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Grassmann and flag varieties lead many lives in pure and applied mathematics. In this talk,  I will focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure of the algebraic complexity is the amount of complex critical points of the corresponding optimization problem. After an exposition of different realizations of these manifolds as algebraic varieties I will present a sample of optimization problems over them and compute their algebraic complexity. This is based on joint work with Hannah Friedman.