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. )

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