Identity, Maths Club of IISER Kolkata

Girls only love men as tall as 6' and above. Socrates, ca. 2023 It is undeniable that heights strongly influence our daily lives. Be it our heights, or the height of a mountain we scale, or the height of all problems - humans. Mathematics too hasn't been able to escape its clutches, with height functions being useful in several fields, including but not limited to - Diophantine Geometry, Automorphic forms and the Weil-Mordell theorem - something you should have heard before if you attend my talks. If you have attended school (or maybe you are a climate activist) - then try recalling the elementary school days when fractions were introduced. Albeit unknowingly, but we had as children classified fractions into proper and improper - based on whether the denominator was larger than the numerator or vice versa. Well, it seems mathematicians have stuck with this classification - giving us the crux of todays discussion - height of a rational number. Given a rational number $x=\frac mn...

A very belated Happy Pi Day to all! Traditionally, this day is celebrated by finding clever and whimsical ways of approximating $\pi$, from using series' such as $$ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots $$ to using out of control cars . One of my favourite employs randomness: take a unit square, and mark off a quarter-circle inside it with unit radius. Now, select a bunch of points uniformly randomly from the square. As you keep sampling more and more points, the fraction of points inside the quarter-circle will approach $\pi / 4$. The ratio of the number of red dots to the total number of dots converges to $\pi / 4$. Indeed, $\pi / 4$ is precisely the area of the quarter-circle. Why does this work? This is all thanks to a very useful result in probability, known as the  Weak Law of Large Numbers . When you generate the $i$th point $\mathbf{x}_i$, consider the indicator function  $\mathbb{1}_S(\mathbf{x}_i)$, which gives $1$ when $\mathbf{x}_i$ li...

Gourav Banerjee, a 21MS student, goes to the main canteen of IISER Kolkata for dinner at some arbitrarily scheduled time between 8 and 9 pm. He frequently meets an anonymous, beautiful girl in the mess and begins to wonder whether the girl is stalking him or if their meeting is just a coincidence. So he tries to compute the probability of meeting that girl in the mess during dinner time given the following constraints: Both Gourav and the girl go to mess for having dinner at some random time between 8 - 9 pm. Because of the Queue at the mess, both stay in the mess for minimum of 30 min. What do you think? Solution Let $x$ denote the time when Gourav enters the mess and let y denote the time when girl enters the mess. Here we take origin to be the 8 pm mark and a distance of 1 unit represents 1 hour on both $x$ and $y$ axis so all possible coordinates within the unit square $ABCD$ represents an event where Gourav and the girl both visit the canteen. Now the favourable coordinates which ...

... or How I Learnt to Stop Worrying and Accept Mathematical Rigour. In Measure Theory, among other things, we try to generalize the concept of 'volume' (or 'area' or 'length' according as which one you prefer). In other words, like some sets come with an inherent 'measure' associated with them (for example, intervals in $\mathbb R$ comes with a length and "well behaved" shapes come with an area or volume depending on how many dimensions you count), we want to assign to any general subset a measure of its own. Since any general subset may not look as structured as the other sets that have their own volume, ideally we are free to randomly associate any number we want to to the set. But, as in all of mathematics, we want these numbers to satisfy certain properties. So, we will impose on these numbers a few properties that are very intuitive and geometric, but to our utter shock, are inconsistent. These properties are so much in tune with our ever...

Here's a fun problem from a course on Curves and Surfaces. Consider the following curve, called the twisted cubic. \[ \gamma\colon \mathbb{R} \to \mathbb{R}^3,\qquad \gamma(t) = (t, t^2, t^3). \] Show that no four distinct points on this curve can lie on the same plane. Method 1 - The scalar triple product The scalar triple product  is a good tool to have at one's disposal (thanks to Satbhav Voleti for sharing this solution); given three vectors $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}$, their scalar triple product $\boldsymbol{u}\cdot(\boldsymbol{v}\times \boldsymbol{w})$ represents the volume of the parallelepiped whose legs are these vectors. Now if these three vectors all lie in the same plane, the corresponding parallelepiped is completely flat with zero volume, thus the scalar triple product is also zero. Coming back to our problem, suppose that the four distinct points on the twisted cubic $P_i = (t_i, t_i^2, t_i^3)$, $i = 0, 1, 2, 3$, all lie on the same plane. ...

A belated Happy Basanta Panchami to all! Once this blog took off, our interest in discussing new ideas grew exponentially ... or is it really exponential ? Let's see. A few days ago, Gourav Banerjee shared this recursion relation (along with the solution after we got tired of trying) with us. \[x_{n+1} = 2x_n(1-x_n).\]  Although this is non-linear and looks intimidating, the solution turns out to be quite elegant. The trick is to rewrite the relation as follows. \[ 1 - 2x_{n+1} = 1 - 4x_n + 4x_{n}^{2} =  (1-2x_n)^2. \] This already looks much nicer! Taking the logarithm of both sides, \[\log (1-2x_{n+1}) = 2 \log (1-2x_n). \]  Set \(b_n = \log(1-2x_n)\) ... whoa! This is a simple geometric progression , with \(b_{n+1} = 2b_n\). Thus, \(b_n = 2^nb_0\) for some initial \(b_0\), which yields \[1-2x_n = (1-2x_0)^{2^n}, \qquad x_n = \frac{1}{2} \left[1-(1-2x_0)^{2^n}\right]. \] Looks easy, right? This is an example of the more general logistic map ,  \[x_{n+1} = rx_n(1-x_...

This post mainly concerns the age old question of classifying all right-angled triangles with integer sides. In other words, we wish to find all integer solutions for the equation \[ X^2 + Y^2 = Z^2. \tag{1} \] An observant reader may have noticed that if $(X, Y, Z)$ is a solution, then so is $(kX, kY, kZ)$ for any integer $k$, and vice versa. This means that we can discard all common factors of $X, Y, Z$, and focus on solving $(1)$, with the added condition $\gcd(X, Y, Z) = 1$. Such solutions are called primitive solutions . You may remember a parameterized expression for these solutions, but do you know how to derive it? This is what we will be discussing here. There are multiple ways of arriving at the parametrization, but the one we'll use traverses the poetic bridge between numbers and geometry (the two divine deities of mathematics) with such ease, that it definitely makes it the best among them all. Rational hunt There is another simplification we c...