Geometric and algebraic constructions in representation theory

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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.