### Welcome to our community

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Hey MHB!

caffeinemachine here. I guess it's about time I introduce myself.

My interest in mathematics piqued when I was in high school and got introduced to Euclidean geometry as part of usual coursework. I sat for the Indian Institute of Technology Joint Entrance Exam (IIT-JEE) and got selected for a 5-year course in mechanical engineering at IIT Kharagpur. IITs are a big deal in India. And there is so much hype about it that any high school student who is good in math/physics is supposed to crack the JEE exam and become an IITian.

I did not find engineering interesting. And I was hungry for learning math. Somehow I got to know that one can solve the Rubik's cube using group theory. So I started reading group theory from Dummit and Foote, and it was a disaster. I had always been fascinated by calculus, which lead me to read real analysis. Back then, MHB used to be MHF, and I remember Deveno, OpAlg, Ackbach(Ackbeet) and many others answering my questions. Without MHF I wouldn't have learnt anything. By the end of my third year I had decided that I would pursue mathematics after I graduate with an engineering degree. It was a difficult decision because I was struggling so hard with almost everything I was trying to learn.

I received my mechanical engineering degree in April 2014, and wrote the entrance exam for the master of science course at the Chennai Mathematical Institute (CMI). I loved the CMI experience. Made some great friends and learnt more math in one sem than what I had learnt throughout my undergrad.

In hindsight, engineering wasn't all bad. I particularly liked the kinematics-of-machines course where we were taught about mechanical linkages. Basically how simple mechanical objects fit together to give rise to cool devices. A mechanical watch would be an iconic albeit too-complicated-to-be-taught-in-a-course example. Mechanical linkages have been studied by mathematicians too, and I would one day like to understand the available mathematical literature on it. It is a great place where concepts from differential geometry, algebraic topology, algebraic geometry, convex geometry naturally come into play. I wouldn't have ever come to know about this if it weren't for my engineering background.

In 2016 I joined the PhD programme in mathematics in the School of Mathematics at the Tata Institute of Fundamental Research (TIFR), Mumbai. Since my PhD has started I haven't been able to participate much on MHB. Some of it is due to my bad health. Another reason is that not many advanced questions in my areas of interest are posted here. I really want MHB to grow and become a powerhouse for experts in all domains of mathematics.

#### Euge

##### MHB Global Moderator
Staff member
Hi, caffeinemachine !

Thanks for sharing your story. What research do you do now at Tata? I hope you get well soon!

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Hi, caffeinemachine !

Thanks for sharing your story. What research do you do now at Tata? I hope you get well soon!
There is this new area of graph limits developed by Lovasz and co. I am not sure what exactly was the original motivation behind this concept, but here is one. We have notions of limits and completions in various domains in mathematics. One could ask the question: What is an appropriate notion of a limit of a sequence of finite graphs. Lovasz and co. defined when a given sequence of graphs converges in terms of homomorphism densities into each member of the sequence form a given graph. Then they talk about appropriate limit objects. It turns out the limit objects are "weighted graphs having the interval as the node set" and called these graphons. May results in graph theory have found cleaner reformulations in this new language of graphons. One such notion is quasirandomness. There is a lot more here. But I only started this in the month of March. As you already know, I was not happy with what I was reading prior to this. I talked to my advisor, who knew that I had an interest in combinatorics and probability. So he suggested this topic. I am liking it.

Thank you for your wishes.

Have you visited TIFR before?

Staff member

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Mechanical linkages have been studied by mathematicians too, and I would one day like to understand the available mathematical literature on it. It is a great place where concepts from differential geometry, algebraic topology, algebraic geometry, convex geometry naturally come into play.
Hey $\ce{C8H10N4O2machine}$!

Nice introduction!
I am wondering though, how does differential geometry or any of the other concepts you mentioned come into play in mechanical linkages?
Until now I always thought that the only real application of differential geometry was in the theory of relativity.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Hey $\ce C_8H_{10}N_4O_2 \text{machine}$!

Nice introduction!
I am wondering though, how does differential geometry or any of the other concepts you mentioned come into play in mechanical linkages?
Until now I always thought that the only real application of differential geometry was in the theory of relativity.
Look at the paper The Rigidity of Graphs by B. Roth and L. Asimov, Transactions of American Mathematical Soc. 1978. This is the oldest paper that I could find which discusses the fundamental aspects of linkages.