Let $S$ be standard graded polynomial ring over a field, $I$ a homogeneous ideal of $S$. A central open question in liaison theory is determining sufficient and necessary conditions for $I$ to be licci. For instance, a question (dubbed "The Dream") raised in the past few years by Huneke, Polini and Ulrich asks whether there exists a characterization of the licci property of $I$ in terms of the graded Betti numbers of $S/I$.

While an example with a negative answer was found about a year ago by A. Boocher, in this talk we show that a strong positive answer holds for ideals generated by the quadrics. A key ingredient is a novel way to identify the Hilbert function of a quadratic licci ideal with a partition of its codimension. We then proceed to characterize all licci ideals whose quotients $S/I$ are Koszul. The talk is based on joint work with Matthew Mastroeni and Jason McCullough.