Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) being used in computer science? Any research I have done on this topic leads me to some sort of applied math concept. I know that there are many pure mathematicians in physics (i.e. in string theory, studying Hamiltonionan mechanics, Quantum Field Theory and so on) and am wondering where the pure mathematicians in computer science fit in.

Edit: I am looking for factual answers, something along the line of, Hamiltonian mechanics is to symplectic geometry, as computer graphics is to ... or Calabi-Yau manifolds are to string theory as Artificial Intelligence is to ... I want to know the pure math concepts being used in current computer science.

Many objects in Computer Graphics are given by polygonal meshes. For processing these, techniques from Discrete Differential Geometry apply. See also:

• The Discrete Differential Geometry Forum

• Polygon Mesh Processing, especially Differential Geometry slides

• (Discrete) Differential Geometry slides

I will echo what lhf said in that Meshes are very important. I am currently building a small workshop type class for mathematicians to learn some 3D design/modeling, with the hope to 3D print some of our objects. To do so, we are using Rhino, which is a really nice program (and in free trial beta for macs at the moment) that uses something called Non-uniform rational B-spline (NURBS) to give polynomial curves to a set of points (in two dimensions) or a surface (in 3). Personally, I do not know all that much about them, but from a computer science view, they cut down on the amount of data required to make a curve and therefore, use less memory. With NURBS, you can edit the control points of a surface and have the program give a smooth approximation quickly.