Title #1: Popescu's nested approximation theorem for polynomial systems with one catalytic variable:

Consider a $(k+1)$-tuple of formal power series ($F(u,t),F_1(t),\ldots,F_k(t)$) completely determined by a Polynomial equtaion $P(F(t,u),F_1(t),\ldots,F_k(t))=0$.
Such a system is called a polynomial equation with one catalytic variable u and often arises in combinatorics when studying Generating functions of combinatorial Objects (e.g. Dyck-Paths or rooted planar graphs). While there are other combinatorial methods, one can use Popescu's nested approximation theorem to show that in this case $F(t,u)$ is an algebraic power series.
The problem is that the general proof for said theorem requires a lot of algebraic geometry and commutative algebra.
In this talk, I will present a special low dimensional case of nested approximation, which is applicable in this setting and has a proof that does not require as many deep results.

Title #2: Background to the Razumov-Stroganov (Ex-)Conjecture

Abstract:
Fully packed loops are certain finite subgraphs of the Z^{^2}-grid. One
wouldn't expect to find parallels to an infinite mathematical object
like a tiling of a semi-infinite cylinder. In this talk we will learn
about some properties of the two objects and use probability theory to
see that they are comparable when talking about noncrossing matchings.