We will present various approaches to constructing interesting tensors through algebraic methods. Our main motivation is the problem of fast matrix multiplication. In particular, we will show several new tensors that prove that one can multiply $n\times n$ matrices in time below $n^{2.5}$. The talk is based on a joint work with Roser Homs Pons, Joachim Jelisiejew and Tim Seynnaeve.
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