=== Group theory reformulation of matrix multiplication algorithms ===
[[Henry Cohn]], [[Robert Kleinberg]], [[Balázs Szegedy]] and [[Chris Umans]] put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different [[group theory|group-theoretic]] context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the [[Triple product property|triple product property (TPP)]]. They also give conjectures that, if true, would imply that there are matrix multiplication algorithms with essentially quadratic complexity. This implies that the optimal exponent of matrix multiplication is 2, which most researchers believe is indeed the case.<ref name="robinson"/> One such conjecture is that families of [[wreath product]]s of [[Abelian group]]s with symmetric groups realise families of subset triples with a simultaneous version of the TPP.<ref>{{Cite book | last1 = Cohn | first1 = H. | last2 = Kleinberg | first2 = R. | last3 = Szegedy | first3 = B. | last4 = Umans | first4 = C. | chapter = Group-theoretic Algorithms for Matrix Multiplication | doi = 10.1109/SFCS.2005.39 | title = 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) | pages = 379 | year = 2005 | isbn = 0-7695-2468-0 | s2cid = 41278294 | url = https://authors.library.caltech.edu/23966/ }}</ref><ref>{{cite book |first1=Henry |last1=Cohn, |first2=Chris |last2=Umans. |chapter=A Group-theoretic Approach to Fast Matrix Multiplication.{{arxiv|arxiv=math.GR/0307321}}.''|title=Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science'', 11–14 October 2003, Cambridge, MA, |publisher=IEEE Computer Society,pp. |pages=438–449 |doi=10.1109/SFCS.2003.1238217 |isbn=0-7695-2040-5 }}</ref> Several of their conjectures have since been disproven by Blasiak, Cohn, Church, Grochow, Naslund, Sawin, and Umans using the Slice Rank method.<ref>{{Cite book | last1 = Blasiak | first1 = J. | last2 = Cohn | first2 = H. | last3 = Church | first3 = T. | last4 = Grochow | first4 = J. | last5 = Naslund | first5= E. | last6 = Sawin | first6 = W. | last7=Umans | first7= C.| chapter= On cap sets and the group-theoretic approach to matrix multiplication | doi = 10.19086/da.1245 | title = Discrete Analysis | year = 2017 | page = 1245 | s2cid = 9687868 | url = http://discreteanalysisjournal.com/article/1245-on-cap-sets-and-the-group-theoretic-approach-to-matrix-multiplication}}</ref> Further, Alon, Shpilka and [[Chris Umans]] have recently shown that some of these conjectures implying fast matrix multiplication are incompatible with another plausible conjecture, the [[sunflower conjecture]].<ref>[[{{cite journal |journal=Electronic Colloquium on Computational Complexity |date=April 2011 |author1-link=Noga Alon |last1=Alon]], |first1=N. |last2=Shpilka, |first2=A. |last3=Umans,[|first3=C. |url=http://eccc.hpi-web.de/report/2011/067/ |title=On Sunflowers and Matrix Multiplication] |id=TR11-067 }}</ref>
