# Groups, Geometry, and Representations

Groups arise naturally as symmetries of geometric, combinatorial and analytic structures and can be used to understand geometry and topology. Representation theory deals with how these symmetries give rise to families of operators on vector spaces. Conversely, one can study abstract groups, other algebraic structures, and their representations by using geometric techniques and ultimately by treating the structures themselves as geometric objects. These complementary perspectives typify the spirit and nature of the work of the *Groups, Geometry and Representations Academic Group*.

Members of the group study and do research on various problems in algebra, geometry and representation theory and are motivated by the interconnections between these fields. We have some overlapping interests with research conducted by other academic groups, particularly the Discrete Geometry and Combinatorics Group, the Matrix Analysis and Linear Algebra Group, and the Coding and Number Theory Group. We welcome collaboration with them. We have several affiliate members coming from these groups.

**Areas of Research:** Finite, discrete and continuous groups. Lie groups and Lie algebras. Algebraic combinatorics. Representation theory. Galois theory. Algebraic geometry and invariant theory. Differential geometry and geometric analysis. Riemannian, complex, symplectic and Kähler geometries. Topology and algebraic topology.