**DP:**You have done your BTech and you are doing MSc part 1 now, right?

**Me:**Yes.

**DP:**So shouldn’t you appear for the interview the next year?

**Me:**Yes sir, but it will give me some practice. (Oops!)

**EG:**What have you studied?

**Me:**In the previous semester, I completed courses in Algebra- Groups and Rings, Linear algebra and Real Analysis including the topology of metric spaces. I have also studied some number theory on my own.

[The fireworks begin, Algebra first]

**DP:**What are all homomorphisms from ZZ//3ZZ to ZZ//5ZZ?

**Me:**(cheap stuff) Trivial ones. (proved it)

**EG:**Do you know the Quarternion group? 1, -1, i, j, k.. What is its centre?

**Me:**+1 and -1 are in the centre.

**EG**: Is there any relation between the centre and the group’s order?

**Me**: ??

**EG**: Order of the group is 8. So what about 8?

**Me:**It is a power of 2. So Q(8) is a p-group and has a non-trivial centre.

**EG**: Can you prove that?

**Me**: (I wrote the class equation and he asked me to prove it) Consider the group action of G by left conjugation.

**EG**: Left multiplication or Conjugation?

**EG**: Blah blah..

**Me**: Sorry conjugation and not left multiplication…(after a lot of effort (and help 😛 ) finally struggled and proved it)

**VT**

**:**Define a ring R by R={f:[0,1]->RR, continuous} Is it a domain?

**Me**: (drew picture of some zero divisors)

**VT**: What are its maximal ideals?

**Me**: (After thinking for quite some time) M={f in R: f(c)=0 for some c in [0,1]\ } (I gave a faulty proof that this ideal is maximal. I also had to show that this is the only possible maximal ideal) Suppose M has a function f which doesn’t vanish at any c in [0,1] . (thinking..) so it has a well defined inverse, g (again, thinking and write the above sentences on the black board). So 1 in M !! (Eureka!)

**VT**: (smiling, apparently happy with me) yes.

**DP**: You must assume that for each c in [0,1], there is a function f_c which does not vanish at c in [0,1].

**Me**: Each f_c being continuous, doesn’t vanish in a neighbourhood of c. The subset [0,1] being compact, has a cover for every neighourhood cover of c. So there are finitely many f_1, f_2, …, f_n etc. Now consider the never-vanishing function f=f_1^2+f_2^2+…+f_n^2 >0 in every [0,1]

**PA**: X and Y are connected topological spaces. Is XxY connected?

**Me**: (For quite some time, I couldn’t follow his accent. But that was the least of my problems. I had to think of some counterexamples. After some thinking, I decided to try and prove it. Finally, proved it.) Blah blah.. maximal connected.. blah blah.. component..etc..

**PA**: Consider RR with the discrete topology (for a long time, I couldn’t make out his pronuncation of the word ‘distance’ he used) What are its compact sets?

**Me**: The finite ones. (Proved correctly. Meanwhile, DP comes in)

**DP**: f:RR->ZZ is continuous. How many such f can you find?

**DP**: Name them.

**Me**: The continuous image of the connected set RR will be a connected set in ZZ, a singleton. But there are infinitely many such elements in ZZ, so. (Yippie!)

**EG**: So what else have you studied? Measure theory?

**Me**: (fart) I am currently studying it in this semester.

**EG**: So what else can we ask you?

**Me**: (huh!) Sir, I have studied some number theory on my own.

**DP**: How much?

**Me**: In a standard text, upto Gauss Reciprocity Theorem.

**EG**: Can you prove it?

**Me**: Yes. (come on dude, ask me and I’ll show you)

**DP**: Do you know prime number theorem?

**Me**: Yes. (on the board:) lim_(n->oo)(pi(n))/((x/ln(x)) )=1

**DP**: (broad smile) Why, you have done a course in Real Analysis.

**Me**: (abbey it took Hardy and Dirichlet and others many years to conjecture it, main kaise prove karu?) Sir it is a highly non-trivial result.

**DP**: For what values of n, is the cube root of unity in an element of F_n?

**Me**: ( kya bol raha hai yeh? How does omega come in a finite field?) Sir I did not get the question.

**DP**: This is incorrect. (x^(p^n)-x) is reducible.

**Me**: (Fart) Ok. Consider F_4. F_4^**~=Z_3 So the generator of Z_3 will satisfy omega^3=1.

**DP**

**:**What about other solutions?

**Me**: F_n^**~=Z_(n-1) so 3|(n-1)

**DP**: Can you prove that the multiplicative group of a finite field is cyclic?

**Me**: No.

————————————————————————————-

cf. Lessons learnt:

- Theorems’ hypotheses must be studied carefully.
- Must know at least one proof of every theorem studied.
- Must solve adequate problems. I am not Grothendick.

## 17 comments

Comments feed for this article

March 17, 2009 at 20:47

Vishnu SrivastavaBRAVO!!

fought the battle with courage tends to +ve infinity….

March 17, 2009 at 22:28

Aditya JainThe way you handled all the questions..You should get selected..

All the best for the result.

It was all greek to me though..:)

March 18, 2009 at 03:33

Ashwatthamanice interview. Though the qs were over the head. 😛

March 19, 2009 at 21:18

ameya“when you really want something to happen, the whole universe conspires so that your wish comes true”

I guess phaaad diya intervieww…hope u achieve what u want..

March 20, 2009 at 22:40

Abhishek ParabThanks vishnu, lala, ameya for your wishes, but I am quite confident that I won’t qualify; there are many blunders which I committed, which I didn’t mention here..

April 5, 2009 at 16:51

anil.iiitmR={f:[0,1]→ℝ, continuous}, can you give an example of zero divisor?

April 5, 2009 at 22:24

Abhishek Parab@anil.iitm: Consider the function f: f(x)=0 for x in [0,1/2] and f(x)=x for x in [1/2,1]. It is continuous and not identically zero.

I claim it is a zero-divisor, because the function g defined by g(x)= (1/2 – x) for x in [0,1/2] and g(x)=0 for x in [1/2,1] is such that f.g is identically the zero element of R.

August 26, 2010 at 23:50

TusharHi abhishek…

I read your interview…its very interesting. Actually, I am going to give the TIFR entrance (for integrated phd). Im currently in my last year(microbiology) in ruia college. My marks in last two years are pretty much bad. So, does that make a large difference in me getting selected in TIFR? Plz reply…

June 15, 2011 at 11:26

chandrumath.wordpress.comHi–

As for the prime number theorem question it should be . Moreover, when DP said, when you have done a first course in real analysis, prove the PNT, you should have replied:

– Sir you have studied, number theory, prove RH…

June 15, 2011 at 18:59

abhishekparabRH is a celebrated open problem.. The PNT is not an open problem.. But it would be amusing to give such an answer 😛

June 19, 2011 at 16:22

AnonymousWhy I was saying that, because: At undergraduate level of INdian education, it would be tough for people to even think abt such celebrated theorems. Suppose DP asked you this question some 60 years before, Erdos and Selberg provided a proof, it would have been open as well. The thing is: such questions doesn’t really test the talent. And I believe strongly that DP shouldn’t have asked this.

June 19, 2011 at 18:47

abhishekparabYes, true. I don’t think these questions test talent.

December 17, 2011 at 15:26

Manjil P. SaikiaThe prime number theorem was proved in 1890s, it wasn’t open in 1950s. 🙂

December 17, 2011 at 21:24

abhishekparabHmm.. Thanks for pointing that out!

June 22, 2012 at 14:30

ankurMy marks in Bsc. are pretty much bad. So, does that make a large difference in me getting selected in TIFR? Plz reply…

December 11, 2012 at 10:08

Preethi Raj NairHi. We are on the same boat 😀

December 11, 2012 at 10:09

Preethi Raj NairAnd on reading your post am actually kind of really glad that i didn’t do my tifr exam that well. *sighs*