Geometric and algebraic constructions in representation theory

在表示论中，人们将代数对象实现为向量空间的线性变换。通过这种方式，抽象的数学结构可以用它们的元素作为矩阵的具体描述来研究。相反，几何和物理中出现的具体结构可以用抽象代数和组合方法来研究。表象理论从代数、几何学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。不同数学领域之间的相互作用已被证明是非常有成效的，在美学上也是令人满意的。我成功地运用了几何、代数和组合学中的构造法来解决表示论中的分类问题。我的研究主要集中在研究无限维李代数的结构和表示。在许多情况下，一种新的结构使人们能够第一次瞥见新的现象，将无限维物体与其有限维对应物体区分开来。

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.