Constantin Kogler, a mathematician at the Institute for Advanced Study, presents joint work with Samuel Kittle on self-similar measures, the stationary distributions that emerge from repeatedly applying randomly chosen contracting similarities in Euclidean space. Delivered at the Joint IAS/Princeton Groups and Dynamics Seminar, the talk tackles the two defining questions of the field: what dimension a self-similar measure has, and when it is absolutely continuous, meaning it has a density with respect to Lebesgue measure. Kogler shows that Hochman's dimension theorem still holds under a weakened Diophantine assumption, then walks through new explicit constructions of absolutely continuous measures, including the first inhomogeneous examples in dimension 1 and 2 and constructions for arbitrary algebraic rotations and translations. He closes by strengthening Varju's result on Bernoulli convolutions and extending Lindenstrauss-Varju's work to dimension three and above. The talk assumes familiarity with ergodic theory and geometric measure theory.