Arun Ram's research centers around the connections between combinatorics, symmetry groups, geometry, and harmonic analysis. The main goal is to understand how symmetry groups can act on vector spaces and whether these actions can be understood combinatorially. Combinatorics is the art of counting, and some of the counting problems that arise in this type of work are questions like:

(a) How many different ways can a symmetry group act on a vector space?

(b) What is the size of the vector space that the group is acting on?

(c) How many pieces can you divide the vector space into and still manage to get an action of the group on each piece?


          Almost by definition, symmetry groups arise from geometry, since, after all, they are defined as the groups of transformations that preserve some geometrical symmetry. Thus, understanding this geometry is an inherent part of understanding how these groups act on vector spaces. Sometimes the geometry that enters into play is very discrete (like the geometry of cubes and tetrahedra) and sometimes it is very continuous (like the geometry of rotating spheres).


          It all comes together in the kind of work that we are trying to do. One day we might be working with cohomology and sheaf theory, the next day with lots of integrals, and the third day by counting the number of graphs that satisfy certain conditions--All with the goal of understanding symmetry groups.