## 31 May My Experience at PhD Interview at the Tata Institute of Fundamental Research

There were a total of 42 students who appeared at the interview date. There were two rounds : the first was a problem solving round and the second was the interview round. The problem solving round was a total of 1.5 hours long and consisted of 9 questions out of which we were to answer as many as we could. There were 8 subjective type questions and 1 objective type question. After the problem solving round, out of the 42 students, 21 were shortlisted for the interview round.

I was lucky enough to get past the problem solving round. There were two interview panels and I was assigned to panel B. The panel consisted of 5 professors. Sitting in the middle was Prof. Nazmuddin Fakhrudin (NF). On his right was a young professor, whom I will refer to as (YP). On NF’s left was a middle aged prof whom I will denote as (MP). The remaining two professors didn’t ask me any questions.

As I entered I wished everybody good afternoon, and was greeted back.

**Question 1.** (NF). So Mr. Khetan, where are you pursuing your MSc from?

**Answer.** I am pursuing my masters in Mathematics at the Chennai Mathematical Institute.

**Question 2.** (NF). And where were you prior to that?

**Answer.** I was a mechanical engineering undergraduate at IIT Kharagpur.

**Question 3.** (NF). Hmm.. So what are the topics that you find interesting?

**Answer.** I am mostly interested in topology and geometry. I also like algebra.

**Question 4.** (NF). So what have you read in topology.

**Answer.** I have done two courses. The first was on general topology with an introduction to the fundamental group. The second course was on homology theory.

**Question 5.** (NF). So can you give me some examples of the fundamental groups.

**Answer.** , . And there is this bizarre space, the shrinking wedge of circles, whose fundamental group is larger than the fundamental group of the wedge of countably many circles.

**Question 6.** (NF). Any other fundamental groups you can create?

**Answer.** Chokes.

**Question 7.** (NF). What about ?

**Answer.** I think if one attaches a two disc on via an attaching map which wraps around three times then the resulting space should have its fundamental group same as . But I am not a 100% sure.

(NF): Okay then, (MP) would you like to ask him some questions.. (turning towards me) though it’s correct what you just said.

**Question 8.** (MP). What about the wedge sum of two circles?

**Answer.** The fundamental group is the free product of two copies of .

**Question 9.** (MP). What about the Mobius band?

**Answer.** Since the Mobius band deformation retracts to a circle, the fundamental group of the Mobius band is .

**Question 10.** (NF). Now let’s do some algebra. Have you studied groups?

**Answer.** Yes.

**Question 11.** (NF). Classify all groups of order 15.

**Answer.** I used Sylow theorems and solved it with a hint. Gave the full details.

**Question 12.** (NF). What about groups of order 6? Does the same idea work?

**Answer.** I tried the same idea and got stuck. Before I would think of something else (NF) cut me off and asked (YP) to ask me something.

**Question 13.** (YP). Can you prove that a ring of order 15 with identity is necessarily commutative.

**Answer.** I choked. Even with many hints I was not able to solve the question.

I was then bid adieu.

After about a month, the results came and I was selected!

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