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!