Final Year Projects

Computer Engineering

Using fast volume approximation for computing Ehrhart polynomials

We propose a new algorithm for computing the Ehrhart polynomial by approximating the volume of dilated polytopes and interpolate.

This project will make use of the C/C++ software developed by GeomScale and JuliaLang. The project has a large experimental part and an algorithm development part.

Learning Bijections

Train a neural network to act as a bijection. Pick two sets of combinatorial objects that are known to have the same cardinality and train a network to act as a bijection, i.e., give a different output for each input.

The goal is to investigate best practices to develop a general framework in the direction of learning bijections.

Machine Learning for Polynomials

Use machine learning for predicting statistics or properties of polynomials. The main goal is to find appropriate encodings of polynomials and answer questions such as the number of real roots, the distance of the closest real roots, the existence of double roots etc.

You have to try and compare different architectures and design an interface that will be easy to use for the predictions you cover.

Mathematics

Sparse ELimination and the BKK bound

Study the proof of the BKK bound in sparse elimination theory.

The polytope algebra

Study Barvinok’s algorithm for short rational function decomposition. This project is about the study of the proof of Barvinok’s algorithm to explain both how it works and why it is polynomial in complexity (if the dimension is fixed)

Using fast volume approximation for computing Ehrhart polynomials

We propose a new algorithm for computing the Ehrhart polynomial by approximating the volume of dilated polytopes and interpolate.

This project will make use of the C/C++ software developed by GeomScale and JuliaLang. The project has a large experimental part and an algorithm development part.

Full Hilbert Series

We study a generalization of the standard Hilbert series. Hilbert series is the generating function of the dimensions of the graded components of a graded structure.

In this project we study a multivariate generalization of the Hilbert series, based on geometric insight.

Learning Bijections

Train a neural network to act as a bijection. Pick two sets of combinatorial objects that are known to have the same cardinality and train a network to act as a bijection, i.e., give a different output for each input.

The goal is to investigate best practices to develop a general framework in the direction of learning bijections.

Resultant Systems

Use machine learning for predicting statistics or properties of polynomials. The main goal is to find appropriate encodings of polynomials and answer questions such as the number of real roots, the distance of the closest real roots, the existence of double roots etc.

You have to try and compare different architectures and design an interface that will be easy to use for the predictions you cover.

The polytope algebra

Study McMullen’s polytope algebra.

topic and applications

The purpose of this project is the study of different types of resultants. Their comparison and their usage in applications are to be studied as well.

Examples of resultants include Sylvester, Macaulay, Dixon, etc.

Master's thesis topics

Solid Partitions on a Cube

Compute the full generating function of the solid partitions on a cube. The problem has a long history, since P. MacMahon more than 100 years ago investigated it.

Theoretically the problem is easy to understand. Computationally is intractable by current software and methods.

In this project we study structural properties that can speed up the implementation as well as using parallel computing for making the computation possible.