On a basic level, tensor triangular geometry (tt-geometry) associates to a (symmetric) tensor triangulated category a geometric invariant, its Zariski spectrum, roughly by treating the category as a commutative ring. On a slightly higher level, the spectrum is the universal object which captures the support theory on the category, shadowing sheaf supports in algebraic geometry. Either way, given a tensor triangulated category $T$, calculating its spectrum \$\operatorname{Spec} T\$ gives an important global information about the category but also appears to be a highly nontrivial problem. I’ll mention some known calculations which exploit the richness of the endomorphism ring of the unit object, \$\operatorname{End}^*_T(1)\$, going back to the work of Quillen, but also highlight some emerging new (and old) cases where \$\operatorname{End}^*_T(1)\$ is rather small and cannot possibly capture the entire spectrum of \$T\$.