AG Diskrete Mathematik - Doppelvortrag

11.06.2024 15:15 - 16:45

Gregor Böhm und Christian Theiner

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

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.

B. Stufler

TU Wien, Dissertantenraum, Freihaus, Turm A, 8. OG., Wiedner Hauptstr. 8-10, 1040 Wien