# Postdocs at the Section for Mathematics

Prasant Singh

The topic of this project is error-correcting codes coming from Grassmann and Schubert varieties. Grassmann and Schubert varieties are algebraic varieties whose points correspond to linear subspaces of a given vector space over a general field satisfying varying intersection conditions. These varieties have been studied over the field of complex numbers since the 19th century and are considered to be classical examples of algebraic varieties. When considering a finite field as field of definition, these varieties come up in algebraic combinatorics.

The study of error-correcting codes coming from Grasmann varieties was started in the late 1980s, while a corresponding study in case of Schubert varieties was initiated in the year 2000. Despite this, many elementary properties of these codes are still unknown especially in the Schubert case. Recently Beelen-Ghorpade-Høholdt (2012), Beelen-Pinero (2016), Pinero (2017)  found that these codes exhibit a structure akin to the highly successful class of LDPC codes, which makes it in principle possible to decode Grassmann and Schubert codes using LDPC-decoding inspired methods. The goal of this project is to increase our understanding of the theoretical properties of these codes as well as to gain insight in their decoding.

Duration: 15 November 2017 – 14 November 2019.
Funding body: HCØrsted COFUND Postdoc programme.

Mrinmoy Datta

Intersection of hypersurfaces - the fundament of algebraic coding theory

For a prime power q, let GF(q) denote a finite field with q elements. Let X be a projective algebraic set defined over GF(q) by r homogeneous polynomials, each of degree d>0, in m+1 variables with coefficients in GF(q). Suppose the polynomials are linearly independent over GF(q). One of the main problem of this project is to determine the maximum number of GF(q)- rational points X may admit. Incidentally, this question is closely related to finding the generalized Hamming weights of a well known family of codes, which is known as projective Reed-Muller codes.

The other main problem which is of interest in this project is to find the the weight distribution of the generalized Reed-Muller codes and projective Reed-Muller codes. Formulating geometrically, this problem is equivalent to finding a the number of hypersurfaces (defined over GF(q) that admit a given number of GF(q)-rational points.

Duration: 1. june 2016 -- 31. may 2018
Funding body: Danish Council for Independent Research: Natural Sciences Postdoc programme

Vincent Neiger

Multi-structured linear systems and uses in decoding algorithms

The aim of this project is to design efficient methods for solving mathematical problems which arises in particular in error-correcting codes, which are used to ensure reliable data transmission over unreliable or noisy communication channels: digital television, Blu-Ray discs, bar codes, etc. The family of problems we focus on is central in symbolic computation, and revolves around solving univariate and multivariate polynomial equations. It can be viewed in linear algebra as multi-structured linear system solving. While recent progress has been achieved, the currently fastest algorithms only take advantage of one level of structure of the input equations. This project aims at better assessing the difficulty of handling several level of structures by drawing links with other problems such as multivariate evaluation or modular composition, and at investigating for first solutions in specific cases. At the same time, an efficient implementation of previously known fast algorithms for the one-level case will be undertaken.

Optimization of energy harvesting systems by scattering methods

This project is aimed to develop and apply techniques from Arnold Diffusion in order to benefit diffusing trajectories in energy harvesting systems based on oscillators. When an oscillator is periodically perturbed, KAM theorem guaranties the existence of invariant curves. These curves cannot be crosses by trajectories and, hence, become boundaries for the growth (and loss) of energy. In other words, an oscillator can only vary its energy on the order of the magnitude of the periodic forcing. However, when two or more oscillators are coupled, these KAM tori do not have enough dimension to separate the space and there might exist trajectories accumulating O(1) energy for arbitrarily small magnitude of the forcing. This is know as "Arnold Diffusion", and is something desirable in energy harvesting system, as it maximizes the energy abortion from a source. Classically, Arnold Diffusion has has been observed in the study of instabilities in celestial mechanics allowing, for instance, to explain the lack of asteroids in certain regions of the asteroid belt of the solar system.

Duration of the postdoc: 23/09/2015 - 22/09/2018.
Funding body: HCØrsted COFUND Postdoc programme

Nurdagül Anbar

Algebraic curves with many rational points

Having many applications in other branches of mathematics, special interest arises on algebraic curves over finite fields with many rational points. In this respect, the main problem is to determine Ihara's constant, i.e. the asymptotic bound for the ratio of the number of rational points and the genus of a curve, for the case of non-square finite fields. Recently Bassa-Beelen-Garcia-Stichtenoth gave a substantial improvement for previously known results on the lower bound of this constant for nonprime cases using special curves on Drinfeld modular varieties. In this project, I am firstly planning to further investigate the varying additional structure of the Drinfeld modules used in this construction. Also, I want to study data related to these special curves and to compute Galois closures, zeta functions and so forth. Furthermore, I am planning to find an alternative to these conventional constructions of curves with many rational points to give an improvement of known asymptotic results.

Duration: 1 November 2014 - 31 October 2016
Funding body: HCØrsted COFUND Postdoc programme

## Contact

Prasant Singh
Postdoc
DTU Compute
+45 45 25 30 38
http://www.compute.dtu.dk/english/research/mathematics/postdocs
23 OCTOBER 2018