In this minicourse, I will present the first examples of separable II$_1$ factors $M$ that admit no non-trivial crossed product decompositions: $M\not\cong B\rtimes\sigma G$, for any trace-preserving action of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. Our approach relies on constructing separable II$_1$ factors $M$ with the strong rigidity property that every embedding of $M$ into $M\overline{\otimes}M$ arises from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$. I will also discuss the main deformation/rigidity and coarseness techniques used in the proof. This is joint work with Adriana Fernández Quero and Hui Tan.