The search for universal models began in the early twentieth century
when Hausdorff showed that there is a universal linear order of
cardinality $\aleph_{n+1}$ if $2^{\aleph_n}= \aleph_{n + 1}$, i.e., a
linear order $U$ of cardinality $\aleph_{n+1}$ such that every linear
order of cardinality $\aleph_{n+1}$ embeds in $U$. In this talk, we will
study universal models in several classes of abelian groups and modules
with respect to pure embeddings. In particular, we will present a
complete solution below $\aleph_\omega$, with the exception of
$\aleph_0$ and $\aleph_1$, to Problem 5.1 in page 181 of \emph{Abelian
Groups} by L\'{a}szl\'{o} Fuchs, which asks to find the cardinals
$\lambda$ such that there is a universal abelian p-group for purity of
cardinality $\lambda$. The solution presented will use both
model-theoretic and set-theoretic ideas.
The talk will take place online via Zoom. It is necessary to register via the web form.