8,752 research outputs found
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
On Degenerate Secant and Tangential Varieties and Local Differential Geometry
We study the local differential geometry of varieties with degenerate secant and tangential varieties. We show that the
second fundamental form of a smooth variety with degenerate tangential variety
is subject to certain rank restrictions. The rank restrictions imply a slightly
refined version of Zak's theorem on linear normality and a short proof of the
Zak-Fantecchi theorem on the superadditivity of multisecant defects. We show
there is a vector bundle defined over general points of whose fibers carry
the structure of a Clifford algebra. This structure implies additional
restrictions of the size of the secant defect. The Clifford algebra structure,
combined with further local computations, yields a new proof of Zak's theorem
on Severi varieties that is substantially shorter than the original. We also
prove local and global results on the dimension of the Gauss image of
degenerate tangential varieties, refining the results in [GH].Comment: Exposition altered according to the helpful recommendations of the
referee. AMSTe
New lower bounds for the rank of matrix multiplication
The rank of the matrix multiplication operator for nxn matrices is one of the
most studied quantities in algebraic complexity theory. I prove that the rank
is at least n^2-o(n^2). More precisely, for any integer p\leq n -1, the rank is
at least (3- 1/(p+1))n^2-(1+2p\binom{2p}{p-1})n. The previous lower bound, due
to Blaser, was 5n^2/2-3n (the case p=1).
The new bounds improve Blaser's bound for all n>84. I also prove lower bounds
for rectangular matrices significantly better than the the previous bound.Comment: Completely rewritten, mistake in error term in previous version
corrected. To appear in SICOM
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