Exapended title: **A rapidly divergent tale about Hurwitz, Lambert, Frobenius, enumerative geometry and quasi modular forms.**

The course starts in April. Dates of the course and course notes are available here

**Indicative program:**

Week 1: Introduction to Hurwitz theory, enumeration of graphs for fixed degree

Week 2: The Lambert Space, Enumeration of Hurwitz numbers for fixed degree

Week 3: Enumeration of graphs for fixed genus, Enumeration of Hurwitz numbers for fixed genus

Week 4: Intermezzo: Review of representation theory of finite groups à la Zagier. Coverings in terms of representation theory and Frobenius theorem.

Week 5: Review of modular and quasimodular forms. The quasi-modularity of mirror symmetry for elliptic curves.

Week 6: q-Brackets and quasimodular forms. The Bloch-Okounkov theorem.

Week 7: Cumulants, moments and the Möller transform.

Week 8: The Eskin-Okounkov evaluation. Asymptotics of very rapidly divergent series.

**Description of the course:**

Hurwitz theory is an extremely rich topic at the frontier of Enumerative Algebraic Geometry, Representation Theory, Topology, String theory, Integrable systems and Topological Recursion.

The course will introduce many useful techniques that there employed to solve enumerative problems around that area (mostly developed by Zagier, Dubrovin, and collaborators). These techniques involve a set of smartly crafted combinatorial tricks, differential equations, saddle/subtle point expansions, quasi-moduli forms and Fock spaces from physics.

Rich of these tools and examples, you might be able to apply properly adapted versions of these techniques in the geometric enumerative problem you’ll be after, during your PhD or in your professional academic career. If instead you’re into problems of a different nature, the course could still serve as a general-culture-type source of inspiration to get the gist of neighbouring subjects.

Based on a course given by Don Zagier at the Institut Henry Poincaré (Paris), in occasion of a thematic trimester on Topological Recursion, Probability and Combinatorics in 2017.