ZURAB JANELIDZE'S BLOG

When the world around us gets too difficult to bear, the mind escapes to a different world where a rainbow is all it takes to brighten up the day. And yes there will be a river of tears, but also a bridge will rise over it. Beyond the bridge, beyond the rainbow, there are bright yellow planes. But you decide to stay where you are since the tears have made the ground on which you stand fertile. NFT

Summary Initiated seven new collaborative research projects within the Mathematical Structures research programme, that includes researchers and postgraduate students from various universities in South Africa: operator semigroups, measure structures, metric frames, canonical extensions, ranked monoids, sum structures, lower topology. Supervised and co-supervised nine postgraduate students (two honors, two masters, and five phd). Represented South Africa at the General Assembly of the International Mathematical Union along with a colleague in Mathematics Education. In collaboration with colleagues and students, developed and delivered a successful math-music theatrical production for the celebration of the International Year of Basic Sciences for Sustainable Development. The production was supported by NITheCS, ASSAf and DSI. Developed and delivered four national postgraduate courses online: SOFiA on python, mathematical structures (in collaboration), introductory set theory (in c...

 

This piece is to remind you of resilience, or toughness. Life is not a straight path and there come moments when the best you can do is endure. This requires bring out the fighter within you. It may also require you to stay focused.  Look out for the following objects in the video, which carry the symbolism as described below: Light sources (light bulbs, lanterns, etc.): ideas that could help you get through the difficult times Switched off TV screen with headphones over it: the feeling of emptiness Guns, glasses and the helmet: self-defense mechanisms  Male and Female characters: your body (male character) and your soul (female character)

Many real-life situations lead us to considering a mathematical problem dealing with finding all possible numbers \(x\) satisfying a certain formula. In most primitive cases, this formula is an equation involving basic arithmetic operations (like the one we considered in Lecture 1). As an example of a formula that does not fall in this category, consider the following one: \(x<y^2\) for every value of \(y\) (Formula A) In other words, the formula expresses the property that no matter what value of \(y\) we pick, we will always have \(x<y^2\). Let us write this purely symbolically as follows (so that it looks more like a formula!): \(y\Rightarrow x<y^2\) (symbolic form of Formula A) In general, the symbol "\(\Rightarrow\)" describes logical implication of statements. Here the implication is: if \(y\) has a specific value then \(x<y^2\). In the symbolic form above, the assumption that \(y\) has a specific value is expressed by just writing \(y\) on the LHS (left-han...

In this blog-based lecture course we will learn how to build mathematical proofs. Let us begin with something simple. You are most likely familiar with "solving an equation". You are given an "equation", say \[x+2=2x-3\] with an "unknown" number \(x\) and you need to find all possible values of \(x\), so that the equation holds true. You then follow a certain process of creating new equations from the given one until you reach the solution: \[2+3=2x-x\] \[5=x\] This computation is in fact an example of a proof. To be more precise, there are two proofs here: one for proving that if \(x+2=2x-3\) then \(x=5\) (Proposition A), and the other proving that if \(x=5\) then \(x+2=2x-3\) (Proposition B). The first proof is the same as the series of equations above. The second proof is still the same series, but in reverse direction. The two Propositions A and B together guarantee that not only \(x=5\) fulfills the original equation (Proposition B), but that there is...

What does a "pure mathematician" do? A shoemaker makes shoes, a musician makes music, an applied mathematician uses mathematics to solve some real-life problems... Each of these job descriptions have some sort of measurable output. What is such output for a pure mathematician?  Some will say that a pure mathematician solves problems in mathematics, i.e., mathematical problems that are not necessarily related to "real life". This does not do justice to the efforts of a pure mathematician: if you are keen to solve problems, rather solve real-life problems! The problem is that the language in which these "pure" mathematical problems are solved is such that it cannot (always) be used to solve the "real-life" problems. A pure mathematician wants to solve only those problems whose solutions are expressed in a pure mathematical language. This does not do justice to the efforts of a pure mathematician either: what a picky attitude! Besides, solve-a-prob...

Background The goal of this project is to build a proof assistant based on the SOFiA proof system, where the capital letters in SOFiA stands for Synaptic First Order Assembler (the purpose of the lower-case "i" will be explained further below) .  The use of terms "synapsis" and "assembler" is a suggestion of Brandon Laing, who wrote an MSc Thesis, "Sketching SOFiA" (2020), where the notion of an assembler was introduced: an assembler is the monoid of words in a given alphabet, seen as a monoidal category. The main result of his MSc Thesis was a characterization of assemblers using intrinsic properties of a monoidal category. An assembler gives a robust theoretical framework which guides the syntactical structure of the SOFiA proof system. The latter has been refined through a series of discussions with Louise Beyers and Gregor Feierabend in 2021, after which the first computer implementation of the SOFiA proof system was produced, based on the Py...

The bracket notion for mathematical proofs is an adaptation of the Fitch notation for Gentzen's natural deduction proof system. It has led to the development of the SOFiA proof assistant. This post brings together some videos explaining the bracket notation and the first-order formal language for mathematics in the context of the bracket notation. 1. General Overview ~ 20 min 2. Building Blocks for Statements ~ 1 hour 3. Examples of Forming Statements ~ 40 min 4. Examples of Forming Statements (Continued) ~ 35 min 5. Concluding Quantified Statements ~ 35 min 6. Concluding from Quantified Statements ~ 1 hour

Here you will find the content for the Category Theory course given under the National Graduate Academy NGA-Coursework of the CoE-MaSS . The lectures are on Saturdays 9:00-11:00.  Register here to receive the Zoom link for joining the lectures There is also a Discord channel for this course, which you can find on the Discord server  of the NGA-Coursework project.  This is a video-based course aimed at post-graduate students and as well academics interested to learn about category theory, with live participation of the audience shaping the content of the course. For a reading course at the South African honors level, see: Category Theory: A First Course  by George Janelidze For an introduction to category theory for non-mathematicians and undergraduate students, see: Elementary Introduction to Category Theory  by Amartya Goswami and Zurab Janelidze  Lecture 1: Categories Lecture 2: Functors Lecture 3: Natural Transformations Lecture 4: Adjunctions Lect...

This page contains resources for a SATACS course at NITheCS that runs over the second semester of 2022. The lectures take place on Zoom on Fridays 17:00-19:00. If you would like to join them, register here . Lecture 1: Magmas Lecture 2: Join Semi-Lattices Lecture 3: Relations Lecture 4: Universes Lecture 5: Posets Lecture 6: Groups Lecture 7: Topological Spaces Lecture 8: Posets II Lecture 9: Posets III

This is the blog post of the 2022 August NITheCS Mini-School. Let us begin with some useful links: Lecture notes on universes of sets (introductory)   Some videos explaining the concepts from the lecture notes above Lecture 1 Lecture 2 Lecture 3 Lecture 4

Guitarist This picture symbolizes a human state when one is working on a routine task, while one's mind looks into the bigger picture of things. While guitarists hands are busy playing on the guitar, his eyes are looking into the open space from a balcony. The fence of the balcony symbolizes the restrictions imposed on us by the necessity of a routine task. Contemplation This picture shows the back of a woman with yellow hair, in a stylish red dress, gazing at the white moon. The hair is blowing in a light wind. Mountains covered in snow are in the background. Her outfit is certainly not a match for the cold weather, but contemplation will keep her body warm. This picture symbolizes that deeper things in life can give us physical strength. Camille This drawing shows two sides of Camille Vasquez, one of the lawyers defending Johnny Depp in the livestreamed trial of defamation of Johnny Depp in 2022. The picture shows a head and a body side-by-side. The head, with a thoughtful expres...

  This freehand digital artwork represents the idea that significant change requires restructuring of foundations. The solid ground on the right middle side of the picture (the foundations) dissolves into an uproar of waves that illustrate the process of restructuring, which may appear to be chaotic. Going back from the left to the right side of the picture the waves subside into (new) foundations.

The inspiration for this composition was a discussion with the Composer Hans Roosenschoon, during which I presented to him some of my musical works, and also had a chance to listen to some of his unpublished works. I composed the piece the next day after the discussion. It was originally intended to be only the first sketch of the composition. This piece signifies beauty of creativity. The video shows two contrasting characters, which represent the mind and the soul in a creative process. The roses are the creation. They appear in different color, form and contexts, to signify diversity of creation. Thus, the video provides a symbolic interpretation of the process and product of creativity. Elaboration of some of the symbolism in the video: The bud with the sun in the background shown at the start of the video represents an idea that starts the creative process.  The rose opening up, which is repeated three times in the video, represents the anticipation of the fulfillment, the ful...

The story begins with an observation made to me by my father, on 22 February 2022, that it is a special date. Afterwards, I found out he was representing this date as 20220222 (year first, month and then day). In my original interpretation, it was 22022022. I was not satisfied just with the fact that this is a palindrome (a number which, when reversed, gives back the same number). So I looked into its prime factors: 22022022 = 2 x 3 x 11 x 2 x 3 x 11 x 333667. After a while of staring at this, I checked what would happen if I reversed 333667. That number (766333) turned out to be a prime as well! I learned afterwards that prime numbers whose reverse (in decimal notation) are prime, are called reverse primes . So then 22022022 is a composite reverse prime , i.e., a natural number who all prime factors are reverse primes. The question was: how special does this make the number 22022022? To be continued.

These are notes in progress for a talk given at the online user group conference of the advanced programme mathematics organized by ieb (19 February 2022) 1. Introduction In my experience, what makes transition from school mathematics to university mathematics hard is the depth of engagement with mathematics that university mathematics requires of students, compared to the depth of engagement that school mathematics requires. Do you agree or disagree with the following thesis: A school learner must understand school mathematics at the same depth that a university student is expected to understands university mathematics. If you do not agree and think that a university student should understand university mathematics more deeply than a school learner understands school mathematics, this means that you expect a learner transitioning to university not only having to learn more advanced mathematics, but to understand it more deeply than they understand its foundation, the school mathematic...

Scroll down for video lectures 1. What is a Mathematical Proof?   Mathematical proof is a method of discourse which allows a human being to:  discover new mathematical knowledge, analyze existing mathematical knowledge, verify truthfulness of a piece of mathematical knowledge.  The ability to construct a mathematical proof is part of human nature. It is closely related to the ability to form thoughts and reason. Mathematical knowledge is knowledge of abstract principles about our universe. As such, it requires use of symbols to represent entities that are inherently abstract. For example, the symbol 2 may represent 2 apples or 2 pears . The number 2 is an abstract entity, since it is not confined to any of these concrete representations.  Mathematics functions at different levels of abstraction too. For instance, we may write a symbol, such as n , to represent any number. In one case we could have n = 2 , and in another case we could have n = 3 . This is...