The Probabilistic Method

This summer, I began studying the Probabilistic Method under Prof. Young from the book The Probabilistic Method by Noga Alon and Joel H Spencer.

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The book is very interesting but equally advanced. I have only managed to complete three chapters in three months. Most of the exercises are beyond me but the subject is fascinating.

The first chapter was about the method itself and a few “basic” examples. Say, we want to find a structure with some desired properties. Sometimes, it’s difficult to find the actual structure. But, what we can do is define a probability space of structures which have these properties. Also, we show that these properties hold in the space with positive probability. Thus, although we can’t find the actual structure(s) with this method, we can confirm its existence.

The second chapter dealt with Linearity of Expectation. Easy enough. The main fact is that if we have some random variables X_1, X_2, ..., X_n and X:= c_1X_1 + ... + c_nX_n, then \mathrm{E}(X) = c_1\mathrm{E}(X_1) + ... + c_n\mathrm{E}(X)_n. The applications this simple result has can never fail to amaze you.

The third chapter was on Alterations. Sometimes, the random structure we deal with may not have all the desired properties. There may be a few “blemishes” as Alon and Spencer put it. Then we tweak the method a bit to remove these blemishes. This chapter is quite non-trivial and requires some higher concepts from general mathematics.

I hope I can finish this book in some time. It will take me more than a year at my current speed and the difficulty seems to be increasing exponentially.

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Back to college – Prof. Jacobi in particular

So, after a vacation of two months, I think I’ll start posting again. My semester has begun and life is beginning to become frustrating again. A friend at college has captured this situation with a perfect WhatsApp status:

“Back to the salt mines!”

We have five courses in this semester: Complex Analysis, Differential Geometry, Statistics, Physics and an elective for which I have chosen Representation Theory. The other contenders were Differential Topology, Computer Science and Economics.

Prof. Jacobi has been assigned to teach us Complex Analysis. He has been teaching this course for a couple of years, and he was our favourite to take this course. He wasn’t in campus the first two weeks. He returned this week only.

On Thursday, he took our first class.Within minutes we were enthralled. We know he’s a very lenient teacher. But that didn’t stop him from frightening us. He told us to pass at one go, in his words, “Pass in the front paper itself.” (His English is not the best) Otherwise,

“If you kill my time (by making me prepare a back paper), then I will kill you! Period.”

Whoa!

He told us how Nevanlinna (and Paatero)’s book Introduction to Complex Analysis is like Edmund Hillary and Tenzing Norgay’s conquest of Mt. Everest. Rolf Nevanlinna‘s book is one of the foremost text books in Complex Analysis. We shouldn’t use it as a text book. But we should keep it under our pillows. Everyday before going to sleep, we should study one page of the book. Then we’ll be good mathematicians, he said.

At the end of the class, he recommended Louis Armstrong’s What A Wonderful World and the movie Good Morning, Vietnam. Now that’s the first time a professor has acknowledged the existence of a life outside of Mathematics.

Today, he took a night class to cover for the classes he missed at the beginning. And he was at his best.

“Don’t come to me and ask questions. I won’t able to solve them. Ask your seniors. They are all geniuses. If you ask me to solve your problems, I will ask you to solve my problems. So be prepared.”

He was trying to decide which day he would be able to take another extra class. And he asked a classic question.

“Which day is Tuesday?”

At the very end, he made another one. We usually have 10 marks for homeworks, assignments, quizzes, etc. while the rest is divided among the mid semester and the end semester exams. He wanted to know what we wanted to do for those 10 marks. He said

“Here, no one wants to do anything. I don’t want to set questions, the grader doesn’t want to check answers and you don’t want to study! But unfortunately, we have to do something.”

ROFL