Topology without tears and a video about homeomorphisms on YouTube

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.

A popular statement in topology (also perceived as a joke on topologists): a topologist cannot distinguish between a coffee mug and a donut; the reason being that these two shapes are homeomorphic, or topologically equivalent. The above-linked video nicely discusses this subtle point.

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.

#topology, #videos

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.

Proposition 1:
Let A be a set of natural numbers and let A_n := A \cap \{1, ..... , n \}. Suppose that \lim_{n \to \infty} \frac{|A_n|}{n} \ne 0 or the limit does not exist. Then the sum \sum_{a \in A} \frac{1}{a} diverges to + \infty.

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.

Problem
Suppose three friends A , B, C have to gather at a place from where they decide to go to a common place. However, B, C does not have any vehicle of their own, and A only has a motorcycle in which he can carry only one person along with him. They want to travel in a way so that the last among them to arrive reaches as early as possible.

(i) If they have a common walking speed of u and if v is the speed of the motorcycle and if v > u then what is an optimal strategy for the three friends to travel?

(ii) Suppose now that A walks at a speed of a, person B walks at a constant speed to b and C walks at a constant speed of c. What can we say about their optimal travelling strategy now? (Of course, this would depend on how large or small a, b, c, are in comparison to the motorcycle’s speed and with respect to each other.)

(iii) Do the above answers change if one takes into account the various effects of special relativity?

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.

#puzzles, #special-relativity, #terence-tao