ZURAB JANELIDZE'S BLOG

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 Pierre-Alain Jacqmin. Work in progress on matrix properties: Open problem on finding an algorithm for implication of basic matrix properties solved - see the working version of the  preprint .  Even for binary matrices, the preorder of implications is quite complex. Some new results on this appear in this work in progress. Python implementation of the algorithm for deducing implication of (basic) matrix properties can be found here . The program needs to be improved in some future.

Summary Elected as the President of the South African Mathematical Society. Papers on exactness properties published in Journal of Algebra and Advances in Mathematics. Invited to give a plenary talk at the BRICS Mathematics Conference. Secured funding for a national research programme in mathematics. First computer implementation of the SOFiA proof system developed. Supervised four postgraduate students (two PhD and two MSc). Two papers on matrix properties submitted. Served as the mathematical sciences programme coordinator and on a university research committee. Taught and/or convened two semester modules and two year modules. Progress made on existing and new research projects and delivered talks on those. Carried out duties in the role as mathematical sciences programme coordinator and member of a university research committee. Carried out refereeing and editorial duties (not listed below). November-December 2021 Finalized marks for Foundations of Abstract Mathematics I, II and ...

Below is a summary of the talk given at the Séminaire Itinérant de Catégories  (8 October 2021), prepared before the talk.  The talk is mainly based on Zurab Janelidze's joint work in progress with Francois van Niekerk, as well as his earlier work on forms with former collaborators. The talk assumes that the listener is familiar with basic ideas and concepts of category theory found in Categories for the Working Mathematician by Saunders Mac Lane (in particular, Chapters I, VII and VIII), as well as with the notions of factorization system and Grothendieck fibration. 1. Biproducts, products, sums and monoidal categories The goal of this talk is to explain the following diagram: The notion of an abelian category brings together various important categories of abstract mathematics, such as the categories of modules, which includes the category of vector spaces as well as the category of abelian groups. In an abelian category, the monoidal structure of product and the monoidal...