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