Tensors
Abstract: A tensor of order d is a element in a tensor product of d vector spaces. One can also think of an order d tensor as a d-dimensional array. For d=1, tensors are just vectors,
and for d=2 they are matrices. Some notions generalize for matrices to tensors. For example, the rank of a tensor generalizes to the notion of tensor rank. But the rank of a tensor
is not as well-behaved and can be difficult to compute. However, there are many applications of the tensor rank. For example, there is a close connection between the rank
of a certain tensor and the complexity of matrix multiplication. In more numerical applications, tensor rank and tensor decompositions have been used in fluorescence spectroscopy,
statistics, machine learning etc. I will discuss the tensor rank and many other notions of ranks such as the border rank, Waring rank, slice rank and more, and some of their applications.