Edit: Disclaimer: These questions were asked during MY interview. I have posted them thinking that they may possibly be of use to anyone preparing for such exams. However, please don’t ask advice from me. I may not be good at advising and also, it is often disinteresting.

Interviewers: Prof Nitin Nitsure (TIFR), Prof C S Rajan (TIFR), Prof Balwant Singh (IITB), Prof Shobha Madan (IITK). Most questions were asked by Prof Nitsure, including the last one. They asked me what I was interested in. I told Algebra; Nitsure sir observed that I had correctly attempted all but one question in the Algebra section of the written exam. I was asked to try that question then but I gave up.

$f$ is a morphism from $\mathbb A^1$ to $\mathbb A^1 \backslash \{0\}$. Then what can you say about $f$?

(I interpreted the question in geometry to algebra, worked out coordinate ring homomorphisms but couldn’t reinterpret the result geometrically.)

State the inverse function theorem.

(I couldn’t answer that one and I personally think, one who doesn’t know InFT doesn’t deserve to get a PhD scholarship. There followed a discussion as to how poor I was at analysis, having scored only two points in the analysis section out of a possible ten. One section was optional, though. “Let us see if you know any Complex Analysis:” )

$f$ is a holomorphic function defined on the whole complex plane and whose image is the complement of the open ball of radius $R$ centered at the origin. Then what can you say about $f$?

(Constant map. Every complex analysis interview question is a variant of the Liouville theorem. )

What is the fundamental group of a hawai chappal?

($\mathbb Z * \mathbb Z$. They asked if I knew the proof. I hinted van Kampen. )