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Abstract: Lattices are fundamental objects in physics, mathematics and computer science. Starting from a cubic lattice, say, we can perturb the structure by linear transformations (shearing, stretching, rotating) to obtain new lattices. In this series of lectures, I will discuss the resulting "space of lattices", the dynamics of group actions on this space, natural probability measures, as well as some powerful applications to long-standing problems in various areas of mathematics and mathematical physics.
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