July-August 2021 Current plan: finialize the binary matrix properties paper. Started writing a book in abstract algebra jointly with Amartya Goswami. You can follow the progress here . May-June 2021 The paper on linear exactness properties , joint work with Pierre-Alain Jacqmin, was accepted for publication in Journal of Algebra (it is scheduled for publication in October 2021 - follow the link ). Started supervision of PhD studies of Brandon Laing on the SOFiA project . Started supervision of PhD studies of Ineke van der Berg. Mostly absent from research due to various circumstances, including administrative/refereeing duties. Started a new (highly ambitious) research project on conceptualizing the form of space-time. March-April 2021 Started research on matches of digraphs : pioneering joint work with Francois van Niekerk and Jade Viljoen (research grew out from her honors project). S
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Publications Z. Janelidze, Characterization of pointed varieties of universal algebras with normal projections , Theory and Applications of Categories 11, 2003, 212-214 Z. Janelidze, Varieties of universal algebras with normal local projections , Georgian Mathematical Journal 11, 2004, 93-98 Z. Janelidze, Subtractive categories , Applied Categorical Structures 13, 2005, 343-350 Z. Janelidze, Closedness properties of internal relations I: A unified approach to Mal’tsev, unital and subtractive categories , Theory and Applications of Categories 16, 2006, 236-261 Z. Janelidze, Closedness properties of internal relations II: Bourn localization , Theory and Applications of Categories 16, 2006, 262-282 Z. Janelidze, Closedness properties of internal relations IV: Expressing additivity of a category via subtractivity , Journal of Homotopy and Related Structures 1, 2006, 219-227 Z. Janelidze, Closedness properties of internal relations III: Pointed protomodular categories
NITheCS Mini-School on Elementary Introduction to Category Theory , a series of lectures for the October Mini-School at the South African Mini-School in Theoretical and Computational Sciences, given jointly with Amartya Goswami Abstract Algebra for the Future Mathematician , a book in progress jointly with Amartya Goswami The Caravan this is an introductory book in progress on foundations of abstract mathematics. My inaugural lecture publication and the lecture . See the mathematics playlist of my youtube channel for my video lectures. Cardinal Arithmetic (Cantor’s theory of cardinality for Grothendieck-type universes) Morphisms of finite spaces (introduction to the basic ideas of category theory via topology and combinatorics) Posets and connections (introduction to Galois connections) Homomorphisms of monoids (includes products, sums and quotients of monoids) Universal algebra (Birkhoff’s variety theorem and some Mal’tsev conditions) Abstract
Matrix properties are a particular type of exactness properties that can be seen as category-theoretic analogues of linear Mal'tsev conditions in Universal Algebra. See this list for relevant papers in this research area. The study of matrix properties led to the theory of "approximate operations" developed jointly with Dominique Bourn, and a general theory of exactness properties developed jointly with Piere-Alain Jacqmin.
Link to a talk on noetherian forms at the PALS semnar: written summary , recording of the talk . Noetherian forms are mathematical structures defined by self-dual axioms, that include all lattices, Janelidze-Marki-Tholen semi-abelian categories and Grandis exact categories. They can be seen as a realization of Saunders Mac Lane's hypothesis from his 1950 paper on Duality for Groups that self-dual axioms can be found to treat isomorphism theorems for non-abelian groups, as this is realised for abelian groups with the notion of an abelian category. Abelian categories are actually given by the overlap of semi-abelian and exact categories. The term "noetherian" refers to the fact that these forms can seen as a fulfilment of Emmy Noether's program to "disregard the elements and operations in algebraic structures in favor of selected subsets, linked to homomorphisms between structures by the homomorphism and isomorphism theorems" - quote from Colin Macl