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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

Geometry and the complexity of matrix multiplication


Author: J. M. Landsberg
Journal: Bull. Amer. Math. Soc. 45 (2008), 247-284
MSC (2000): Primary 68Q17
Published electronically: January 7, 2008
MathSciNet review: 2383305
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Abstract: We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.


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Additional Information

J. M. Landsberg
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: jml@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0273-0979-08-01176-2
PII: S 0273-0979(08)01176-2
Keywords: Border rank, complexity of matrix multiplication, secant varieties
Received by editor(s): November 17, 2006
Received by editor(s) in revised form: March 12, 2007
Published electronically: January 7, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.