### A Day @ISI-Kol

Interviews always make me nervous. And that nervousness starts precisely 30 minutes before my call. Dunno why (about the timing)! Anyway, the interview was for the NBHM MA/MSc Scholarship. Well I already get a handsome scholarship from the Govt Of India (INSPIRE) but I wanted just to get a feel, whether I am still grounded after having plenty of courses. The interview was scheduled to be at 4 pm on 18th November and the venue was ISI, Kolkata. I reached Kolkata in the morning and stayed at my friend’s room – thanks to him. I was also accompanied by my friend and classmate who was also called for the interview.

ISI Kolkata is a big place, and has numerous departments. From studies in social sciences to research advances in mathematics and theoretical statistics. No wonder why this institute has been conferred the title of institute of national importance. Coming back to point, we waited for almost the whole day, till our turn at around 5:30 pm. Till then we met students from Vivekananda University, and guys from IMA; and interacted with them about there courses etc. We took lunch at the ISI Canteen, where the food rate is incredibly subsidized – 50 “paise” for 5 chapatis and curry. We waited, waited and waited. Not to mention I had cold, and a bad headache.

Finally I was called, and already my nervousness has reached a saturated point. But it declined appreciably once I entered the lecture room. There were four professors, and one among them was Prof SC Bagchi, whom I had already acquaintance with; the rest looked familiar but I didn’t know the names. We exchanged smiles, and I eased a lot. The interview started with the obvious question – “what courses have you studied?”. Needless to say I would have easily spent 10 minutes just recounting the name of the courses. I named a few, and mentioned along that there were many in which I wasn’t comfortable with. Maintaining truthfulness is necessary to avoid questions from an area which one doesn’t know well. The first question was asked by Prof Bagchi, who asked me to prove that the closed interval [0,1] is a connected set in the real line. I said that the question boils down to proving that the only subsets of the real line which are both open and closed are the empty set and the whole real line. He asked me to prove this. Now this requires a little thinking. And I’ll leave it to you to figure it out or take the help of a book [Topology-Munkres]. I did succeed only after considerable hints from him.

The next question was an easy one – to give an example of a non-holomorphic function. I shot back saying $f(z)=\overline{z}$;  coz it didn’t satisfy the Cauchy Riemann equations. Immediately the next question was when we can say that the function is analytic – which is again simple – that in addition to satisfying the Cauchy-Riemann equations, it should also have continuous partial derivatives. Then I was asked that given a point in a domain where a function is analytic (=holomorphic), what can I say about the radius of convergence. I said that we can get the radius of convergence by the formula $R^{-1}=\limsup |a_n|^{1/n}$. They clarified that what I can say readily by just looking at the domain, and hinted Cauchy’s theorem. I told that by the theorem, it will be the radius of the largest circle that lies completely in the domain. They asked me whether I can put this in topological terms. I thought a bit, and then it dawned. “Yeah, distance from the boundary”. This was exactly they wanted me to tell.  Another guy from the committee asked me whether I was familiar with the notion of analytic continuation. Yes was the reply, and then I explained the rudimentary idea of how one can analytically continue a holomorphic function which agrees with another holomorphic function on a common domain. I was then asked to write a function which is the analytic continuation of $\sum_{n=0}^{\infty} z^n$, which of course if $\frac{1}{1-z}$ but didn’t hit me then.

Finally the linear algebra part. I was asked how can assign a meaning to $e^A$, where $A$ is a $m\times n$ matrix. I didn’t know so told them flatly. They then modified the question that suppose the matrix given is a symmetric matrix, and hinted some diagonalisation theorem. I got it immediately, so the point is that if $A$ is a symmetric matrix, it is similar to a diagonal matrix via real orthogonal matrix. This means $A=ODO^{-1}$, where $\latex D$ is diagonal. Then the computation is easy and the expression $e^A$ is also well-defined via $e^A=O e^D O^{-1}$, where $e^D$ is nothing but the diagonal matrix obtained by taking the exponential of the diagonal entries of $D$. Then they asked – what if $A$ is nilpotent. In that case the expansion contains only finite number of terms and hence $e^A$ is readily defined. So finally what can I say about a general matrix, and hinted “semi-simple”. Though I had read semi-simple operators, but I couldn’t recall then. Actually the result is that on a finite-dimensional vector space over the complex field, semisimple is equivalent to diagonalizable, and every operator on that space may be written as a sum of a semisimple operator and nilpotent operator which commute. So using this fact we may define $e^A$ for general operators.

With that I had spent almost half an hour, and according to me the interview was quite OK. After me, was my friend’s turn. Finally we had a nice chat with the professors, and got some good advice. Altogether it was a good experience.