With their discovery of superconductivity in the cuprates some twenty years ago, Bednorz and Muller set in motion new research directions in transition metals oxide materials with many promising applications. But the most promising of all, high temperature superconductivity in the cuprates, has still fallen far short of expectations. The development of new superconducting materials has been hampered by a lack of theoretical guidance, and despite two decades of intensive research there is still no agreement on the basic mechanisms of superconductivity in the cuprates. The recent staggering advances in high-performance computing, however, open new possibilities to investigate this problem. In this talk, I will review recent progress made with simulations of superconductivity in the Hubbard model. I will report on insights we have gained from attempts to link these simulations with first principles electronic structure calculations, and I will discuss possible paths forward towards performing materials specific simulations of superconductivity in the cuprates.

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