ZURAB JANELIDZE'S BLOG

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...