we will discuss some new density results about the $2$-primary part of class groups of quadratic number fields and how they fit into the framework of the cohen-lenstra heuristics. let $\mathrm{cl}(d)$ denote the class group of the quadratic number field of discriminant $d$. the first result is that the density of the set of prime numbers $p\equiv -1\bmod 4$ for which $\mathrm{cl}(-8p)$ has an element of order $16$ is equal to $1/16$. this is the first density result about the $16$-rank of class groups in a family of number fields. the second result is that in the set of fundamental discriminants of the form $-4pq$ (resp. $8pq$), where $p\equiv q \equiv 1\bmod 4$ are prime numbers and for which $\mathrm{cl}(-4pq)$ (resp. $\mathrm{cl}(8pq)$) has $4$-rank equal to $2$, the subset of those discriminants for which $\mathrm{cl}(-4pq)$ (resp. $\mathrm{cl}(8pq)$) has an element of order $8$ has lower density at least $1/4$ (resp. $1/8$). we will briefly explain the ideas behind the proofs of these results and emphasize the role played by general bilinear sum estimates. \newline\newline note: the speaker will give a prep-talk for graduate 2022年亚洲世界杯预选赛 in ap&m 7421 at 1:15pm. all graduate 2022年亚洲世界杯预选赛 interested in number theory are strongly encouraged to attend.