Research & Training

2025

COGENT WInter School in Galway

Course	Description	Lecturer(s)	Training materials
Introduction to GAP	The mini-course by Bettina Eick gave an introduction to computational group theory with the computeralgebra system GAP. It contained surveys of algorithms for permutation groups, matrix groups, finitely presented groups and polycyclic groups. The latter included some examples on computing the second cohomology group of a p lycyclic group with a finite module and and its applications in computing extensions of polycyclic groups. The mini-course by Graham Ellis gave an introduction to computing cohomology of groups in GAP. The first lecture explained and illustrated algorithms for computing the cohomology of finite groups (via Sylow subgroup calculations) and the cohomology of crystallographic groups (via perturbation calculations). The second lecture explained and illustrated algorithms for computing the cohomology of Bianchi groups (via a method of Swan) and Hecke operators on classical modular forms (via the Eichler-Shimura isomorphism).	Bettina EICK (Technische Universität Braunschweig, Germany) Graham ELLIS (University of Galway, Ireland)	Group Theory With GAP [EICK Bettina] Group Theory With GAP [ELLIS Graham]
Introduction to Oscar	The aim of the OSCAR course is to present the Computer Algebra System OSCAR, which is free open source software written in the Julia language. The first talk provides a general overview over OSCAR, its capabilities, the design decisions and the internal structure of the system. The second talk gives an overview of Group Theory functionality in OSCAR, with many examples. The third talk explains the bidirectional interface between OSCAR and the GAP system. The fourth talk shows examples of the interaction between different areas of mathematics in OSCAR. A list of exercises for the course is available.	Thomas BREUER (RWTH Aachen University) Max HORN (RPTU University Kaiserslautern-Landau)	OSCAR Computer Algebra System [BREUER Thomas, HORN Max]
Introduction to algebraic number theory with PARI/GP	This course is a tutorial for the algebraic number theory functionalities of PARI/GP. We explain, with numerous examples, how to compute with number fields, rings of integers, ideals, class groups, units, Galois groups, ray class groups, and class fields.	Bill ALLOMBERT (CNRS, Université de Bordeaux, France) Aurel PAGE (INRIA, Université de Bordeaux, France)	Algebraic number theory with PARI/GP [ALLOMBERT Bill, PAGE Aurel] Lecture Part 1 [PAGE Aurel] Lecture Part 2 - [PAGE Aurel]
Regulators	For solving Diophantine equations, it is often advantageous to pass to a "number ring'' R larger than the ring of integers. A difficulty when doing so is that uniqueness of factorisation is no longer guaranteed in R. One can restore such a uniqueness when passing to ideals, but in doing so one then needs to also understand the ideal class group (which measures the deviation from being a UFD) and the unit group of the underlying number ring. An amazingly beautiful result by Dirichlet relates two arithmetic invariants, more precisely its class number and its regulator (the covolume of the units in logarithmic space) to a special value of an associated Dedekind zeta function zeta_F(s), its residue at s=1. Higher regulators arise from the covolume of higher logarithms evaluated at higher units, a.k.a. elements in higher algebraic K-groups of R, with explicit candidates in the form of higher Bloch groups. The corresponding relation to special values of zeta_F(s) for s=n is known for n<=4.	Herbert GANGL (Durham University, United Kingdom)	Regulators [GANGL Herbert] DiophaNT2 Song covering the training materials [GANGL Herbert]
Interactive theorem proving with Lean	We start with a basic overview of formalizing mathematics via a proof assistant. Next, we turn to the practice of interactive theorem proving, using Lean. Finally, we give an overview of recent developments, including automated theorem proving and the role of AI.	Sander DAHMEN (Vrije Universiteit Amsterdam, The Netherlands) Alain CHAVARRI VILLARELLO (Vrije Universiteit Amsterdam, The Netherlands)	Interactive theorem proving with Lean [DAHMEN Sander, CHAVARRI-VILLARELLO Alain]
Quantum information and computations with Qiskit		Mehdi MHALLA (CNRS, Université Grenoble Alpes, France)	Quantum Information and computations with Qiskit [MHALLA Mehdi] Lecture Part 1 - MHALLA Mehdi Lecture Part 2 - MHALLA Mehdi
Chow-Witt groups	In these lectures, we introduce Chow–Witt groups, which are enriched versions of the classical Chow groups. We mainly focus on real realization, namely group homomorphisms between the Chow–Witt groups of smooth schemes over the real numbers and the integral cohomology groups of the real manifolds underlying these schemes.	Jean FASEL (Université Grenoble Alpes, France)
Cohomology of arithmetic groups	Arithmetic groups and their cohomology occupy a central position in modern number theory. In this lecture series, we start with the notion of an arithmetic group. We introduce the symmetric spaces associated with them and discuss their cohomology groups. Once this is done, we turn towards understanding the asymptotic behaviour of two invariants of cohomology of arithmetic groups, namely, the rank and the size of torsion part. We introduce the notion of deficiency of a semisimple Lie group and present Lueck's approximation theorem. We end the lecture series with a discussion of the torsion growth conjectures of Bergeron and Venkatesh.	Philippe ELBAZ-VINCENT (Université Grenoble Alpes, France) Haluk SENGUN (University of Sheffield, United Kingdom)
An overview on cybersecurity and cryptographic engineering	This course introduces the fundamentals of industrial quantum cybersecurity through the perspective of a practicing cryptographer. It explains how technologies such as Quantum Random Number Generation (QRNG) and Quantum Key Distribution (QKD) support secure key generation and exchange, and why they are valuable for building information-theoretically secure systems. It also highlights the practical limits of these technologies in real deployments—such as bandwidth, distance, authentication, and operability—and shows how classical and post-quantum cryptography can be combined with quantum tools to create robust, usable, and high-assurance security solutions for industry.	Thomas CAMUS (ID Quantique, Switzerland) Philippe ELBAZ-VINCENT (Université Grenoble Alpes, France)
Jewellery from tessellations of hyperbolic space	The group PSL(2,O_d), for O_d the number ring of an imaginary quadratic number field F=Q(sqrt(-d)), gives rise to a tessellation of hyperbolic 3-space. If one passes to a commensurable group (contained in the associated extended Bianchi group) one finds an ``ideal'' such tessellation, and for a decent proportion of d (=squarefree integer) we find a rich symmetry giving rise to beautiful polytopes with all their vertices lying on a sphere. These polytopes give rise to wireframe models that can be designed via OpenSCAD and subsequently produced in many different materials using a 3D printer. After the talk, we provided models of several of these, many of which were manufactured in precious metals.	Herbert GANGL (Durham University, United Kingdom)	Jewellery from tessellations of hyperbolic space [GANGL, Herbert]
A practical introduction to tessellation origami		Rachel QUINLAN (University of Galway, Ireland)	A practical introduction to tessallation origami [QUINLAN Rachel]