Theses & research projects for students
Open topics from both nodes, from a first undergraduate project to a PhD. Pick something that excites you and write to the supervisor or the node lead.
Athens · 9 open
Research projects for undergraduate mathematics students who want a first taste of research: reading a paper in depth, running computational experiments, or contributing to the lab's software.
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)
Study and implement arithmetic operations on continued fractions following Gosper's algorithm.
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...
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...
Implement and benchmark Euclidean, subresultant, and modular GCD algorithms for univariate and multivariate polynomials.
Study and implement algorithms for isolating the real roots of univariate integer polynomials.
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.
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...
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...
Istanbul · 6 open
Graduation projects (bitirme projesi) for final-year undergraduates. Projects usually last two semesters and combine some theory with a working implementation.
Count the real roots of a polynomial from the signature of its Hermite matrix, and compare with Sturm sequences.
Compare integer programming formulations of graph coloring on benchmark graphs.
Reconstruct missing parts of images using multivariate polynomial interpolation.
Build a small interval-arithmetic library and use it to certify roots found by numerical solvers.
Implement Learning With Errors (LWE) encryption and study how polynomial multiplication affects its speed.
Study and implement optimization methods, such as integer linear programming, for graph coloring problems.
Athens · 13 open
MSc theses in mathematics, computer science or computer engineering. Co-supervision across the two nodes is possible.
Study the chamber complex of a parametric integer program and compute it algorithmically.
Use Gröbner bases to algorithmically extend complementary sequences - the split, fill, and expand algorithms that reverse the Equating/Killing Lemma - toward new orthogonal designs.
Study and implement arithmetic operations on continued fractions following Gosper's algorithm.
Implement Graver basis computation for structured matrices and connect to Hilbert bases and integer programming augmentation.
Bridge the gap between Lean/Mathlib's abstract algebra and executable computer algebra - polynomial arithmetic, Gröbner bases, and certified computation.
Formalize polyhedra, cones, and linear programming duality in Lean 4 and Mathlib, connecting to the lab's Polyhedral Omega work.
A system for representing, storing, and sharing mathematical objects and benchmarks.
Study and implement algorithms for partial fraction decomposition of rational functions, with applications to coefficient extraction from generating functions.
Implement and benchmark Euclidean, subresultant, and modular GCD algorithms for univariate and multivariate polynomials.
Efficient arithmetic on multivariate rational functions arising in generating function computations.
Geometry in symbolic dimension to solve families of integer programming problems parametrically.
Study McMullen’s polytope algebra.
Study the tropical semiring, tropical linear algebra, and tropical linear programming; connect to classical LP via the min-plus structure.
Istanbul · 4 open
MSc theses in mathematics, computer science or computer engineering. Co-supervision across the two nodes is possible.
Extend symbolic-numeric certified eigenvalue location to symmetric, tridiagonal and polynomial eigenvalue problems.
Implement certified Hermite matrices from approximate roots for zero-dimensional polynomial systems.
Test the Polyhedral Omega optimization package on standard ILP benchmark families. Co-supervision with the Athens node is possible.
Certify polynomial non-negativity with semidefinite programming and convert numerical certificates into exact rational ones.
Athens · 2 open
Doctoral topics, usually connected to one of our funded projects. Write to the node lead before applying to the graduate programme.
Formalize polyhedra, cones, and linear programming duality in Lean 4 and Mathlib, connecting to the lab's Polyhedral Omega work.
Geometry in symbolic dimension to solve families of integer programming problems parametrically.
Istanbul · 2 open
Doctoral topics, usually connected to one of our funded projects. Write to the node lead before applying to the graduate programme.
Certified isolation of the real solutions of polynomial systems. Co-supervision with Elias Tsigaridas (Sorbonne Université & Inria Paris) is possible.
Algorithms and their complexity for polynomial multiplication in the rings used by lattice-based cryptography.
Topics students are working on right now. They are taken, but show the kind of work we do.
Athens
Istanbul
Finished theses and projects.
Athens
Nothing listed.
Istanbul
