Representation theorists seek to quantify, analyze, and predict how a group of motions---like rotations or reflections---act on physical and geometric objects. The fundamental problem in geometric representation theory is to construct representations using geometric tools. The fundamental problem in combinatorial representation theory is to understand a representation explicitly in terms of essential combinatorial parameters: for instance to decompose a given representation into the fundamental building blocks of irreducible representations. Both kinds of representation theorists seek to understand representations, but a problem that may seem answered to geometric representation theorists, such as `find a natural representation', is only partially answered to combinatorial representation theorists, who seek to `analyze explicitly a given representation'.
This proposal addresses several problems in geometric and combinatorial representation theory, using tools from combinatorics, algebraic geometry, and topology. The broader impact includes work in many other areas of mathematics, including knot theory and commutative algebra, as well as fields like mathematical physics. Educationally, the PI will establish programs for graduate student retention and excellence, in both research and teaching.
