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...Geometry and complexity theory this material will be published by cambridge university press as j m landsberg prepublication version is free to view download for personal use only not redistribution resale or in derivative works c copyright contents preface ix chapter introduction matrix multiplication separation of algebraic classes how nd hay a haystack the problem explicitness role i rst lower bounds multi linear algebra strassen s equations needed generalization koszul attenings rank ii asymptotic upper facts denitions from bini capovani lotti romani sch onhage laser method cohn umans program v vi iii explicit decompositions via symmetry two decomposition families mhni n revisited invariants associated hni cyclic z invariant hi alternating least squares als secant varieties additional geometric language border iv tensors more classical indirectly dened substitution coppersmith winograd ranks structure algebras valiant hypothesis permanent determinant polynomials circuits vp vnp fla...