17th March, 09, 1200-1300 hrs. @TIFR
Committee: Dipendra Prasad, Eknath Ghate, Pablo Ares Gastesi, Vijayalaxmi Trivedi, Amitava Bhattacharya (he didn’t ask any questions)
In the chair: Dipendra Prasad 

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]

amath
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?
Me: Left multiplication. (come on, I was confused seeing so many mathematicians all at once)
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]
[Topology]
(DP walks out, Pablo takes charge)
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?
Me: Infinitely many. (really?)
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 
Sir, I cannot prove it. 
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. 
(He repeats adding that omega^3=1, as if i didn’t know)
(I start writing something like)
F_(p^n)~=(F[x])/((x^(p^n)-x))
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.
(After some mutual consensus, I was told to leave.)
endamath
————————————————————————————-
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.
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