Related papers: How arithmetic and geometry make error correcting …

In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call them Cartier Codes since their construction involves the…

Error-correcting codes over the real field are studied which can locate outlying computational errors when performing approximate computing of real vector--matrix multiplication on resistive crossbars. Prior work has concentrated on…

Following our joint work arXiv:1003.4578 with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also…

In the past few years, substantial progress has been made in the understanding of the algebra of kappa classes on the moduli spaces of curves. My goal here is to provide a short introduction to the new results. Along the way, I will discuss…

Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in…

Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…

Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…

This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Aut\`{o}noma de Barcelona in July 2025. In the first lecture we introduce tropical…

This is an overview of math.AG/0310186, math.AG/0309290, math.AG/0501247, math.AG/0401002 and math.AG/0504584 written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005.

This is a survey describing recents developments in enumerative geometry of curves on projective varieties. Various methods to arrive at results such as Kontsevich's formula for plane rational curves, or Caporaso-Harris's formula for plane…

This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS.

Algebraic geometry codes or Goppa codes are defined with places of degree one. In constructing generalised algebraic geometry codes places of higher degree are used. In this paper we present 41 new codes over GF(16) which improve on the…

Since Serre gave his famous Harvard lectures in 1985 on various aspects of the theory of algebraic curves defined over a finite field, there have been many developments. In this survey article, an overview will be given on the developments…

In this talk the main features of the operator formalism for the $b-c$ systems on general algebraic curves developed in refs. [1-2] are reviewed. The first part of the talk is an introduction to the language of algebraic curves. Some…

Program correctness used to be the main concern of computer software in the early days when formal semantics was a hot topic. But, the word "correct" was afterwards replaced by reliable, robust and trustworthy etc., a tradeoff situation…

We introduce several modifications of conic fitting in Geometric algebra for conics by incorporating additional conditions into the optimisation problem. Each of these extra conditions ensure additional geometric properties of a fitted…

We extend coded distributed computing over finite fields to allow the number of workers to be larger than the field size. We give codes that work for fully general matrix multiplication and show that in this case we serendipitously have…

We present new constructions of quasi-cyclic (QC) and generalized quasi-cyclic (GQC) codes from algebraic curves. Unlike previous approaches based on elliptic curves, our method applies to curves that are Kummer extensions of the rational…

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…