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)

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. 😛