I have never been very fond of the mathematical topic called topology, however recently I came across this very interesting video on YouTube which, in my opinion, gives by far the best demonstration of homeomorphisms.

I usually like to keep advertisements off my blog, but this video is so nicely prepared, that I thought that sharing it with others would be beneficial, hence this blogpost.

Another very nicely written text on topology which I still remember liking a lot when I was learning the subject is the text titled “Topology without tears” authored by Sidney A. Morris. The book is freely available on the internet, at his website topologywithouttears.net; once again this goes against my usual practice of not-advertising specific sources to read a subject, but this time it is because I feel that this is the most appropriate textbook one can read and understand even if the person does not have an innate interest or an innate ability to study the subject.

# Random walk on networks: talk at Symposium on Combinatorics and Probability

Recently (on April 30, 2022) I delivered a talk on random walks on networks at the Symposium on Combinatorics and Probability which was held in a hybrid mode at Indian Statistical Institute, Kolkata.

This was the first ever research talk of mine in the field of probability theory and also the second research-level talk given by me (the first one was a talk on combinatorial nullstellensatz given on the occasion of Professor Noga Alon’s birthday at a graduate level combinatorics colloquium).

The videotaped lecture of my talk along with the other talks can be found here.

#symposium-on-combinatorics-and-probability

# A rich collection of multiple choice quiz questions

I typically abstain from positing advertisements on my weblog, unless of course I believe that such a thing would help a significant proportion of my readers. However, recently I came across a collection of Multiple Choice Quiz from a friend who found it embedded on the official website of Professor Terence Tao, and going through the questions I really found them quite interesting and a few were quite challenging too. Thus I would violate my advertising principles this time and instead encourage all my readers to go through the questions. The quiz is available at this page.

#multiple-choice-quiz

# Density and divergence of sum of reciprocals

In a previous blogpost I discussed about the divergence of the sum of reciprocals of the prime numbers. Interestingly, although that problem does not have any direct connection with the contents discussed here, but a friend of mine asked me the following problem which could be solved using similar arguments as used in the one with prime numbers.

Proof: By passing on to a subsequence $(k_n)$ of natural numbers going to infinity in $n$ we may assume that $\lim_{n \to \infty} \frac{|A_n|}{n}$ exists and is positive, let $\lim_{n \to \infty} \frac{|A_n|}{n} =: d (>0),$ thus for all $\varepsilon >0$ there is a natural number $N_{\varepsilon}$ such that for all natural numbers $n > N_{\varepsilon}$ one has $d - \varepsilon < \frac{|A_n|}{n} < d + \varepsilon.$ Take $\varepsilon = d/2$ then it follows that for all natural numbers $n > N_{d/2}$ one has $|A_n| > \frac{nd}{2};$ now taking $n$ sufficiently large so that $nd/2 >1$ it follows that there is some $m \in \mathbf{N}$ such that $|A_m| > 1.$ By an exactly parallel argument it will follow that for all natural numbers $k$ one has $|A_{km}| > k,$ whence if $A = \{ a_1 , a_2 , a_3 , a_4 , ...... \}$ where $a_j < a_{j+1}$ for all $j \in \mathbf{N}$ then we must have $a_1 \le m , a_2 \le 2m , ..... , a_k \le km$ for all natural numbers $k,$ whence $\sum_{a \in A} \frac{1}{a} = \sum_{j \in \mathbf{N}} \frac{1}{a_j} \ge \sum_{j \in \mathbf{N}} \frac{1}{jm} = +\infty ,$ and the proof is complete. $\square$

I do not see a direct relationship between this result and the result on divergence of the sum of reciprocals of prime numbers, except at the level of their proofs. Thus anyone who can find a connection between these is encouraged to share their thoughts through comments below.

#easy-beautiful-proofs

# A puzzle inspired by Professor Tao’s airport-inspired-puzzle

More than a decade ago, Professor Terence Tao posted a nice puzzle here on his weblog. Recently, I was talking to my friend about this and he being a physics student was inspired by this puzzle and we together came up with another nice problem which I intend to discuss over here.

Note. Here an optimal strategy would be of the form: one person carries one of his friends on the motorcycle till a certain distance before their destination and drops him/her there while letting the third friend walk till that time, then the first person turns the motorcycle and travels in the opposite direction until he/she finds his/her third friend (who has walked some distance by this time) and picks him/her up on the motorcycle and travels to the destination, or drop him/her a certain distance ahead of the destination and so on and so forth. Assume that the pick-up and drop-down actions take zero time to occur.