Sunday, November 28, 2010

The last proofs of Gagan Das

Gagan Das, or Gagan Babu, as he is called most often, had a strange feeling on Sunday morning.

He knew that his life is coming to an end.

It is not that he was suffering from any terminal illness. He also had a moderately optimistic view of life around him. He was unmarried, smoke once and never touched liquor. His physique is same as any average clerk in the government office. Little protruding belly, thin hair-line, a general facial expression not revealing any trait Freud had documented. His height was 5 feet 4 inches. He lived life sending and receiving letters in a zonal post office of north Kolkata. In short, his life was average and so his death should be. But, the eerie feeling of approaching death was something Gagan Babu could not explain.

Nevertheless, it was similar to the situation he faced few times before. He felt Samir Raha is going to die. Samir Raha was his childhood friend and sole companion in his morning tea session at a roadside shop. Just one day, when Samir was lifting his underweight body slowly, Gagan Babu thought he would fall and die. He was wrong. Samir Raha walked back home. In a few days, however, Samir Raha actually dropped and died. It also happened in case of Raju. Gagan Babu tutors a few needy students at home. Raju was among them. He explained how he jumps in the Ganges after Durga puja immersion and collects the ornaments of the Goddess to sell those for next year Puja. He was ecstatic on the day of Navami, the day before the immersion. Gagan Babu had a uncanny feeling that Raju is not going to come back from the river. It need not be like that. Raju did it many times before. He was an expert swimmer. But, strangely his premonition matched the reality. Raju died by hitting too deep in search of the fake ornaments. Gagan Babu felt terribly sad for few days, more because he did not warn Raju to not venture this time. But, he kept all these to himself.

And now, today morning, he had this same feeling about him. He knew, somehow, that his end has come.

Gagan Babu used to wake up early. It is 5:30 AM now, his schedule for morning tea at the shop of Bechu. He misses Raha but, still goes there. Today, he did not feel like. He thought of spending the day doing something worthwhile. He started thinking, what could be done. In an absent-minded way, he proceeded through a cold water bath, breakfast with puffed rice and watching the crows gather on top of the multi-storied nursing-home opposite to his balcony.

At 9 AM, 4 students arrived. They are all children from nearby slums. He teaches mathematics to them, for almost 10 years. The students are brought to him by members of local club or sometimes the maid servants. Mostly they are very good students, studying in standards 5 to 10. They hardly study beyond secondary level. Passing standard 8 suffices for them to get a job in many places. Mihir is different. Mihir is studying in class 9. He wants to study further, possibly with a mathematics major. He came today with his brother and two friends. Gagan Babu felt good after seeing them. He asked them to take a seat and eased himself on the chair. Their exams were over last week. Mihir brought the question paper. He glanced through the questions. Those were covered very well by him. He asked if there was any problem. There was. A geometry problem based on theorems needed to be solved. Mihir could not identify the special drawing that would drop the problem in place with known theorems. Gagan Babu quickly explained it. Mihir was sad for that only mistake. Gagan Babu thought of consoling him but, the next question from Mihir left him dumbfounded.

Mihir asked, "Sir, I tried to solve it with algebra because, I knew it depends on Pythagorean theorem."

Gagan Babu, even after being confident of his favourite student's level, did not expect that.
Mihir asked again, "Sir, can I do that ?"

"Yes, sure, why not?" - Gagan Babu replied.
"How?" - was the question again.

"Can you come tomorrow morning, at 9 AM ? I will have to think about it a little." Gagan Babu answered.
"OK Sir."

The students left.

After closing the door, the death feeling occurred again to Gagan Babu's mind. And then he thought about this small problem mixing algebra and geometry. The approach itself is novel for such basic problems. Gagan Babu did things like that in his leisure. He liked mathematics. Always. He followed the developments of mathematics from few popular science magazines. He had a friend named Santanu Mitra, a topper of his college, teaching higher mathematics at ISI, Kolkata. He visits him sometimes to ask questions. Last time it was with a request - to explain the solution of Andrew Wiles of Fermat's Last Theorem. Santanu hesitatingly mentioned that he himself did not understand beyond few pages. But, he referred an excellent book called Fermat's Last Theorem. Gagan Babu liked the book immensely. He liked the characters in the world of mathematics. Like Euler, who commented after losing an eyesight that he will now have less distractions to do mathematics. But he liked the genius of Galois most. Galois was a prodigy. He died young due to the injuries from a duel and on his last night of life, he kept awake, scribbling mathematics and solving some important problems.

Death. It again flashed in the mind of Gagan Babu. But, this time with more conviction. That he can do something better than to just die average. He can try to solve some important problems. Perhaps, his lifelong mathematical fascination will culminate, on the premises of the simple question Mihir has left.

Gagan Babu quickly finished his lunch and started working on the problem at his desk. He figured it out very easily. More easily than he thought it could be done. It required projecting the original drawing in a co-ordinate space. The corresponding arithmetic and algebraic derivation was straightforward. In algebra, it needed exactly 7 steps to reach the Pythagorean Theorem.

Gagan Babu was ecstatic that he could solve that. It was his second most important discovery, he thought. The first was in his college days, where he found an important property of linear Boolean functions. He showed it to Santanu then, who suggested a proof. They went to Father Jacob, a mathematician of repute in their college. Father Jacob congratulated them on that discovery, only to mention that it was discovered and proved 2 decades ago.

It was only 1 PM. Gagan Babu thought of going to Santanu's home with his new approach of proof. Santanu lives in south Kolkata. It will take no more than 1 hour by metro. But then, he thought, what did he want now. He is going to die anyway. He can rather die like Galois, his hero of mathematics. Gagan Babu can leave his little findings for posterity to ponder on.

Gagan Babu took a new exercise copy. He wrote his name with bold face on top. And named the exercise copy as "Gagan Das: Last Proofs". He first cleanly copied the solution of the finished geometric problem. He knew what to work on further. He had to map the algebraic derivations of polynomial equations to geometry and find their roots using geometry. It would be interesting to do it. Galois also worked on Quintic equations. Gagan Babu started drawing neatly. The derivations has to be legible and clear. For everyone to know and understand.

It was 6 pm when Gagan Babu cracked the polynomial of degree 2 or quadratic equation. He remembered his awe when his school teacher explained the Sridharacharya's rules for deriving it. He stood in front of his proof with the same awe. If not more. This must have been his most important work ever. Gagan Babu suddenly discovered that he could do more, much more than spend the entire life as a clerk. He liked the beauty of mathematics but, never knew that he could explore so deep in that.

Gagan Babu took a cup of tea and started pondering again. His goal was to go till quintic equation before his clock ticks last. He knew that could come anytime. Like sitting in a vast examination hall, all alone, without any clock. He had to finish his most important examination paper. To make a mark out of his dull, average life.

Gagan Babu woke up once more around 9 PM, to shut the window. The sounds were distracting.

When Mihir came to visit Gagan Babu next morning, there was no one to open the door. Mihir had to call the local club members to break the wooden door. Gagan Babu was sitting with his head on the desk. He died sometime in the night. His face revealed a peaceful death.

Amidst the mayhem, Mihir picked up the copy to find the solution to the problem he posed a day earlier. It was followed by more complex drawings and derivations, which he could not decipher. After few pages, there was a small note requesting the copy to be sent to a Prof. Santanu Mitra, ISI, Kolkata.

It took few hours of hard effort for Mihir and his younger brother to locate the office of Prof. Mitra. It was a small office, with lots of books and an old computer. Prof. Mitra welcomed the young kids with a smile. Mihir revealed in short about his maths teacher and his death. Prof. Mitra was sad to hear that. "Gagan was a nice fellow" - he commented. Then Mihir talked about the exercise book Gagan Babu left at his desk.

Santanu eagerly took the copy and glanced through the results. It provided some known derivations relating algebra and geometry. What's the use of it ? Then, he remembered their joint venture years ago to Father Jacob in St. Xavier's. They were overjoyous thinking they have discovered something. Gagan hit it over the same known problems again. He almost cast aside the exercise copy when, he happened to read a sentence like - "I reached Galois". It was the derivation around quintic equations.

Suddenly, Prof. Mitra figured out something more profound in that. These are known and simple proofs but, unknown to Gagan, his friend. He reached the summit of his life, achieved his individual triumph, without learning higher mathematics, without studying any text that covers these proofs.

Mihir was looking towards Prof. Mitra with expectant eyes. His brother was little bit restless. Prof. Mitra looked towards them. And told, "This is a very very significant result in mathematics. I am proud of my friend to have achieved this. I will make a copy of these results and give the originals to you. OK ?

After a pause, he said.
"Someday, these results will be taught to you in lectures. You will be proud to think of your Sir then, too."

Mihir shook his head, smiled and promised to himself that he will understand the proofs some day.

For mathematically inclined readers, here are the papers which are interesting to read.


  1. Though you asked this from K Elst ,you can see these links for Aryan Invasion Theory..

  2. ignorance is bliss - in this particular case it led to bliss. nice.