Higgs spaces, loop crystals and representation of loop Lie algebras

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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.