Continued Fraction Arithmetic
Study and implement arithmetic operations on continued fractions following Gosper's algorithm.
Theses & research projects for students
Topics
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.
Undergraduate
Athens · 4 open
Undergraduate research
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.
Polynomial GCD Algorithms
Implement and benchmark Euclidean, subresultant, and modular GCD algorithms for univariate and multivariate polynomials.
Real Solving
Study and implement algorithms for isolating the real roots of univariate integer polynomials.
Sturm Sequences
Implement exact real root counting via Sturm chains and compare with Descartes' rule.
Istanbul · 8 open
Undergraduate thesis
Graduation projects (bitirme projesi) for final-year undergraduates. Projects usually last two semesters and combine some theory with a working implementation.
Barvinok's algorithm
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)
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...
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....
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...
Resultants and their 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.
Sparse Elimination and the BKK bound
Study the proof of the BKK bound in sparse elimination theory.
The polytope algebra
Study McMullen’s polytope algebra.
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...
Master's
Athens · 21 open
Master's thesis
MSc theses in mathematics, computer science or computer engineering. Co-supervision across the two nodes is possible.
Chamber Complex
Study the chamber complex of a parametric integer program and compute it algorithmically.
Computer Algebra for Orthogonal Designs
Use Gröbner bases to algorithmically extend complementary sequences - the split, fill, and expand algorithms that reverse the Equating/Killing Lemma - toward new orthogonal designs.
Continued Fraction Arithmetic
Study and implement arithmetic operations on continued fractions following Gosper's algorithm.
Formalization of Symbolic Algorithms
Formally verify symbolic computation algorithms (GCD, Euclidean algorithm, real root isolation) using Lean 4 and Mathlib.
Graver Bases
Implement Graver basis computation for structured matrices and connect to Hilbert bases and integer programming augmentation.
LLL Lattice Basis Reduction
Study and implement the LLL algorithm for lattice basis reduction; apply to short vector problems and polynomial factoring.
Lean and AI
Use Lean 4 as a training and evaluation ground for machine-learning-guided theorem proving, and study what makes a Lean proof state learnable.
Lean and Computer Algebra
Bridge the gap between Lean/Mathlib's abstract algebra and executable computer algebra - polynomial arithmetic, Gröbner bases, and certified computation.
Lean and Polyhedral Geometry
Formalize polyhedra, cones, and linear programming duality in Lean 4 and Mathlib, connecting to the lab's Polyhedral Omega work.
Machine Learning for Mathematics
Apply machine learning techniques to mathematical objects such as polynomials and polytopes.
Math Data
A system for representing, storing, and sharing mathematical objects and benchmarks.
Optimization via Polyhedral Omega
Use PO-based exact counting/enumeration to design exact and hybrid optimization methods for integer programs.
PO-derived Bounds for Knapsack Families
Study bounds on knapsack-type integer programs derived from Polyhedral Omega counting and compare to branch-and-bound.
Partial Fraction Decomposition
Study and implement algorithms for partial fraction decomposition of rational functions, with applications to coefficient extraction from generating functions.
Polynomial GCD Algorithms
Implement and benchmark Euclidean, subresultant, and modular GCD algorithms for univariate and multivariate polynomials.
Rational Function Arithmetic
Efficient arithmetic on multivariate rational functions arising in generating function computations.
Rational Function Arithmetic (Program)
Research program on fast arithmetic for rational functions in counting and generating function contexts.
Real Solving
Study and implement algorithms for isolating the real roots of univariate integer polynomials.
Sturm Sequences
Implement exact real root counting via Sturm chains and compare with Descartes' rule.
Symbolic Polyhedral Omega
Geometry in symbolic dimension to solve families of integer programming problems parametrically.
Tropical Arithmetic and Optimization
Study the tropical semiring, tropical linear algebra, and tropical linear programming; connect to classical LP via the min-plus structure.
Istanbul · 1 open
Master's thesis
MSc theses in mathematics, computer science or computer engineering. Co-supervision across the two nodes is possible.
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...
PhD
Athens · 11 open
PhD thesis
Doctoral topics, usually connected to one of our funded projects. Write to the node lead before applying to the graduate programme.
Computer Algebra for Orthogonal Designs
Use Gröbner bases to algorithmically extend complementary sequences - the split, fill, and expand algorithms that reverse the Equating/Killing Lemma - toward new orthogonal designs.
Formalization of Symbolic Algorithms
Formally verify symbolic computation algorithms (GCD, Euclidean algorithm, real root isolation) using Lean 4 and Mathlib.
Graver Bases
Implement Graver basis computation for structured matrices and connect to Hilbert bases and integer programming augmentation.
Lean and AI
Use Lean 4 as a training and evaluation ground for machine-learning-guided theorem proving, and study what makes a Lean proof state learnable.
Lean and Computer Algebra
Bridge the gap between Lean/Mathlib's abstract algebra and executable computer algebra - polynomial arithmetic, Gröbner bases, and certified computation.
Lean and Polyhedral Geometry
Formalize polyhedra, cones, and linear programming duality in Lean 4 and Mathlib, connecting to the lab's Polyhedral Omega work.
Machine Learning for Mathematics
Apply machine learning techniques to mathematical objects such as polynomials and polytopes.
Optimization via Polyhedral Omega
Use PO-based exact counting/enumeration to design exact and hybrid optimization methods for integer programs.
Rational Function Arithmetic (Program)
Research program on fast arithmetic for rational functions in counting and generating function contexts.
Symbolic Polyhedral Omega
Geometry in symbolic dimension to solve families of integer programming problems parametrically.
Tropical Arithmetic and Optimization
Study the tropical semiring, tropical linear algebra, and tropical linear programming; connect to classical LP via the min-plus structure.
Istanbul · 1 open
PhD thesis
Doctoral topics, usually connected to one of our funded projects. Write to the node lead before applying to the graduate programme.
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...
In progress
Topics students are working on right now. They are taken, but show the kind of work we do.
Athens
- Interpolation Eugenia Maria Betsani · supervised by Zafeirakis Zafeirakopoulos
Istanbul
Nothing listed.