NBHM Interview Experience for MSc Scholarship

I appeared for the NBHM test for the MSc Scholarship in 2024 when I was in the first year of my MSc Mathematics. The written test was held on January 20. For the Master’s Interview Shortlist, the M-cutoff was 38. I scored 42 and hence, was one of the 51 candidates to be shortlisted for the interview from all over India. The interview was held online, and I received the link for the same via email a day before it was supposed to be held.

My interview was in the morning shift, and I was supposed to report at 11:20 am. I did so but was not admitted until 12 noon. At first, I was greeted by the head of the interview panel and was asked to introduce myself. He then asked me my favorite topic. I told him Algebra and the topology of metric spaces.

Prof: How much have you studied algebra?

Me: Group Theory including Sylow’s, Solvable Groups, Nilpotent Groups, and Ring Theory up to prime and maximal ideals.

P: Define Prime and Maximal Ideal.

P: Prove that in a Commutative Ring with Unity (CRU), every maximal ideal is prime.

P: Give an example of a prime ideal that is not maximal.

P: Give an example of a ring with exactly two maximal ideals.

P: Have you studied Principle Ideal Domains (PID)?

Me: No, sir. I just know the definition.

P: That’s enough for me. Prove that ℤ (the integers) is a PID.

P: Is ℤ[x] a PID?

P: Prove that ℚ[x] (the ring of polynomials with rational coefficients) is a PID. You may use the same logic used in proving ℤ to be a PID (Euclid’s Algorithm).

Another Prof: Give me a group with exactly three conjugacy classes.

Me: ℤ₃

P: Can we say in general that ℤₙ has n conjugacy classes? Why?

P: Can you give me another group with exactly three conjugacy classes?

Me: S₃ (the symmetric group on 3 elements)

P: Another?

Me: I can’t think of any other right now. I guess these are the only two.

P: Can you prove your claim?

Me: Let G be a group with three conjugacy classes. Then |G| = 1 + a + b, where a and b are the cardinalities of the other two classes. Now, using the fact that a|(1+b) and b|(1+a), we conclude that either a=b=1 or a=1, b=2 (or vice versa). And hence, ℤ₃ and S₃ are the only such groups.

P: Why does a|(1+b)?

Me: Because |cl(a)| = |G|/|N(a)| and hence, the claim.

Another Prof: Give me a continuous map from a unit circle S¹ to ℝ which is injective.

Me: There doesn’t exist one, as the existence of such a map would imply a homeomorphism between S¹ and ℝ or a subset of ℝ, which is not the case.

P: Why is it not the case? First, tell me what is a homeomorphism?

M: Why is it so important? What does it tell us about two homeomorphic objects?

P: Coming back to the original question. Why aren’t S¹ and ℝ homeomorphic?

M: If we remove one point from S¹, it remains connected, but ℝ doesn’t have this property. Since connectedness is a topological property, they can’t be homeomorphic.

P: What about the subsets?

M: The union of disjoint intervals can’t be homeomorphic because they are disconnected. For an interval, the same argument of removing a point works.

P: Only if the interval is open. If the interval is closed, you can remove an endpoint, and it’ll still be connected.

M: True.

P: So, what if the interval is closed?

M: (after thinking for a while) If we choose any other point except the endpoints, it would become disconnected. But we can remove any point from S¹ and it will still be connected.

Another Prof: If f(x) is a non-negative continuous function on a closed interval such that its Riemann integral is zero, what can you say about that?

Me: It would be identically zero. (Proved it.)

P: What if we remove the condition that it’s non-negative?

M: Then we can’t make that claim. For example, an odd function on [-a, a].

(Around 50 minutes had passed by now.)

Then she asked me if I had done uniform convergence. I told her although I had done it in my curriculum, I wasn’t much comfortable with the topic. She then gave me a sequence of functions, and I told her it’s a pointwise limit, but couldn’t comment on its uniformity. After giving multiple hints in multiple steps, she practically solved the problem herself. This went on for around 10 minutes, and with that, my interview was over.

A few days later, the results were announced, and I had made it to the final list.

PS: If you are preparing for the NBHM interview, just remember that they won’t ask you to solve any complicated problems. They just test your foundation and the basics. Happy learning!