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In recent years the study of tensors, i.e. multiarrays, became a topic of an extensive research in applied and pure mathematics. Many problems in tensors are variations of problems in matrices, and many numerical and theoretical approaches to solve these problems using tools and results from matrices. In this talk I will survey some results and open problems mostly for 3-tensors. We will discuss the following topics.

  1. Rank and border of tensors.
  2. Results and conjectures.
  3. Matrix multiplication.
  4. Approximations of tensors : (a) Rank one approximation, (b) Perron-Frobenius theorem, (c) Rank $(R_1,R_2,R_3)$ approximations, (d) CUR approximations, (e) Low rank approximation of tensors.
  5. Scaling of nonnegative tensors.

A variant of this talk is available at

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46