Mathematical Structures for Foundations and Innovation

The Focus Area, "Mathematical Structures for Foundations and Innovation", of the National Institute for Theoretical and Computational Sciences (NITheCS), is an ambitious scientific platform that aims to advance the understanding of foundational mathematical structures through deep and rigorous theoretical and computational research. Its primary objectives include the following.

Exploring Fundamental Mathematical Structures: to investigate core mathematical structures across diverse theoretical and computational (sub)disciplines, including but not limited to: analysis, algebra, combinatorics, logic, topology, category theory and theoretical computer science.

Fostering Interdisciplinary Structural Insights: to look for synthesis of ideas through unifying mathematical structures across subdisciplines of mathematics and beyond, to strengthen existing relationships between different areas of research and discover new relationships between independent areas of research, thereby generating innovative perspectives.

Developing Future Researchers: to nurture the next generation of mathematicians by recruiting and mentoring postgraduate students and postdoctoral fellows, with a focus on empowering them to become key role-players in the international mathematical arena and ensuring integration of researchers from historically disadvantaged groups and institutions to ensure inclusivity in the growth of mathematical research capacity.

Theoretical research in the focus area is within the traditional boundaries of pure mathematics. More applied mathematical research is integrated in various other focus areas of NITheCS. Research topics with significant pure and applied mathematical components can be pursued by means of collaboration across focus areas. In addition to the specific objectives described above, an overarching objective of the focus area is to grow the presence and leadership of South African mathematicians in leading international mathematical communities. This is to be achieved by building on existing research strengths as well as widening the scope of present research specializations.

The work done in the focus area is organised into five work packages: Consolidated Research Programmes, Emerging Research Directions, Transformation and Inclusion in Pure Mathematics, Mathematical Innovation and Commercialisation, Focus Area Coordination and Operations. Each of these work packages has an allocation of a certain percentage of the overall funding of the focus area.

To uncover and elucidate mathematical structures central to future innovation in mathematics and human development, strengthening human and research capacity and positioning South Africa (and Africa) as a global leader in the generation of foundational mathematical ideas.

This grand challenge is a high-leverage investment in the knowledge economy: when a country strengthens its capacity to generate new mathematical structures, it strengthens the intellectual infrastructure on which modern science, computation, and advanced technology depend. Foundational mathematical advances repeatedly act as "general-purpose innovations": they create new languages for modelling, new methods for reasoning, and new computational paradigms that later propagate across many domains—often with effects far beyond what can be anticipated at the time of discovery.

The value proposition is twofold and measurable. First, it directly builds human and research capacity by producing highly skilled graduates and early-career researchers trained in rigorous abstraction, structural reasoning, and computational experimentation—capabilities that are scarce, transferable, and essential for a resilient, innovation-driven society. Second, it positions South Africa (and Africa) to contribute at the level of first principles rather than primarily as adopters of external advances, increasing scientific sovereignty, attracting and retaining top talent, strengthening international partnerships, and raising the global visibility of South African research. This programme translates resources into concrete outcomes—trained researchers, research networks, workshops and schools, high-quality publications, open educational resources, and reusable mathematical/computational frameworks—while maintaining a clear strategic focus: to make South Africa (and Africa) a global leader in generating foundational mathematical ideas that shape long-term trajectories of innovation and human development.

Associate Professor Eric Andriantiana (Rhodes University), Coordinating Assistant, Email: e.andriantiana at ru.ac.za

Associate Professor Maggie Aphane (Sefako Makgatho Health Sciences University), Deputy Coordinator, Email: maggie.aphane at smu.ac.za

Associate Professor Amartya Goswami (University of Johannesburg), Scientific Chair, Email: agoswami@uj.ac.za

Professor Zurab Janelidze (Stellenbosch University), Coordinator, Email: zurab@sun.ac.za

Dr Cerene Rathilal (University of KwaZulu-Natal), Coordinating Assistant, Email: RathilalC@ukzn.ac.za

Mr. Okhwatisa Nemunzhele (Stellenbosch University), is the Operations Manager of the Focus Area, who assists the coordinating team in daily operations within the focus area. Please feel free to contact him for any enquiry:

Email: 29894530@sun.ac.za

Mobile: 0637919481

The Mathematical Structures Almanac is a regularly updated publication detailing activities of achievements of the focus area, as well as its structure and aims. Follow the link below to access the publication of the focus area detailing its activities and achievements:

https://www.overleaf.com/read/bbfgpxtzwfxv#3be2b2

Approximate numbers: 15 papers published, 15 papers submitted for publication, 15 postgraduate students, 15 presentations. Below is an overview of the highlights of activities in 2025. Members of the coordinating team are members of editorial boards of 4 international journals in mathematics, of which two are Q1 journals. Most of the members of the coordinating team have NRF ratings, with one B1 rating.

Furthering Collaborations through Algebraic Structures

Prof. Goswami initiated collaborations with Dr. Vaino Shaumbwa from the University of Namibia and Prof. David Blecher from the University of Houston. New fundamental connections and concepts have been explored at the interface of rings, modules, Banach spaces, semirings, quantales, multiplicative lattices and general universal algebras. Prof. Goswami gave his first talk at an international conference on rings and modules in Austria in July, where he was warmly welcomed. 12 papers co-authored by him were published in 2025.

A Conference on Ordered Structures

Dr. Wen-Chi Kuo co-organised the “Conference on Ordered Structures with Applications (COSA 2025)” held at Wits Rural Facility from March 31 to April 4 at Kruger National Park in South Africa and included guest speakers from the FA as well as their collaborators from northern Africa (Tunisia).

Advances in the study of Higher Structures

Singificant advances were made in 2025 in the study of higher categorical structures, especially in the direction of developing foundations for higher dimensional homological algebra. International collaboration and networking took place with several Italian categories theorists based in Italy and the UK. Prof. Janelidze gave an invited talk at the ItaCa Fest online webinar in April. Several members of the work package presented talks at the International Category Theory Conference held in the Czech Republic in July.

Discrete Structures Retreat

A 3-day research and writing retreat was held at Mpekweni in the Eastern Cape in January 24-26.

The participants of the retreat were: Prof. Eric Andriantiana (Rhodes University), Dr Adriana Roux (Stellenbosch University), Dr Valisoa Rakotonarivo (University of Pretoria), and Dr Zekhaya Benard Shozi (University of KwaZulu-Natal). Their work builds on the innovative work in [E.O.D. Andriantiana and Z.B. Shozi, The number of 1-nearly independent vertex subsets, Quaestiones Mathematicae 2024, 47(12): 2353-2373], where a generalised notion of independence in a graph is proposed.

Topological Structures Research

During March 24-28, Dr. Cerene Rathilal received a visitor, Dr Ana Belèn Avilez Garcia (Chapman University, USA), at the University of KwaZulu-Natal, who joined her in the investigation of connectivity in the context of point-free spaces.

Prof. Dharmanand Baboolal and Dr. Simo Mthethwa are also part of Dr. Rathilal's research group.

Broader Community

Joint Meeting of the London Mathematical Society, the South African Mathematical Society, and the Southern African Mathematical Sciences Association, took place in 2025. This historic networking event held 1-5 September was co-sponsored with the funds of the focus area. It led to successful collabotaitons and promising future partnerships. The event showcased some of the work on mathematical structures and their applications done in South Africa, Southern Africa and the UK. Several members of the coordinating team of the focus area were invited speakers at this meeting. See https://futuremath.net/imi for more information.

The purpose a "Pan-Disciplinary Workshop on Mathematical Structures" held in September 2025 was engagement with researchers and students at the historically disadvantaged univesity (HDI), Sefako Makgatho Health Sciences University (SMU).

The meeting included presentations by students, a session for identifying mathematical structures studied at SMU for the purpose of future integration of SMU in the focus area.

The meeting also included explorative research sessions, which led to some interesting ideas of how the applications of graphs in chemistry and optimisation could potentially be extended to applications of more nuanced structures: categories. This workshop brought together all of the work packages of the focus area.

Website of the meeting: https://futuremath.net/prmsw

The activities of the Future Mathematicians Progamme was introduced to the attendees of the Mega Mathematics Festival held in July at the University of the Western Cape, in a dedicated special session held in partnership with the South African Mathematics Foundation and South African Mathematical Society. This programme aims to introduce school learners and university students to abstract mathematical thinking and deductive reasoning.

The third edition of the theatrical, musical, and scientific production Fundamano explored the concept of a biological merger between the human brain and artificial intelligence, confronting audiences with a question of profound societal significance. Through the integration of theatre, music, and scientific imagination, the production created a space for reflection on the ethical, psychological, and social implications of human–AI integration.

https://youtu.be/QSa_cfBAdNQ?si=59yUQIunWJEAiNeA&t=6667

https://youtu.be/UOltCuz-YbM

In 2025, a unifying theme in research on algebraic structures was the transfer of different notions of ideals between algebraic settings, sometimes bridging distinct mathematical disciplines. A particularly significant result was obtained by Amartya Goswami and David Blecher, who extended the notion of an M-ideal—originally studied for Banach spaces in functional analysis—to non-commutative rings (preprint). Another major achievement was the extension of the Cohen structure theorem, a central result in commutative algebra, to a broader class of rings. This work was undertaken by Amartya Goswami together with Stellenbosch University postdoctoral researchers Elena Caviglia and Luca Mesiti, both active participants in the focus area (preprint). Altogether, this research theme produced 11 published papers in 2025, with a further nine submitted for publication.

Research on categorical structures also yielded substantial results in 2025. Five papers were published in three leading international category-theory journals, three of them ranked Q1 or Q2, while a further six papers were submitted for publication. Among the notable publications is a paper by Michael Hoefnagel and Emma Theart (journal article), which reveals a new conceptual connection between categories of topological and algebraic structures by transferring the notion of extensivity from categories to individual morphisms. Another significant paper, co-authored by Zurab Janelidze, Amartya Goswami and Graham Manuell (journal article), establishes unexpected links among an abstract treatment of principal ideals, the modularity of Galois connections in distributive homological algebra, protomodularity and its formal extension in categorical algebra—whose origins lie in the lattice isomorphism theorem for groups—and the Frobenius reciprocity law of categorical logic. A further highlight was the joint work of Zurab Janelidze, Elena Caviglia and Luca Mesiti, which developed two-dimensional perspectives on two distinct topics in categorical algebra (first preprint; second preprint).

Progress was also made in graph theory and theoretical computer science. Eric Andriantiana and Audace A. V. Dossou-Olory, a close collaborator from Benin, studied induced subgraphs containing exactly one edge, known as 1-nearly independent vertex sets, and established fundamental properties of their average size (preprint). In a different direction, Brink van der Merwe and collaborators from Sweden constructed a novel and efficient automaton for byte-pair encoding tokenisation, a method for decomposing strings into substrings that plays an important role in large language models (conference publication).

A principal achievement in topology was the extension of Kuratowski’s connectedness property of unicoherence from point-set topology to point-free topology. This result was obtained by Cerene Rathilal in collaboration with Elena Caviglia and Luca Mesiti (preprint); the paper has been accepted for publication in Algebra Universalis.