Details of Grant 

EPSRC Reference:	EP/I02610X/1
Title:	Higgs spaces, loop crystals and representation of loop Lie algebras
Principal Investigator:	Pouchin, Mr G
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Sch of Mathematics
Organisation:	University of Edinburgh
Scheme:	Postdoc Research Fellowship
Starts:	01 September 2011	Ends:	31 August 2014	Value (£):	232,411
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 15 Feb 2011 PDRF Maths Interview Panel Announced 01 Feb 2011 PDRF Maths Sift Panel Announced
Summary on Grant Application Form
The notion of group comes from the consideration of the set of symmetries of a given object. Conversely, given a group, we can ask which objects have a set of symmetries corresponding to this group. Such an object is called a representation of the group, and representation theory is about solving the problem of finding all these representations.My work concerns geometric representation theory. Namely, I am interested in constructing algebraic objects such as Lie algebras and algebraic groups in terms of convolution algebras of functions on geometric objects. This method has proven to be very fruitful in the 90s, when many combinatorial objects associated to groups and their representations, such as characters, were interpreted in terms of geometric invariants of some varieties. They were then used to prove several important conjectures.The main purpose of my research is to introduce these kind of results to a new set of algebras called loop Lie algebras, and to relate them to another set of geometric objects called Higgs fields. A new combinatorial object, which I call a loop crystal, should be the crucial link between the algebraic and geometric parts. This loop crystal, which I have already constructed in the simplest possible case, should lead to a new approach to conjectures in geometry. Conversely, this should provide powerful new tools to study representation theory.All these results have many connections to other flourishing domains such as cluster algebras, and is part of the Langlands Program philosophy, which involves a lot a different areas of mathematics, from geometry to number theory.
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Organisation Website:	http://www.ed.ac.uk