Today, Maria gave a talk for the department in connection to the seminar “Meet DTU Compute”. She talked about her work algebraic curves and their uses in coding theory. Title and abstract are given below.
Title: Algebraic curves in coding theory
Abstract: Whether colors in the rainbow or notes in a musical scale, there is a natural human desire to categorize objects, and classifying shapes by their geometric properties has always been a fundamental mathematical research area. Here shapes are grouped into basic geometric objects such as points, lines, ﬁgures. However, most geometrical objects occurring in science are so complex that they defy this traditional mathematical description making it hard to classify them. One solution is to use algebra to see what geometric objects have in common, which entails describing them as a set of equations. In this light, algebraic curves are the simplest objects one can study, as in the plane they are given by one single equation.
For a long time, the study of algebraic curves was the province of pure mathematicians. But then, in a series of three papers in the period 1977-1982, Goppa found important applications of algebraic curves over finite fields, and especially of those having many rational points, to coding theory (reliable communication). Since then, many other applications followed naturally.
In this presentation, after a general introduction to coding theory and algebraic curves, the main ideas behind Goppa’s construction will be described together with some more recent applications of algebraic curves in Information Theory.