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Recently (22 March 2013) I took my advanced topics exam. In spirit of the Princeton Generals, I wrote out my interview questions for use to anyone in these special topics.

Topics: linear algebraic groups, class field theory

Committee: Freydoon Shahidi (chair), David Goldberg

Linear algebraic groups:

Goldberg decided to start with Linear algebraic groups. What is meant by ‘split’? (I told I had prepared only the algebraically closed case. They were okay). What is parabolic? If Q is parabolic in G and P is parabolic in Q, prove that P is parabolic in G. (Went totally blank. They gave hints – use the other equivalent definition).

State Bruhat decomposition. (I started defining terms – maximal torus, root system, Weyl group etc.) How is the Weyl group related to the torus? (Another moment when I blanked out). How does W act on T? (By conjugation. So W is the quotient of the normalizer of the maximal torus by its centralizer). When is the centralizer equal the torus? (When it is maximal). Shahidi objected and corrected me twice when I wrongly pronounced “veil” for Weyl group – correct is “vile”. There is another important group after Andre Weil.

Bruhat decomposition for GL_n. What is it’s Weyl group? (S_n). What is the length of an element? (minimal length of the decomposition in terms of reflections). What is the polynomial the big cell satisfies? (I had prepared this one. Told the answer and that the proof goes by induction but Shahidi was not satisfied. He told something I didn’t quite understand. I gave the decomposition explicitly and he told it generalizes the well-known LU decomposition to classical groups).

What is the critical step in the classification of semisimple groups of rank 1? (Uses Bruhat decomposition). Give a sketch of proof. (I couldn’t. But turns out, they really wanted me to state the theorem. SL_2 and PSL_2).

Given that the normalizer of a parabolic is itself, show that it must be connected. (Missed a step in that one can choose the conjugating element inside the connected component).

[ That was all about algebraic groups. Nothing about Dynkin diagrams, finding parabolics, simple connectedness and fundamental groups of classical groups. ]

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Number Theory:

Class group of Q(\sqrt -7). (Minkowski bound works).

Class group of Q(\sqrt 21). (I saw that 25 – 21 = 4 and started considering the factorization of the prime 2 above using quadratic reciprocity but got it wrong. It took me almost 10 grueling minutes to figure out that 21 was not a prime! Embarrassment).

Define the Hilbert symbol. State the norm condition. Prove the bi-multiplicativity property. (Consequence of a group homomorphism). State and prove Hilbert’s reciprocity. (I had prepared a classical proof from Serre but he insisted on proving via Class field theory. Proved that the local p-Artin maps glue to give a global Artin map and the theorem follows elegantly).

Goldberg asked if I knew anything about the generalized symbol. (I stated and mentioned the skew-symmetry property). He said it’s not in your syllabus anyways.

Show that p = a^2 + ab + b^2 precisely when p is 1 (mod 3). (I was looking at the field of cube roots of unity but was blundering some norm calculation. After many frustrating minutes for everyone, it was discovered that the problem was wrong and should have p = a^2 – ab + b^2. I commented, “I have been doing this calculation since tenth grade” on which Prof. Shahidi quickly retorted, “still you haven’t memorized it!”

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The exam lasted for 90 minutes. They told me to wait outside and after a stressful 5-minute period for me, they came out and congratulated me.

Shahidi: Good, congratulations.

Me: Thanks.

Shahidi: Don’t see me for a month now.

Me: Yeah, the past two weeks were tough on me too.

Shahidi: Let us not meet each other for a while now. (All smile in acknowledgement).

I get many emails, some from my friends and some that are redirected from my blog posts asking advice to prepare for interviews for higher studies in Mathematics, most often TIFR and NBHM interviews. For questions asked during my interview, see the “interview” tag. Here are a few tips, some quite obvious.

• The first question they will ask would most likely be what your favourite subject is. A good answer is, perhaps, the subject you are most comfortable with. It goes without saying that you must prepare as many “standard” tricky questions in that area.
• Usually there is a “qualifying” written test and it is best to completely solve the questions thoroughly well-in-advance before the interview. I would also recommend studying theorems and solving problems related to those questions.
• Rather than test what and how much mathematics you have studied, interviewers are keen in testing the way you think after a question is asked, and how you tackle it. They usually keep asking questions with increasing difficulty until you fumble. So for most of your interview, you are intentionally kept uncomfortable. I mean, mathematically uncomfortable. It does not mean that your performance was bad.
• If you haven’t studied a particular specialized topic or if you aren’t comfortable with some topic, say Galois theory or Measure theory, it is best to let them know. The last thing you would want is to let them see you writing crap on the board. But such an excuse cannot be given for fundamental topics like groups and rings, basic real analysis, linear algebra and perhaps complex analysis.

Hopefully, this list shall be updated regularly!

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

• There exist $n \times n$ matrices $A$ and $B$with real entries such that $[I-(AB-BA)^n]=0$. True or false?

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

• $A$ is symmetric and positive definite $n\times n$ matrix such that $(\text{tr} A)^n \geq n^n \text{ det } A$. True or false?

(This is the AM-GM inequality applied to the (positive) eigenvalues of $A$. 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. )

• $\mathbb F_q$ is the field of $q$ elements. What is the cardinality of $M(n, \mathbb F_q)$? (This is a standard interview question) Define

$S_r = \{ V: V \text{ is an } r- \text{dimensional subspace of } \mathbb F_q^n \}$

What is the cardinality of the set $S_r$?

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

• (Topology) $f:S^1 \to \mathbb R$ is a continuous function. Can it be injective? Surjective? Conider the set

$T = \{ (x,y) \in S^1 \times S^1 : x\neq y \text{ but } f(x) = f(y) \}$.

Show that $T$ is nonempty. (Showed.) What is the cardinality of $T$? (Re-framed the question as) Prove that $T$ is uncountable. (I couldn’t solve it then, but later I remembered that $f(S^1)$ will be a loop in $\mathbb R$ and hence the image will contain an interval traced twice. )

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

($\mathbb A^1$. 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 $n\times n$ orthogonal matrices compact? Connected? (as a subset of $\mathbb R^n$ with the induced Euclidean topology)

(They didn’t mention $\mathbb R$ or $\mathbb C$. I proved they are not bounded. For connectedness over $\mathbb R$, 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!

Edit: Disclaimer: These questions were asked during MY interview. I have posted them thinking that they may possibly be of use to anyone preparing for such exams. However, please don’t ask advice from me. I may not be good at advising and also, it is often disinteresting.

Interviewers: Prof Nitin Nitsure (TIFR), Prof C S Rajan (TIFR), Prof Balwant Singh (IITB), Prof Shobha Madan (IITK). Most questions were asked by Prof Nitsure, including the last one. They asked me what I was interested in. I told Algebra; Nitsure sir observed that I had correctly attempted all but one question in the Algebra section of the written exam. I was asked to try that question then but I gave up.

$f$ is a morphism from $\mathbb A^1$ to $\mathbb A^1 \backslash \{0\}$. Then what can you say about $f$?

(I interpreted the question in geometry to algebra, worked out coordinate ring homomorphisms but couldn’t reinterpret the result geometrically.)

State the inverse function theorem.

(I couldn’t answer that one and I personally think, one who doesn’t know InFT doesn’t deserve to get a PhD scholarship. There followed a discussion as to how poor I was at analysis, having scored only two points in the analysis section out of a possible ten. One section was optional, though. “Let us see if you know any Complex Analysis:” )

$f$ is a holomorphic function defined on the whole complex plane and whose image is the complement of the open ball of radius $R$ centered at the origin. Then what can you say about $f$?

(Constant map. Every complex analysis interview question is a variant of the Liouville theorem. )

What is the fundamental group of a hawai chappal?

($\mathbb Z * \mathbb Z$. They asked if I knew the proof. I hinted van Kampen. )

I had my TIFR PhD interview today. Inspired by the Princeton Generals, where students who get through their Qualifying exam write about the questions asked, even I am writing here describing my interview experience. But I am not very enthusiastic about writing this, since the questions were very basic and not from all subjects, a very unlikely event during a Maths PhD interview. Nevertheless, here is the interview–

Chair: Prof. A. Sankaranarayanan

Other interviewers: Prof. Indranil Biswas, Prof. Raja Sridharan, Prof. Ravi Rao

I do not remember who asked which question, so in much of what follows, I am referring to all as “interviewer”. Most questions were asked by RR.

AS: So you work at Larsen & Toubro?

Me: No sir, I was working at L&T for an year from 2007 to 08. Then I joined IIT Bombay in 2008 till now.

RR: Under whom have you taken courses at IIT?

(I name Limaye sir and other teachers)

Me: Sir I do not have a particular interest as such, but I do like Algebra.

RR: $K$ is a field. Is $K[X^2,X^3]$ a UFD when considered as a subring of $K[x]$?

Me: Yes sir, zerodivisors in $K[X^2,X^3]$ would mean zerodivisors in $K[X]$ which is a domain. (Mistake.)

RR: So it is a domain. But is it a UFD?

Me: Ohh.. Yes.. I mean no. Consider $X^6$, it can be factored in two ways. (I proved that the two were irreducibles and not unit-times-each-other.)

RS: $f:\mathbb R \to \mathbb R$ is smooth and $\lim_{x\to\infty} f(x)=0$ Then can you say if $\lim_{x\to\infty} f'(x)=0$?

(I tried for a long time to prove, then tried to give a counter-example. $\displaystyle\frac{\sin x}{x}$ didn’t work. He suggested $\displaystyle\frac{\sin x^3}{x}$ and it worked, though I goofed up in differentiating and taking limits)

(In the meanwhile, there was an interesting discussion among them if they should finish off Algebra before switching to Analysis.)

RR: Is $\displaystyle\frac{\mathbb C[X,Y]}{(X^4+X^3Y+Y^4)}$ a domain?

Me: If it is to be a domain, then the polynomial should be prime, hence irreducible. (thinking) Eisenstein.. won’t work!

Interviewer: Hint — The polynomial is homogeneous.

(Solved it.)

Interviewer: A linear transformation $T:\mathbb R^n \to \mathbb R^n$ sends straight lines to themselves. What can you say about $T$?

(I wrote, wrongly, a diagonal matrix with different diagonal entries. Later corrected the mistake after they asked me to repeat the question.)

Interviewer: $V$ is a $n$-dimensional vector space and $W$ is its subspace such that for every isomorphism $T:V\to V$, we have $T(W)\cap W \neq (0)$. What can you say about $W$?

(Proved that the dimension must be $\geq n/2$.)

Interviewer: Consider $S_p$, the symmetric group on $p$-symbols and an element of order $p$ in it.

(I wrote the element as a $p$-cycle. There followed a long discussion as to why it should be a $p$-cycle, and orders of commuting elements, their lcm’s etc. It finally ended with me group-acting $S_p$ on those $p$-symbols and ‘proving’ that order of disjoint cycle types is the lcm of their orders.)

RR: Find a $p$-subgroup of $\text{GL}_3(\mathbb F_p)$.

(This took the longest time of all. The question was interesting, involving some fundoo linear algebra. In spite of many hints from them, I was not able to solve it. Finally, RS told the answer.)

Me: (excitedly) Ohk, not just the elements of order $p$-power, I have found the  group of order $p^3$!!

RS: No, I have found it!

Me: (embarassed) yes sir..

(There followed a discussion if they should ask more questions. Finally, RS asked me to prove that the additive group of a field of characteristic zero is not cyclic. I was halfway done when I was told to go.)

Note: The interview lasted for an hour. The interviewers were pleasant but and did give subtle hints when I got stuck up. But they stopped me whenever they thought I was not giving the right justification or hand-waving proofs.

PS: I was not offered the fruit juice the others before me were offered. 😛

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)

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.
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
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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.

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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### Quotable Quotes

ABHISHEK PARAB
“Do not think; let the equation think for you”

PAUL HALMOS
”You cannot be perfect, but if you won’t try, you won’t be good enough”

ALBERT EINSTEIN
“Don’t worry about your maths problems; I assure you, mine are greater”

THE BEST MATH JOKE
"A comathematician is a device for turning cotheorems into ffee"

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