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Theory Seminar

Tensor rank and stability in representation theory

Daniel Barter University of Michigan, Mathematics
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Matrix multiplication is one of the fundamental operations in mathematics. From a computational perspective, it would be nice to have formulas which multiply n by n matrices using fewer than n^3 multiplications. One such example is Strassen's formula which expresses the product of 2 by 2 matrices using 7 multiplications. The existence of such formulas is intimately related to tensor rank, which has been studied for more than 100 years by algebraic geometers. The majority of the talk will be spent explaining the relationship between matrix multiplication and tensor rank. No background in algebraic geometry will be assumed.

Despite tensor rank being very old, it is not well understood in high dimensions. In 2010, Church and Farb uploaded a paper titled "representation theory and homological stability" to the Arxiv. The basic slogan is this:

Geometry gets harder as dimension increases, unless there is a stable family of groups acting. Then geometry stays the same difficulty a dimension increases

In tensor rank problems, there is always a stable family of groups acting, and in the past few years, some theorems have been proved about tensor rank by exploiting this. For example, see Snowden's paper "syzygies of segre embeddings and delta modules" or Draisma and Eggermont's paper "plucker varieties and higher secants of Sato's Grassmanian" .

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