy theory - EP/H018743/1

Start Date: 2010-10-01T00:00:00+00:00

End Date: 2012-09-30T00:00:00+00:00

Organization: Queen's University of Belfast

Description: 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 i ...