Martin Widmer

Let $\ell$ and $d$ be a positive integers. By the Cohen Lenstra heuristics

the $\ell$-torsion part of the class groups of degree $d$ number fields should

be "very small" in terms of the discriminant for "almost all" such fields.

However, non-trivial such bounds for all $\ell$ are known only for

$d\leq 5$ due to recent work of Ellenberg, Pierce, and Wood. We explain 

their strategy, and how one can improve their bounds 

for $d=4, 5$. If time permits we also present analogous results for certain

families of arbitrarily large degree. (This is joint work with Christopher Frei).