I went to IMSc Chennai for their PhD interview. Following questions were asked.

- There exist matrices and with real entries such that . True or false?

(This is not possible. Observe that trace of is the same as that of and trace of a nilpotent matrix is zero. )

- is symmetric and positive definite matrix such that . True or false?

(This is the AM-GM inequality applied to the (positive) eigenvalues of . These two questions were a part of the only question in the Algebra section of the NBHM paper that I was not able to answer. Rest 9 questions were correctly answered. )

- is the field of elements. What is the cardinality of ? (This is a standard interview question) Define

What is the cardinality of the set ?

(I couldn’t solve this question completely. I did things like showing that is a finite set. It turns out that acts on and this is used in computing the order of . They asked me to try it later. 🙂 )

- (Topology) is a continuous function. Can it be injective? Surjective? Conider the set

.

Show that is nonempty. (Showed.) What is the cardinality of ? (Re-framed the question as) Prove that is uncountable. (I couldn’t solve it then, but later I remembered that will be a loop in and hence the image will contain an interval traced twice. )

- Give an example of a compact set that is not Hausdorff.

(. We agreed that any infinite set with the co-finite topology would do. I could make out that my answer surprised them though they didn’t ask any questions on Algebraic Geometry. )

- Is the set of all orthogonal matrices compact? Connected? (as a subset of with the induced Euclidean topology)

(They didn’t mention or . I proved they are not bounded. For connectedness over , the determinant function works. )

**Tailpiece:** I have qualified the interview. I plan to join IMSc. Advice — Do solve the NBHM question paper thoroughly including all sections. Most questions for me were asked from that paper. Unfortunately, I hadn’t solved the paper after the exam. Fortunately, it didn’t matter!

## 12 comments

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April 28, 2010 at 22:45

Anshuman GauravCan you tell me for which post these questions are i.e. Int PHD (physics), computer Science, etc

June 29, 2010 at 00:30

VarishCongrats ! Good luck for PhD !

June 29, 2010 at 09:19

abhishekparab@Varish, thanks! Good luck to you too.

March 3, 2011 at 11:33

vijayaraghavan.Dhi I passed in nbhm written test . how to prepare for interview please tellme

January 24, 2012 at 00:55

Dinesh valluriA is an orthogonal matrix if AA^(T) = I let A=(a_ij). We can see that Trace(AA^(T))= sum of the squares of all entries of A. which is indeed the norm of A in the given metric. but Trace(I)=n so norm is always n. Hence Orthogonal matrices are bounded.(At lest in the real case). But in your last question you said that it is unbounded?

January 24, 2012 at 21:06

Abhishek ParabHi Dinesh,

I am sorry in that I typed wrongly on the blog above. The question was about matrices with determinant 1, not orthogonal matrices. Hope that clarifies. Sorry for the possible trouble.

January 30, 2012 at 11:39

SubhoHi Abhishek.I am a BSc student and rescently gave my NBHM PhD Scholarship Test . I think my score is 19 out of 40.Do you think it is good enough to be

called for the interview at IMSC and around what time shall I get to know if I have

qualified or not ? The exam was on 28th January.

Thanks.

March 8, 2012 at 22:45

SubhoYou were absolutely correct Abhishek . Set of orthogonal matrices with complex entries are indeed unbounded ahd hence not compact , though it is compact when the entries are real.

April 15, 2014 at 00:12

BIKASH CHAKRABORTYI was asked at NBHM P.hd interview at ISI,KOL that is an open annulus homeomorphic to an infinite cylinder?then they told me to find the covering space of infinite cylinder.

Then amites dasgupta asked to give an complex valued function which takes the interior of a disc to its exterior.

finally they told me to find the transcendental extension of the complex field.

April 15, 2014 at 22:23

Abhishek ParabAn open interval is homeomorphic to the Real line; the given spaces are products of these with the circle. The covering of the infinite cylinder is the complex plane, via the exponential map.

If the disc map need not be bi-holomorphic, just translate the disc. I cannot think of a bi-holomorphic map off hand.

The function field of a curve over C is transcendental of tr.deg 1.

November 28, 2014 at 23:37

sagar kumar sahoocan any tell me, what marks to score in nbhm phd test for calling for interview???plz plz plz tell me……………

December 18, 2014 at 10:07

Neeraj DatarSir, I am a 2nd yr engineering student. I wish to do masters n research in maths. I want to give NBHM PhD exam. Plz suggest me some books fr this exam’s preparation. Please.