I had my TIFR PhD interview today. Inspired by the Princeton Generals, where students who get through their Qualifying exam write about the questions asked, even I am writing here describing my interview experience. But I am not very enthusiastic about writing this, since the questions were very basic and not from all subjects, a very unlikely event during a Maths PhD interview. Nevertheless, here is the interview–

Chair: Prof. A. Sankaranarayanan

Other interviewers: Prof. Indranil Biswas, Prof. Raja Sridharan, Prof. Ravi Rao

I do not remember who asked which question, so in much of what follows, I am referring to all as “interviewer”. Most questions were asked by RR.

AS: So you work at Larsen & Toubro?

Me: No sir, I was working at L&T for an year from 2007 to 08. Then I joined IIT Bombay in 2008 till now.

RR: Under whom have you taken courses at IIT?

(I name Limaye sir and other teachers)

IB: What are your interests?

Me: Sir I do not have a particular interest as such, but I do like Algebra.

RR: K is a field. Is K[X^2,X^3] a UFD when considered as a subring of K[x]?

Me: Yes sir, zerodivisors in K[X^2,X^3] would mean zerodivisors in K[X] which is a domain. (Mistake.)

RR: So it is a domain. But is it a UFD?

Me: Ohh.. Yes.. I mean no. Consider X^6, it can be factored in two ways. (I proved that the two were irreducibles and not unit-times-each-other.)

RS: f:\mathbb R \to \mathbb R is smooth and \lim_{x\to\infty} f(x)=0 Then can you say if \lim_{x\to\infty} f'(x)=0?

(I tried for a long time to prove, then tried to give a counter-example. \displaystyle\frac{\sin x}{x} didn’t work. He suggested \displaystyle\frac{\sin x^3}{x} and it worked, though I goofed up in differentiating and taking limits)

(In the meanwhile, there was an interesting discussion among them if they should finish off Algebra before switching to Analysis.)

RR: Is \displaystyle\frac{\mathbb C[X,Y]}{(X^4+X^3Y+Y^4)} a domain?

Me: If it is to be a domain, then the polynomial should be prime, hence irreducible. (thinking) Eisenstein.. won’t work!

Interviewer: Hint — The polynomial is homogeneous.

(Solved it.)

Interviewer: A linear transformation T:\mathbb R^n \to \mathbb R^n sends straight lines to themselves. What can you say about T?

(I wrote, wrongly, a diagonal matrix with different diagonal entries. Later corrected the mistake after they asked me to repeat the question.)

Interviewer: V is a n-dimensional vector space and W is its subspace such that for every isomorphism T:V\to V, we have T(W)\cap W \neq (0). What can you say about W?

(Proved that the dimension must be \geq n/2.)

Interviewer: Consider S_p, the symmetric group on p-symbols and an element of order p in it.

(I wrote the element as a p-cycle. There followed a long discussion as to why it should be a p-cycle, and orders of commuting elements, their lcm’s etc. It finally ended with me group-acting S_p on those p-symbols and ‘proving’ that order of disjoint cycle types is the lcm of their orders.)

RR: Find a p-subgroup of \text{GL}_3(\mathbb F_p).

(This took the longest time of all. The question was interesting, involving some fundoo linear algebra. In spite of many hints from them, I was not able to solve it. Finally, RS told the answer.)

Me: (excitedly) Ohk, not just the elements of order p-power, I have found the  group of order p^3!!

RS: No, I have found it!

Me: (embarassed) yes sir..

(There followed a discussion if they should ask more questions. Finally, RS asked me to prove that the additive group of a field of characteristic zero is not cyclic. I was halfway done when I was told to go.)

Note: The interview lasted for an hour. The interviewers were pleasant but and did give subtle hints when I got stuck up. But they stopped me whenever they thought I was not giving the right justification or hand-waving proofs.

PS: I was not offered the fruit juice the others before me were offered. 😛

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