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Spin droplets in the quantum XXZ model in one dimension: In this talk I will describe a recent result proved by Bruno Nachtergaele and myself regarding the existence of droplet states for the quantum XXZ model at zero temperature. The XXZ model is a deformation of the classical Ising model whose ground states are a single interval of down spins in a chain otherwise filled with up spins. We proved that the ground states for the XXZ model have similar structure, but with quantum fluctuations at the endpoints of the interval. Our proof generalized to the quantum spin ring, and the infinite spin chain in sectors of finite number of down spins. It is also a first step towards proving that droplets exist in two dimensions as ground states of the quantum model. Some interesting facts which emerged from our proof are a recovered quantum group symmetry in the thermodynamic limit (it was broken by ++ boundary conditions), as well as the fact that droplets have mass which increases exponentially with their diameter. This will be a thirty minute talk.