### Description

Chair: Seunghyun Seo

Numerical semigroups are cofinite additive submonoids of the natural numbers motivated by the study of linear Diophantine equations. Through a simple injection to Young diagrams, researchers have used known results about numerical semigroups to answer questions about core partitions. In this talk, we will explore connection between integer partitions and numerical semigroups with a focus on...

Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. Recently, this extremal problem has been studied for different groups $G$ and subsets $J$. For example, using the linear algebra methods, Alon showed the upper bound $D_{G}(J,N)\leq...

For a graph H the reconfiguration/mixing versions of the extension problem for H asks, of a given graph G with a partial H-colouring p, if one can move between different homomorphisms extending p, by changing the image of one vertex at a time.

We characterise the graphs H for which one can do so for any choice of G and p and any pair of homomorphsims extending p.

We also consider the...