IMPORTANT PRELIMINARY NOTICE: The recruitment process will begin on 3 March, and will continue until all positions are filled.

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COGENT is a European Doctoral Network, funded by the European Commission (EC) as part of the MSCA programme and by UK Research and Innovation (UKRI)

Network coordinator: Université Grenoble Alpes (UGA).

Academic EU beneficiaries: Stichting Vrije Universiteit Amsterdam (VUA), Technische Universität Braunschweig (TUB), Université de Bordeaux (UBx), University of Galway.

Academic UK beneficiaries: University of Durham, University of Sheffield (UFSD).

Partner organizations worldwide: Colorado State University (CSU), University of Massachusetts (UMASS), University of Michigan, University of North Carolina-Greensboro (UNCG), University of Oklahoma, ID Quantique (IDQ, Switzerland), MSM Programming (Croatia).

Doctoral candidates 1 to 10 are funded by the EC as part of the MSCA program.

Doctoral candidates 11, 12 and 13 are funded by UKRI.

In the case of Vrije Universiteit Amsterdam (doctoral candidate 3), the PhD position will be extended to 4 years. The fourth year of this DC is funded by VU Amsterdam. 

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The COGENT project (Cohomology, Geometry, Explicit Number Theory) will advance the state of the art in an active area of interplay between algebra, geometry and computer science. It aims to promote game changing approaches to explicit and exact calculations underlying longstanding conjectures in number theory.

COGENT has the ambition to stimulate interdisciplinary and intersectorial knowledge exchange between number theorists, algebraists, geometers, computer scientists and industrial actors facing real-life challenges in symbolic computation in order to bridge key knowledge gaps. To this end, the project will address the urgent need for computer assisted investigations of several longstanding number-theoretic conjectures, and EU industry’s need for workers with an advanced mathematical and computational skill set.

COGENT brings together international experts with relevant diverse and complementary expertise in mathematical and computational techniques. They will work on topics such as the development of methods for the efficient application of the cohomological machinery to various classes of arithmetic groups and applications to number theory. COGENT will broaden theoretical knowledge aimed at extending the scope of computer aided symbolic and exact calculations, with a view to potential industrial applications in disparate areas ranging from cryptography to applied topological data analysis. The project will contribute new blood to, and help maintain the critical mass of, the research community underpinning and reliant on EU supported open source scientific software such as GAP and Pari/GP. The main training goal of the project is to form a new generation of young researchers with a unique expertise, including high performance computations in geometry and topology, proof-assistant and formalized mathematics, machine learning and quantum computing techniques.

The training will incorporate innovative boot camps, exposure to new research collaborations, and a wide-range of transferable skills.

The project consists of a relatively large consortium (5 academic EU beneficiaries, 2 academic UK beneficiaries funded by UKRI and 7 partner organizations worldwide).

Beneficiaries provide supervision of at least one Doctoral candidate project, while the Partner Organisations provide co-supervision, secondments, short visits or training.

The members of the COGENT network are experienced supervisors with more than 90 successful PhD students supervised working either in academia, in the public sector or in industry. Moreover, they have been part of more than 200 PhD committees and been involved in more than 50 PhDs as referees. 

Doctoral candidates will participate in a range of online seminars, in-person workshops, conferences and other training activities organised as part of this COGENT Doctoral Network.

Research goals

The COGENT project aims to tackle a range of scientific challenges including:

• Obtaining full explicit information on the cohomology of several families of arithmetic groups.
•  Circumventing the exponential complexity of current computational methods. 
•  Implementing the action of Hecke operators.
•  Explicitly studying algebraic K-groups and related conjectures, and connecting elements in algebraic K-groups and in homology.
• Testing and refining the integral Langlands Program.
•  Formally verifying intricate proofs and computer-aided computations.
• Identifying suitable applications of motivic cohomology and group theory to cryptography.

PhDs positions

DC1: Integral cohomology of finite p-groups 

• Location: Technische Universität Braunschweig (TUB), Germany
• Supervisors: Bettina Eick
• Short description: The first aim of this project is to develop and implement new approaches towards the investigation of the integral cohomology of a finite p-group G. A key idea is to combine the methods developed by Eick & King (2016) for mod-p cohomology rings with the nilpotent quotient algorithm for integral associative algebras by Eick & Moede (2020).  The second aim is to apply the newly developed methods in explicit investigations. In particular, it is planned to investigate the conjecture that the integral cohomology rings of the (infinitely many) p-groups in a coclass family can be described by a single parametrised presentation. A recent result by Symonds (2021) supports this in the case of mod-p cohomology rings.
• Planned secondments: The project is seconded by University of Galway and will include one or more visits to this university.

DC2: Constructing equivariant CW-complexes for arithmetic (and related) groups

• Location: University of Galway, Ireland
• Supervisors: Graham Ellis & James Cruickshank
• Short description: For many topological computations on a group G, it is useful to have an equivariant CW-complex on which the group acts with finite stabilizers. The archetype of such complexes is the Voronoi complex formed by the perfect forms of rank n which allows computing (in theory) the cohomology of the arithmetic group GL(n,Z). But this complex is known to grow exponentially with the rank of perfect forms and is already difficult to grasp in rank 8. The objective is to build new complexes for groups where such complexes are not known and to build smaller complexes when some already exist.
• Notice: This DC’s project is deeply connected with DC8 and DC7. The results of DC2 are expected to be used by DC7 for computing Hecke operators. 
• Planned secondments: The project is seconded by Université Grenoble Alpes and UMASS and will include one or more visits to these universities.

DC3: Formal verification of theory and algorithmic output for modular forms

• Location: Stichting Vrije Universiteit Amsterdam (VUA), Netherlands
• Supervisors: Sander Dahmen and Assia Mahboubi
• Short description: The goal is to formally verify essential aspects of the correctness of algorithms and output related to the classical modular symbols method, and possibly trace formulas, used to compute spaces of classical modular forms by most computer algebra packages, such as, Magma, SageMath and Pari/GP. This involves employing proof assistants, e.g. Lean. After the classic modular forms case, we consider extending it to cover cases of Bianchi modular forms for which the modular symbols method applies almost verbatim. Finally we explore extending it to cases where methods more sophisticated than that of modular symbols are required, such as (other) modular forms for GL(2) over number fields and GL(3) over the rationals.
• Notice: The fourth year of this DC will be funded by VU Amsterdam. For further info, click here. -> https://www.few.vu.nl/~sdn249/vacancy.html
• Planned secondments: This project is seconded by University of Sheffield and will include one or more visits to this university.

DC4: Effective algorithms for space groups

• Location: Technische Universität Braunschweig (TUB), Germany
• Supervisor: Bettina Eick
• Short description: Space groups are the symmetry groups of repeating patterns in an n-dimensional space and are linked to chemistry and physics problems. Space groups also appear in various research projects in mathematics, and naturally arise in the stabilizers of the cells in the chain complexes associated to arithmetic groups, and as such are closely related to cohomology of groups and to number theory within the COGENT project. The first part of this PhD project is to produce a unified database of all space groups of dimension at most 6 in the spirit of the SmallGroups library (used in the software GAP and Magma), following ideas of Besche, Eick and O’Brien (2002). The second part is to design an effective method to determine all isomorphism types of maximal subgroups for any given space group of dimension at most 6.  The third part of the project will be to explore the minimal generator problem for space groups (in particular its decidability in few cases where it is unknown). The quantum computing aspect will be explored as well.
• Planned secondments: This project is highly connected to DC6 and will be co-supervised with the Bordeaux team (UBx). There project will include regular visits to this university. There will be strong interaction with Colorado State University.  UGA will provide complementary expertise for the quantum computing part.

DC5: Homological perturbation theory for Hochschild cohomology and applications

• Location: University of Galway, Ireland
• Supervisor: Emil Skoldberg
• Short description: This project would work on developing and implementing efficient algorithms for the computation of the Hochschild cohomology through homological perturbation theory who has been used in other areas of algebraic topology, as well as using discrete Morse theory to implement mappings from which the algebra structure of the cohomology can be computed. Thereby allowing for a complete description of the cohomology algebra. The algebra structure of the Hochschild cohomology allows for the detection of elements in the ring structure in the cohomology of groups. In the particular case of arithmetic groups it is related to Quillen’s conjecture. We expect the information of the Hochschild cohomology of group algebras over finite fields to detect elements in the mod p cohomology of the associated arithmetic groups, and as such yielding calculations that support or disprove Quillen’s conjecture for this group. As first step the work will focus on finite groups (and then extend to arithmetic groups). We expect full explicit applications in the case of finite groups (and low degrees), with the goal to get further information on arithmetic groups. The approach will be based on works of Skoldberg (2005,2006), Generalov et al. (2023) and Hayami (2007). Moreover, with the expertise of VUA, we will explore the possibility to formalize part of the techniques.
• Planned secondments: This project is seconded by Stichting Vrije Universiteit (VUA) and will include one or more visits to this university. 

DC6: Algorithms for Polycyclic groups associated to number fields and cryptographic applications

• Location: Université de Bordeaux, France
• Supervisor: Karim Belabas
• Short description: Each extension of the maximal order O with the unit group U of an algebraic number field K defines a polycyclic group. The arising groups are metabelian and have a highly interesting structure. The works by Eick et al (2017,2019,2022) suggest that these groups are an important key example for practical algorithms. In particular, the search for an efficient solution of the conjugacy problem and the isomorphism problem, as well as the study of their complexities, are fundamental. Moreover, those groups, for well-chosen parameters, could provide an interesting setup for the Anshel-Anshel-Goldfeld cryptosystem with potential applications to post-quantum cryptography.
• Planned secondments: The project will be co-supervised by B. Eick (TUB). The project will include one or more visits to Technische Universität Braunschweig. We expect also discussions with UGA and IDQ on the cryptographic applications.

DC7: Constructive contractibility of CW-complexes associated to arithmetic groups and applications to Hecke operators

• Location: University of Galway, Ireland
• Supervisor: Graham Ellis and James Cruickshank
• Short description: For many topological computations on a group G, it is useful to have a contractible CW-complex on which the group acts with finite stabilizers. Many such CW-complexes are available in the literature, and further examples will be provided by complementary projects in this network. The primary objective is to use discrete Morse theory to encode on a computer the contractibility of a given CW-complex, and use this encoding to achieve machine calculations of Hecke operators and other cohomology operations. The project will focus on encoding contractibility of G-equivariant CW-complexes for discrete groups such as G = SL(n,O) over the ring O of integers and imaginary quadratic integers. The second objective is to implement these constructive proofs as part of user-friendly open source software for computing cohomology operations in the cohomology of finite index subgroups of G with twisted coefficients in finite dimensional modules over rational or finite fields. The third objective is to use the software to carry out systematic experiments concerning Hecke operators on automorphic forms associated to finite index subgroups of some arithmetic groups aimed at helping to understand the torsion Langlands program described by Calegari and Venkatesh.
• Planned secondments: There will be regular videomeeting of the Doctoral candidate with M. Dutour Sikirić (MSM). This project is connected to DC2 and DC8, and there will be regular interaction with UGA, UMASS and UNCG.

DC8: Advanced computing and heuristics for higher dimensional Voronoi complexes and applications

• Location: Université Grenoble Alpes, France
• Supervisor: Philippe Elbaz-Vincent
• Short description: The Voronoi complexes associated to G=GL(n,R) or SL(n,R), with R the integers or a Euclidean imaginary quadratic ring, or a congruence subgroup of G, are important geometric models for the computation of the cohomology of G and has been used extensively  during the past 15 years. But its inherent complexity prevent full computations beyond n=7. The goal of this project is to revisit the algorithms through the perspective of parallelism and GPGPU computing, combined with mathematical strategies such as quotient methods, and different types of filtrations in order to reduce drastically the size of complexes (at the cost of several simple  spectral sequences computations). We also expect to be able to combine the methods of DC2 and DC7 based on discrete Morse theory in order to further reduce the complexity. We expect also to use machine learning techniques on a large set of data collected by Elbaz-Vincent and others in order to detect specific patterns on the cycles of the Voronoi complexes.  Combining together those methods, we expect to detect high p-torsion classes in the cohomology of G for n>11 and p>n+1 with applications to the K-theory of integers and the torsion Jacquet-Langlands program. We will use  C and OpenCL as programming languages under a Linux environment, and it will be built upon software developed by Elbaz-Vincent (for specific  targets) and Dutour Sikirić (for more general applications). We will also explore the quantum computing aspects of those methods and investigate the possibility to apply the methods to other geometric models.                                                                                                                                                        
• Planned secondments: The project is seconded by M. Dutour Sikirić (MSM), Université de Bordeaux (expertise on parallelism and programming aspects) and Université Grenoble Alpes (expertise on the quantum computing aspect). The Doctoral candidate will interact with M. Dutour Sikirić (MSM) on a regular basis through online meetings. The project will include one or more visits to the Université de Bordeaux. Collaborations with DC2, DC7 and DC9 are expected, as well as DC10 for relationships with Hermitian K-theory. The Doctoral candidate will discuss with UMASS and UNCG in order to explore potential applications of the setting to congruence subgroups, and with UMICH and UOK for potential applications to their geometric models.

DC9: Computing isometries and automorphisms of lattices: Classical vs Quantum Computing and Applications

• Location: Université Grenoble Alpes, France
• Supervisors: Philippe Elbaz-Vincent and Mehdi Mhalla
• Short description: The problem of computing isometries between lattices is both an emblematic algorithmic problem and a crucial tool when performing classification of families of lattices or computations of the Voronoi complexes associated to finite index subgroups of GL(n,Z). The reference algorithm is the one of Plesken and Souvignier (1997), with several implementations such as the ones of Pari/GP and Magma.  Most of these implementations are inadequate in order to deal with various families of lattices and we would like to have a certified implementation, with a precise complexity study and a free implementation with fast running time on modern CPUs, which will integrate advanced backtracking, as the one proposed recently by Jefferson, Pfeiffer, Wilson and Waldecker (2021). We will also investigate the possibilities of improved heuristics in order to speed up the methods. The DC will study both its classical and quantum complexities (the quantum complexity has never been explored so far), optimized C implementations are expected (as well as an integration to Pari/GP).
• Planned secondments: Collaborations with DC2, DC7 and DC8 are expected.  The project will be seconded by Université de Bordeaux and will include one of more visits to this university.

DC10: Twisted characteristic classes, symplectic groups and Hermitian K-theory of number fields

• Location: Université Grenoble Alpes, France
• Supervisor: Jean Fasel
• Short description: As emphasized by the COGENT project, cohomology theories are central tools in both algebraic topology, algebraic geometry and number theory. One of their common natural playgrounds is the so-called stable homotopy category after Morel and Voevodsky (1999) which gives rise to bi-graded cohomology theory associated to smooth k-schemes (k a field). In this setting, cohomology with coefficients in a Thom space arises naturally and it is important to be able to understand this cohomology in order to perform explicit computations. An important special case is given by the so-called oriented cohomology theories, which are associated to GL and can be extended to its symplectic subgroups Sp(2n), and obtain the so-called symplectically oriented theories, or SL.  The former behaves formally as the case of oriented theories, possessing a well-behaved theory of Borel classes. The first goal of this proposal is to extend the definition of Borel classes to the case of twisted symplectic bundles, drastically improving their range of applications. Then apply it to the computation of cohomology of classifying spaces associated to Sp(2n) for reasonable cohomology theories, such as the Hermitian K-theory of number fields after Karoubi, Schlichting and others.  This construction leads to twisted Borel classes that we expect to be easier to handle than their non-twisted analogues. We expect explicit computations in the case n=2,3.  The second part of the proposal is to explore potential applications, for instance in connection with the theory of formal ternary laws as considered by Déglise, Fasel and others (2023).
• Planned secondments: We expect potential interactions with DC2, DC7 and DC8. Université Grenoble Alpes will provide complementary expertise on the computational aspects.

DC11: Cohomology of SL2 over the Hamiltonians

• Location: University of Sheffield, United Kingdom
• Supervisor: Haluk Sengün
• Short description: Using the Voronoi approach, the goal is to develop a computer program that will compute the cohomology of arithmetic groups for GL(2) over a definite rational quaternion algebras together with Hecke action. The second objective is to use this program to carry out systematic experiments concerning growth of torsion in the homology, automorphic forms and modularity of Galois representations. These will be the first experiments in the literature concerning automorphic forms for GL(2) over the Hamiltonians.
• Notice: This DC will be funded by UKRI.
• Planned secondments: P. Gunnells (UMASS) will act as one of the co-supervisors. The project will also be seconded by Université de Bordeaux. The project will include one or more visits to UMASS and Université de Bordeaux. 

DC12: Cryptographic Applications of Motivic Cohomology Groups

• Location: University of Durham, United Kingdom
• Supervisor: Herbert Gangl
• Short description: Explicit versions of algebraic K-groups, like higher Bloch groups for number fields, could potentially provide the basis of a cryptographic signature scheme where elements in such a group would provide a high degree of ambiguity needed to obscure a given message, whereas the correctness of the signature would be tested by the cancellation of an associated "symbol". The project will start with the classical Bloch groups of adequate (explicit) number fields analyzing a proposed signature scheme and show that it could be translated to an NP-hard problem (both for classical and quantum complexity). A security proof of the scheme will be provided (with the same security properties than ML-DSA), as well as optimized implementations (in C). Analogues for higher Bloch groups will be also investigated. The last goal will be to work on finite analogues of such setting by exploring the possibility to design a signature scheme based on the modular dilogarithm for finite fields (Schroeppel and Beaver, 2002) and see potential links with finite polylogarithms and previous works of Elbaz-Vincent and Gangl (2002, 2015).
• Notice: This DC will be funded by UKRI. It is expected that this PhD will be a joint degree between University of Durham (UK) and Université Grenoble Alpes (France).
• Planned secondments: The DC will be co-supervised by P. Elbaz-Vincent (UGA). The project will include one or more visits to UGA to work on the cryptographic aspects. The project will also be seconded by IDQ. The project will include one or more visit to IDQ to test potential cryptographic primitives in a real world setting.

DC13: Torsion elements in higher K-groups

• Location: University of Durham, United Kingdom
• Supervisor: Herbert Gangl
• Short description: One topic is to gather experimental data on the rational number relating (as in Lichtenbaum’s conjecture) the Beilinson regulator and the leading coefficient in the expansion of the zeta function of k at negative integers, for k a number field. In particular, it encodes the size of a specific higher K-group. Because (primes dividing) the order of K_2 (O_k ), O_k being its ring of integers, appear to be related to the homology of congruence subgroups of GL_2 through the work of Calegari and Venkatesh, we want to compute the rational numbers in the Lichtenbaum conjecture experimentally also for n>2, or n = 2 and k of higher degree.
• Notice: This DC will be funded by UKRI. 
• Planned secondments: The DC will be co-supervised by R. de Jeu (VUA). The project will include one or more visits to Stichting Vrije Universiteit Amsterdam, Netherlands. The project will also be seconded by University of Sheffield and Université de Bordeaux. The project will include one or more visits to these universities. The work with University of Sheffield will be related to the relation, proven by Calegari and Venkatesh, between K-groups and the “trivial part” of the homology of GL(2) as a Hecke module for imaginary quadratic number rings. The work with Université de Bordeaux will relate to the improvement of the computational aspects.