EPSRC Reference: 
EP/K036696/1 
Title: 
Monopole moduli spaces and manifolds with corners 
Principal Investigator: 
Singer, Professor M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
UCL 
Scheme: 
Standard Research 
Starts: 
01 January 2014 
Ends: 
31 December 2017 
Value (£): 
354,199

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Nonabelian magnetic monopoles are particlelike solutions of a differential equation that live on ordinary threedimensional space. It is known that some of the solutions correspond to widely separated particles, but that as the separations decrease, the particles lose their individual identity and can no longer be distinguished. These monopoles are part of a physical theory (YangMillsHiggs theory) which cannot be solved exactly but which predicts that monopoles will interact nontrivially with each other when they move. When they move at low speeds, their motion is well approximated by geodesics on a certain curved space, much as particles in Einstein's theory of general relativity move under the influence of gravity. This project will study the distance function (metric) underlying this curved space M, and will study in particular how it looks at large distances.
By undertaking a detailed study of the metric, we shall be also be able to study natural differential equations which are defined on the space M. This is of interest in the study of differential equations generally, since the largescale structure of M is rather complicated, and can only be built up recursively. The behaviour of differential equations on M is important since they are necessary for the understanding of the quantum theory of the original YangMillsHiggs theory. The important ingredient here again is the largescale structure of M and its metric.
This research will be important not only for understanding monopoles and features of the YangMillsHiggs theory of which they are a part, but also for the development of a suite of sophisticated tools for understanding differential equations in this type of setting. These tools will be useful in similar problems and can be expected to have an importance beyond their applications to the theory of monopoles.

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