last semester, we introduce three different approaches to solve lattice walk problem, namely, the kernel method, the Tutte invariant method and the Riemann Hilbert problem. There are three main approach in the study of lattice walk problem in algebraic combinatorics. The job of this semester is to introduce some other methods, which are not that commonly use, but they connect to other aspect of mathematics.

Here is the current plan: We will start with the connection between kernel method and cycle lemma in traditional enumerative combinatorics, and the jump to the problem of winding angle, which demonstrates a deep connect between lattice walk and elliptic function. Then we will discuss how to use elliptic function to solve models in M-quadrant cones and obtain a integral representation. Finally, we will go back to the kernel method and show how to find an explicit algebraic expression.