Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties

These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.Comment: To appear in the proceedings of the 2008 Srni Winter School on Geometry and Physic

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 $X^n\subset \Bbb C\Bbb P^{n+a}$ 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 $TX$ 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