Description:

My plan is to discuss a fairly broad spectrum of ideas relating to different kinds of combinatorial questions, often originating from mathematical physics or from the study of moduli spaces.  Exactly which topics will be covered will depend to some extent on the interests and level of the participants, so the list below should be taken only as an indication of some of the possibilities.  My hope is that some parts of the course will be understandable to everybody, even Qiuzhen College students near the beginning of their studies, while others will be more advanced, but hopefully still profitable even to hearers who cannot follow them completely. One possibility, if the hearers like the idea and it turns out to be practical, could be to discuss some topics in two lectures in the same week, with the Tuesday lecture being more elementary and giving some of the necessary background and with the Thursday talk going deeper.

Here are some of the topics that will be (or may be) covered:

* General principles about counting (use of generating functions; always take symmetries into account; Lagrange inversion formula)

* Combinatorial applications, starting with Cayley's formula for the number of trees with a given number of vertices,  then counting trivalent and more general kinds of graphs; Lambert ring.

* Counting coverings of surfaces.  Hurwitz numbers.

* A crash course (maybe two 1 1/2 hour lectures) on the representation theory of finite groups and in particular of symmetric groups.

* A short introduction to modular and quasimodular forms.

* Applications of quasimodularity in combinatorial problems such as the Dijkgraaf-Kaneko-Zagier formula for ramified coverings of a torus; its generalization by Bloch-Okounkov; counting problems on flat surfaces.

* The moduli space of curves, its Euler characteristic, and the method of matrix models (though this may be treated more briefly since it is a somewhat more advanced topic).

It is my hope that the course will be at least to some extent interactive and that the participants will feel free to ask many questions during or after the lectures.

Registration: https://www.wjx.top/vm/rermSLK.aspx#