Wolf Barth (1942--2016)

In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his fundamental contributions to the theory of algebraic surfaces and moduli of vector bundles, and include his later work on algebraic surfaces with many singularities, culminating in the famous Barth sextic.Comment: accepted for publication in Jahresbericht der Deutschen Mathematiker-Vereinigung, obituary, 17 pages, 2 figures, 1 phot

Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.Comment: 17 pages, 4 figure

Semantic spaces

Any natural language can be considered as a tool for producing large databases (consisting of texts, written, or discursive). This tool for its description in turn requires other large databases (dictionaries, grammars etc.). Nowadays, the notion of database is associated with computer processing and computer memory. However, a natural language resides also in human brains and functions in human communication, from interpersonal to intergenerational one. We discuss in this survey/research paper mathematical, in particular geometric, constructions, which help to bridge these two worlds. In particular, in this paper we consider the Vector Space Model of semantics based on frequency matrices, as used in Natural Language Processing. We investigate underlying geometries, formulated in terms of Grassmannians, projective spaces, and flag varieties. We formulate the relation between vector space models and semantic spaces based on semic axes in terms of projectability of subvarieties in Grassmannians and projective spaces. We interpret Latent Semantics as a geometric flow on Grassmannians. We also discuss how to formulate G\"ardenfors' notion of "meeting of minds" in our geometric setting.Comment: 32 pages, TeX, 1 eps figur

Construction of Rational Surfaces Yielding Good Codes

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.Comment: 20 pages, 7 figure

Algebraic geometric codes

The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes

On products and powers of linear codes under componentwise multiplication

In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author's talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that could otherwise probably not fit in a regular research paper. On the other hand, more advanced results and applications are only quickly mentioned with references to the literature. We also point out a few open problems. Our presentation alternates between two points of view, which the theory intertwines in an essential way: that of combinatorial coding, and that of algebraic geometry. In appendices that can be read independently, we investigate topics in multilinear algebra over finite fields, notably we establish a criterion for a symmetric multilinear map to admit a symmetric algorithm, or equivalently, for a symmetric tensor to decompose as a sum of elementary symmetric tensors.Comment: 75 pages; expanded version of a talk at AGCT-14 (Luminy), to appear in vol. 637 of Contemporary Math., AMS, Apr. 2015; v3: minor typos corrected in the final "open questions" sectio

Anyons in Geometric Models of Matter

We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.Comment: 22 pages LaTe