Combinatorics,

geometrY,

Optimization and

Number theory

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All topics are available to students from both computer engineering and math departments. The separation in the lists below is only a suggestion.

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.

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.

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.

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

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)

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.

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.

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.

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.

Study McMullen’s polytope algebra.

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.

If you are interested in working with me for your master's, you should contact me before the application deadline.

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.