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 ...