Higgs spaces, loop crystals and representation of loop Lie algebras

群的概念来自对给定对象的对称性集合的考虑。相反，给定一个群，我们可以问哪些对象具有对应于该群的一组对称。这样的对象被称为群的表示，表示理论是关于解决寻找所有这些表示的问题。我的工作涉及几何表示理论。也就是说，我感兴趣的是用几何对象上的函数的卷积代数来构造代数对象，如李代数和代数群。这种方法在90年代被证明是非常有成效的，当时许多与群有关的组合对象及其表示，如特征，被用一些变种的几何不变量来解释。然后，它们被用来证明几个重要的猜想。我的研究的主要目的是将这些结果引入一组新的称为环李代数的代数中，并将它们与另一组称为希格斯场的几何对象联系起来。一种新的组合物体，我称之为环状晶体，应该是代数和几何部分之间的关键纽带。我已经在最简单的情况下构建了这个环状晶体，它应该会导致几何猜想的一种新方法。相反，这将为研究表示论提供强有力的新工具。所有这些结果都与其他蓬勃发展的领域如簇代数有许多联系，并且是朗兰兹计划哲学的一部分，该哲学涉及从几何到数论的许多不同的数学领域。

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.