In many of my papers, we have been inspired by and found use for geometric methods, as have many other roboticists. One reason I like geometry is the naturalness of that language for describing properties of dynamics – again something physicists have long known and roboticists have likewise come to understand. However, a constant frustration has been the difficulty of efficiently computing with geometric objects, especially when the focus is on abstraction (e.g., global properties of manifolds) rather than, say, efficient location of points in a planar region. In our most recent paper, at R:SS 2014, we found that the computational barriers begin to come down when you bring in more tools from algebra – we used matrix computations based on an algebraic topological formulation of what we were after, and that became quite efficient.
In this context, I found the following quote very interesting:
Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, it will answer any question you like. All you need to do is giving me your soul: give up geometry and you will have this marvellous machine.”
– M. Atiyah, “Mathematics in the 20th century,” in Mathematical Evolutions. Providence, RI: Mathematical Association of America, 2002.