ZURAB JANELIDZE'S BLOG

The Merriam-Webster Dictionary defines the word "monotony" as "tedious sameness". For a creative soul, monotony is on par with torture. This piece conveys the struggles of the soul that has been trapped in monotony and is looking for ways to break free from it. The video shows a man performing various free style dance moves as an expression of the struggles. Behind him is a wall on which, in the first part of the video, projects a movie of clouds moving through the sky. This is a symbolism of one's creative aspirations before one becomes a prisoner of monotony. In the second part of the video, the projection changes to showing the dancer in action, from multiple angles. This symbolizes that after a while of monotony, the aspirations dissolve into excessive awareness of one's own self. At this stage, an illusion of having broken free from monotony may arise, which is symbolized by the scene with the chair. The chair is the illusion of freedom, illustrated by ...

The process of creation of mathematics has the following hierarchically dependent components: Coming up with a concept. Coming up with a question dealing with a relationship between concepts (this includes formulating a hypothesis, as well as finding an example or a counterexample of a concept/phenomenon). Answering a question dealing with a relationship between concepts (this includes proving theorems as well as solving problems without being given the recipe for solution). Applying the answer to a question dealing with a relationship between concepts to answer another such question (this includes solving problems by applying a given recipe for solution). Modern mathematics education (both at the school and at the university levels) focuses mainly on the last two points. What is regarded as a low quality mathematics education would focus only on the last point. For a more whole mathematics education, the first two points must receive as much attention as the last two points do....

work in progress I propose here a theory of human function, which I have been developing based on introspection. In this theory, human function is represented in terms of exchange of information of four agents, which I call the spirit , the mind , the soul and the body . Although these are surely familiar terms, having a variety of scientific, pseudo-scientific, religious, philosophical and other usage, I do not assume any insight derived from such usage. The essence of each of these agents will be revealed through the roles that they play in human function. Matching of this essence with any of the existing definitions of these entities is unintentional and may well be coincidental. The four agents are organizes in the following directed graph: We call it the Human Function Scheme ( HFS for short). The arrows represent directions of information flow from one agent to another. Postulate 1.  Human function is marked by internal information processing within each of the four agents...

Link to a plenary talk on noetherian forms at a BRICS conference (2021): slides of the talk , recording of the talk . Link to a talk on noetherian forms at the PALS seminar (2020): 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 realized 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 ...

These are notes for my online discussion with Georgian Student Parliament on Tuesday 7 December, 2021, organized by Nina Tsatsanashvili. All photos in this post are from Wikipedia.  If you look up the word "structure" on the Google Dictionary , you will find the following definition: the arrangement of and relations between the parts or elements of something complex. When the "parts or elements" are subject to specific interpretation, we have at hand an "abstract structure". For example, consider a painting, which can be seen as an arrangement of colors. For instance, Leonardo da Vinci's Mona Lisa : This is not an abstract structure, since its constituents are specific colors that can be found on a specific poplar wood panel that currently resides in a gallery of the Louvre Museum in Paris. In contrast, Ludwig van Beethoven's Für Elise  is an abstract structure, since the sounds that make up this musical piece are dependent on the interpretation of...

Notes for the talk given at the 2021 Congress of South African Mathematical Society on 29 November.  1. Finitely complete categories A finitely complete category is a category that has finite products and equalizers (and hence, all finite limits). Not every category is finitely complete, but most categories of mathematical structures are.  There is a representation theorem for finitely complete categories (Yoneda embedding), which allows to present any category as a full subcategory of a (larger) category of presheaves of sets, which is closed under all limits that exist in the category. This means that a lot of times, proving a theorem in a finitely complete category involving finite limits, reduces to proving the same theorem in the category of sets.  For instance, the fact that the product of objects is commutative, up to a canonical isomorphism, can be deduced from the fact that the same is true for the cartesian product of sets. Or, the fact that the composite of two...

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...

Calculus  (notes in progress in Real Variable Calculus) 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’...