The Mathematics that I wanted to learn: The trajectory of an average mathematician ( Part-2)

I am back again to tell you the stories about my strange mathematical education. I was finally able to escape and arrive the Indian Institute of Technology, Kharagpur, (IITKGP for short) which was about 120 kms south of Kolkata. It was way back in 1991. Kharagpur was essentially a railway town and other than that the only thing that it can boast of was the IIT.

The IITs were mainly built for technical education but they also provide degrees in the sciences which were much effective then the one’s given by the universities. I found my way into the hostel which would be my residence for the next two years. I was quite baffled by the huge campus of IIT Kharagpur. The first thing that I faced there was ragging which was done by our seniors but of a milder version than the stories I have heard.  I was told that 2 year Masters students of science are indeed considered as second class members of the IIT student community. It was the B.Techs who were the kings. But this did not bother me at all. I just wanted to learn mathematics get access to the great library. As per my other needs I was very happy that Kharagpur was also an important center for my other great love —the railways. So anyway I would have a nice time and indeed I had a nice time.

I thought that the mathematics that I would learn would be exciting. I also thought that I would be taught by fine researchers who would show us the excitement of the subject. But this just remained a dream and did not happen that way.  I had wanted to learn more mechanics, more probability, more analysis and more linear algebra. I also thought that I will learn topology and differential geometry.

Alas, things did not happen in that way. The subjects that were listed for the first semester were strange for some one who had a rigorous undergraduate training in mathematics. We had to learn PASCAL programming, Numerical Analysis, Measure Theory, Complex Analysis and Fluid dynamics.

We came out of the Fluid dynamics course without learning the Navier-Stokes equation. We were taught from an uninteresting book by Chorlton. The mess was more severe in the measure theory course. It was  taught in a way as if it was history and modern mathematics did not depend upon it. It was taught by someone who had worked in elasticity and possibly learned measure theory from the book by Tishmarch on function theory. That was the prescribed text when much modern texts were already in the market. The instructor in the measure theory course was also the Head of the Department of Mathematics spent most of his time during the day in the HOD’s office and did so even during his alloted class hours. He used to come to the class and ask us to read from some portion from Tischmarch. One day he came in asked us to prove that the Cantor set is of measure zero. Then he said ” this problem does not worth a cowbell, what to say of a Nobel”. He smiled and left the class.

Classical Mechanics the subject that was so close to my heart in my undergraduate days was taught in IIT Kharagpur in the most boring way. It was done in such a way that I lost almost all interest in the subject. We were taught from Goldstien’s book on classical mechanics without even bothering to tell us why we need to study mechanics in that particular way. At the end things must follow the F = ma principle. We were just blindly doing things and to my surprise I got a good grade. I then understood that we were not being taught by researchers and instead being taught by people who had actually lost all contact with serious mathematics. The only glimmer of hope was a course on the Calculus of Variations taught by Professor A. S. Gupta a leading researcher in Fluid dynamics. Then I realized how we have been taken for a ride. Analytical mechanics taught without telling us even the term , calculus of variations.

The most surprising the course that I faced during my masters course was the General Theory of Relativity. Albert Einstein has been on my mind since I picked up Nigel Calder’s book Einstein’s universe. Though I did not understand much of what was written there but I felt a thrill about making an attempt to understand relativity. I was very keen to know how E=mc^2 was derived. Luckily in the library of St. Xaviers  College, Kolkata ( Calcutta at that time) I had found a book on Special relativity by A. P. French. Through that book I first learned how mechanics is actually done. I learned first time about the Galileo’s principle of relativity and the notion of a frame of reference and was surprised that these were not at all taught in our mechanics course for mathematics students. I had enjoyed the book by French and developed a life-long love for mechanics and a wonder for the theory of relativity and Albert Einstein. When I told our teacher that no one in our class really knows anything about Special Relativity he smiled and told that he assumes that we know it. What an irony. Somehow I managed to go through the course on the General Theory of Relativity and of course without understanding and I secured a decent grade and I really have no idea how. Such was my dismal training and I finally could build up some decent idea only when I was doing my doctoral and post-doctoral work.

Functional Analysis was taught in a much better fashion and we learned upto the Closed Graph Theorem and did not move beyond. The course was linear algebra was not very interesting either. In fact I also had to go through a course on Systems Programming which was taught from the book by Donovan. I was completely taken aback since we were leaning a assembly language and I wanted to learn mathematics.

I shudder at the very thought that today I am a Professor of Mathematics and I have such a dismal background. But I realize that it is my love for mathematics that has possibly carried me through.

Again as I have told before that through this blog I wanted to share my excitement of the parts of mathematics which I love. I doing so I also want to refine and relearn the subjects that I love. Euclidean Geometry, Classical Mechanics and Convexity. Albert Einstein will also be a part of our discussion.


Joydeep Dutta





The Mathematics that I wanted to learn : The trajectory of an average mathematician ( Part-1)

One of the most incomprehensible thing for me is that I am in the mathematics profession. With a little apprehension I would like to call myself an average mathematician. In fact I was possibly never cut out to study mathematics. I was the worst student of mathematics in my batch in school till the eighth standard. I did not even know what a factor was, what is the meaning a multiple of a number and so forth. I had failed most of my exams on mathematics till then. But possibly as “Don Corleone” had said everyman has but one destiny, I was destined to fall in love with mathematics. In my eighth standard I was lucky to have Mr Nilachal Samanta as my teacher. He told me that he used  to fail in mathematics and later on  excelled in it and had the chance to attend the lectures of the great theoretical physicist S. N. Bose at the University of Calcutta. He used give a lot of time to me and spoke to me about mathematics as if he was telling me a story. He largely spoke about geometry. At that age when kids are more bothered about their next football match I learnt about the Babylonians, about Euclid, about Archimedes, about Apollonius, about Brahmagupta and so on. One afternoon I suddenly felt that I was understanding what my teacher was telling me and was even enjoying it. That evening I went back to my  hostel room and started looking at the exercises on geometry. They were called riders in our days. To my surprise I found that I could effortlessly apply the the correct theorems, do the correct constructions and solve the riders. I went back the  next day and told my teacher that ” I have fallen in love with mathematics”. He patted my back and said, ” now you can go your own way”. I remain forever indebted to him. My love for mathematics continues to remain as strong as it was on that fateful day and seems to grow more with every passing day. To my surprise I shunned every other choice of a profession and went to study mathematics at the St. Xavier’s College of Calcutta University.  There I learned modern algebra, real analysis, linear algebra, analytic geometry, probability and statistics, linear programming and even the mathematics of astronomy.  However two subjects actually caught my fancy. One was Newtonian mechanics and the other was real analysis. I still remember Professor Ram Ghosh starting his lectures on real analysis with the sentence…let epsilon be greater than zero no matter how small. Mechanics was fun. You essentially had to set up the differential equations and solve them and these differential equations are nothing but simply the Newton’s second law of motion. What we really lacked while studying mechanics was the stress on the physical intuition. We simply resolved the forces and never asked questions about why we ignore some types of forces and not others, while setting up the differential equations of motion. Our prescribed text was by a local mathematician and the reference text was the famous ( or dreaded) book on mechanics by S. L. Loney. The book by Loney was more that 100 years old and the problems were really challenging. The challenge was not in the physics but in the algebraic manipulations. I was bored of that text and thus one day went in search of a good text on mechanics in the famous book district of Calcutta called the ” College street”. I found one and that was by Smith and Smith. I had enough money to buy that from Sarat book house which probably is the best address in India for books on maths and physics. I really loved that book. It gave the physical intuition, and studied everything using vectors, and that fascinated me. I truly learnt about Newtonian mechanics from that book, and it indeed saved me from being thrown out from  the list of the  select number students who retained their honours in mathematics in their B.Sc. ( Bachelor of Science ) Part-I examination. This was way back in 1990.  The book is still on my bookshelf. I faired poorly in the examination. Some of my friends were surprised looking at my marks since they knew among them I was probably one of the most passionate about mathematics. One my friends Sandip Banerjee ( now a faculty in mathematics at the Indian Institute of Technology, Roorkee) told me that I did not take a proper approach to the University examination. The technique was to try and solve the question papers that appeared in the last 10 years of the University examination and one was bound to get questions which would get repeated and one can just vomit the answer without any thinking. So in mathematics there was no credit for thinking. It reminded me what Richard Feynman said of the education system in Brazil in the late fifties: ” I cannot understand how could anyone get educated by this self propagating system of education where everybody was passing the examination and teaching others how to pass the examination”.  We were basically in the same state even in the early nineties and  probably we are still the same now. It goes without saying that I did heed Sandip’s advice and prepared properly for my Part-II examinations. In fact during the first two years when my friends were solving the question papers of the last ten years I was busy reading ” Topology and Modern Analysis” by G. F. Simon’s and Mechanics by P. C.  Smith and R. Smith. Using my pocket money I had bought some excellent texts from a store in College street which sold very cheap books by Mir Publishers, Moscow. Those books were written by excellent Russian and erstwhile Soviet mathematicians and were of very high quality. I spent most of my time with these books rather than really preparing seriously for the exams. I had a much smoother run in my Part-II and finally got out from the Calcutta University with a second class honours in Mathematics. I understood that the most inevitable thing for me was to flee from Calcutta University. I was really lucky that I got through the examination for a 2 year master’s degree in IIT Kharagpur.  However St. Xavier’s college was indeed fun and my teachers were truly kind. I fondly remember my teachers there and specially Dr T. K. Chakravarty. The mathematics environment of St. Xavier’s was filled with the legend of Father Goreaux, a renowned mathematician , and was also said to be one of the last students of Albert Einstein.  Now I know that Father Goreaux  may not have been a Phd student of Einstein since according to Abraham Pais, Einstein’s biographer, Einstein, did not have a Ph.d. student. I remember talking to him several times and he also gifted me an analysis book my the famous mathematician De-La-Valle-Poussion. I felt very elated that day but I never had any courage to ask Father Goreaux about his interactions with Einstein. Of three central themes of my blog two of them would be from my school and college. One would be to again feel the hidden pleasures of Euclidean Geometry and the other would be to regain the lost joys of classical mechanics. In the second part of this article I will tell you the my experiences as a graduate student in IIT Kharagpur.