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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

A Gap in Mathematics Education

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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.  It is

Musical Works

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Introduction My musical works, which I have started creating in 2014, explore deeper abstract emotions, such as those arising from an intellectual pursuit, that may be connecting oneself with their dream world and touch the essence of existence. The works can be classified in the following categories: Impromptus  and ballades : these works are based on once-off improvisations, which may be edited for fixing accidental notes, adjusting suitable playback tempo, and introducing repetitions/transpositions. The difference between impromptus and ballades lies mainly in the playback length of the musical piece, with ballades being the longer musical works. A ballade is meant to tell a "musical story", while an impromptu paints a "musical picture".  Esprits : these musical pieces are combined with improvised performance of theatrical nature as a visual interpretation of music. Esprits belong to an artform defined by "mindful improvisation" (which I call "espr

Metaphysics of Human Function based on a Mathematical Structure

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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 as

Noetherian Forms

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

Abstract Structures in Mathematics and Music

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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

A surprising story of how a computer was taught to prove some theorems in finitely complete categories

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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 monomorphisms

Matrix Properties

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.