Tensor rank and secant varieties 1000-1M25TRSW

Contents of the course.
(1) Definition of secant variety in projective geometry. As examples, some different notions of rank, especially tensor rank, Waring rank and border rank. The notion of general rank.
(2) Definition of defective varieties and Terracini’s lemma. Defectivity of curves and surfaces.
(3) The powerful tools of scheme theory: Castelnuovo’s exact sequence, the Horace method and Alexander-Hirschowitz’ theorem for symmetric tensors.
(4) From symmetric to partially symmetric tensors: defectivity of Segre-Veronese varieties.
(5) A modern result: defectivity of varieties that are equivariant under a group action.
(6) Commutative algebra in action: the apolarity lemma.
(7) Computing the Waring rank: Sylvester’s algorithm for binary forms, the rank of monomials, the rank of some cubics and quartics.
(8) Open problems: the complexity of matrix multiplication, Strassen’s conjecture and Comon’s conjecture.