Geometric and algebraic constructions in representation theory

在表示理论中，人们将代数对象理解为向量空间的线性变换。这样，抽象的数学结构就可以用矩阵的具体描述来研究。相反，几何和物理中的具体结构可以用抽象的代数和组合方法来研究。表示理论从代数、几何和组合学中汲取思想和方法，并反过来为这些领域提供应用。数学的不同领域之间的相互作用已经被证明是非常富有成效和审美满足的。我已经成功地运用了几何学、代数和组合学的结构来解决表示理论中的分类问题。我的大部分研究集中在研究无限维李代数的结构和表示。在许多情况下，一个新的结构允许我们第一次瞥见将无限维对象与有限维对象区分开来的新现象。

In representation theory one realizes algebraic objects as linear transformations of vector spaces. In this way abstract mathematical structures can be studied using a concrete description of their elements as matrices. Conversely, concrete structures arising in geometry and physics can be studied by abstract algebraic and combinatorial methods. Representation theory draws ideas and methods from algebra, geometry, and combinatorics and in return provides applications to each of these areas. The interplay between different areas of mathematics has proved extremely fruitful and aesthetically fulfilling. I have successfully employed constructions coming from geometry, algebra, and combinatorics to classification problems in representation theory. Most of my research is focused on studying the structure and representations of infinite dimensional Lie algebras. In a number of cases a new construction allowed a first glimpse into new phenomena distinguishing the infinite dimensional objects from their finite dimensional counterparts.