Appears in collection : Combinatorics and Arithmetic for Physics : Special Days

The algorithmic method of Creative Telescoping turns out to be an extremely use- ful tool in experimental mathematics, when dealing with concrete mathematical problems. As striking examples, it can be used to compute and prove automati- cally: a recurrence satisfied by any binomial sum (like the Ap ́ery numbers), the equality of two period functions (in the sense of Kontsevich and Zagier), or a re- currence for the moments of a measure. In this talk, I will explain some theory behind Creative Telescoping, and show how it can be applied in practice on a problem originating from biological physics. The problem concerns the shape of biomembranes, such as blood cells, and examines the uniqueness of the variational Helfrich problem in the case of genus 1 with a prescribed isoperimetric ratio. This question boils down to computing the surface area and volume of a projection of the Clifford torus in terms of Gaussian hyper- geometric functions. We tackle this using Creative Telescoping, and then prove that the rescaled ratio of these functions is monotonically increasing. The talk will be based on joint work with Alin Bostan and Thomas Yu