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

  • There exist n \times n matrices A and Bwith 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!