# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, June 08, 2004, 12:15 pm

**Duration**: This information is not available in the database

**Location**: This information is not available in the database

**Speaker**: Andreas Meyer

## An Introduction to Bilinear Complexity Theory and on a new Approach using Geometric Invariant Theory

In order to find new lower bounds in bilinear complexity, e.g., for the matrix multiplication, we want to combine former ideas due to Strassen with the new approach Mulmuley and Sohoni suggested in 2001. Strassen proved that a tensor has border rank *r* if and only if it lies in the projective *SL*_{r}*SL_{r}*SL_{r}-orbit closure of the unit tensor of rank *r*. Hence, the crucial problem for finding new lower bounds is to show that such a tensor does not lie in a specific orbit closure, which means that this problem is reduced to the orbit closure problem analyzed in Mumfords book on Geometric Invariant Theory.

The new approach is to prove this by constructing explicit representation theoretic obstructions, that is, irreducible modules whose multiplicities in the coordinate ring of the orbit closure of a tensor *t* exceeds that of the orbit closure of the unit tensor. This is a wild problem in the general case but seems to behave better in some special cases, i.e., if the tensors are stable resp. partially stable.

The talk will give an introduction to bilinear complexity theory and a short illustration of the idea mentioned above.

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