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

Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."

This is a technical report on the proceedings of the workshop held July 21 to July 25, 2008 at the American Institute of Mathematics, Palo Alto, California, organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We include a list of open problems coming from applications in 4 different areas: signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and holographic algorithms, and entanglement and quantum information theory. We emphasize the interactions between geometry and representation theory and these applied areas

Counting degree-constrained subgraphs and orientations

The goal of this short paper to advertise the method of gauge transformations (aka holographic reduction, reparametrization) that is well-known in statistical physics and computer science, but less known in combinatorics. As an application of it we give a new proof of a theorem of A. Schrijver asserting that the number of Eulerian orientations of a $d$--regular graph on $n$ vertices with even $d$ is at least $\left(\frac{\binom{d}{d/2}}{2^{d/2}}\right)^n$. We also show that a $d$--regular graph with even $d$ has always at least as many Eulerian orientations as $(d/2)$--regular subgraphs

Life as an Explanation of the Measurement Problem

No consensus regarding the universal validity of any particular interpretation of the measurement problem has been reached so far. The problem manifests strongly in various Wigner's-friend-type experiments where different observers experience different realities measuring the same quantum system. But only classical information obeys the second law of thermodynamics and can be perceived solely at the holographic screen of the closed orientable two-dimensional manifold implied by Verlinde's and Landauer's mass-information equivalence equations. I conjecture that biological cell, as a dissipative structure, is the smallest agent capable of processing quantum information through its holographic screen and that this mechanism have been extended by natural evolution to endo- and exosemiosis in multicellular organisms, and further to language of Homo sapiens. Any external stimuli must be measured and classified by the cell in the context of classical information to provide it with an evolutionary gain. Quantum information contained in a pure quantum state cannot be classified, while incoherent mixtures of non-orthogonal quantum states are only partially classifiable. The concept of an unobservable velocity, normal to the holographic screen is introduced. It is shown that it enables to derive the Unruh acceleration as acting normal to the screen, as well as to conveniently relate de Broglie and Compton wavelengths. It follows that the perceived universe, is induced by the set of Pythagorean triples, while all its measurable features, including perceived dimensionality, are set to maximise informational diversity.Comment: This research is incomplete and partially incorrec