| Résume | On the complex projective line, for any configuration of n distinct marked points and n integers whose sum is -2, there is a meromorphic differential, unique up to rescaling, whose zeros and poles coincide with those marked points and have order prescribed by the integers. Working up to the complex 3-dimensional group of projective transformations, this means that the number of meromorphic differentials with prescribed order of zeros and poles is finite if and only if n=3. Since the sum of the orders needs two be -2, only two cases are left: two zeros and one pole or two poles and one zero. These numbers of differentials can be enriched by allowing extra poles, but with the constraint that their residue vanishes. Problem: compute these two families of integer numbers. In joint works with A. Buryak and D. Zvonkine we provide the answer to this problem and a certain higher genus generalization, together with and intriguing relation to integrable systems. |
