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

Date and Time: Tuesday, June 07, 2005, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Andreas Meyer

Geometric Complexity Theory: The Bilinear Case

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. Since the multiplication map of a given associative algebra (e.g., a matrix algebra) induces a bilinear map, finding lower bounds for the complexity of an algebra means finding lower bounds for the rank of the tensor induced by the multiplication map of this algebra.

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.

Mulmuely and Sohoni suggested 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.

On the other hand, one is also (more generally) interested in the question whether a given tensor lies in the orbit closure of another tensor or not. Another approach to solve this is to use the stability of certain tensors and prove incomparability of such tensors by using Luna's étale slice theorem. Roughly spoken, it says that the orbit of a stable from locally "looks" like a fibre bundle.

We will give a short introduction to both methods and name some results we were able to show by using the latter one.

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