This post will cover more than what its heading says.

When my CSIR-NET result published, I got 47th rank. I contacted my professor. I was informed that now I have JRF, getting into IITs will not be a big deal. So I decided to get into IISc, IIT-Madras, IISER-Pune, IIT-Bombay in this order. This order was based on my subject in which I was interested to do PhD. I was thinking that to get IISc will be somewhat tough but IITs will be easy since I am a JRF qualified(it was true in early days). Thus I applied for this four institutes only. To my surprise I was below cutoff for IISc to appear for interview. Cutoff was 35 rank in JRF and 850 score in GATE. Later I came to know that IITs and IISERs don’t care if you have qualified JRF or GATE or INSPIRE. I got surprised when I came to know that IISER Kolkata has also set cutoff. Two of my friends with *not much good score* in JRF are not called for written test/interview. Although they were called at IIT-B, IIT-M, IISER-P, IIT-Indore, IIT-Guwahati, IIT-Gn, etc…

Now I was having written test/interview at IIT-B on 9th-10th/may, at IIT-M on 15th-16th/may and at IISER-Pune on 17th/may .

**IIT Bombay:**

Written test was in morning on 9th May. I did some silly mistakes and my name was not there in the list published on the same day 2pm and consisting of 33 students (out of around 150 who came for written test) who were shortlisted for interview. 😦

If you are looking for getting into PhD in IIT Bombay then you should look at written test papers of previous years. Here it is with syllabus. Written test of IIT-Bomaby consists only Linear algebra(40%), Real analysis(30%) and Probaility(30%). Probability is for those who applied for Stats.

**IIT Madras:**

Written test started at sharp 9:30am of 15th May. Result declared at 14:00. There were around 100 students. I was in those list of 40 who was selected for Interview. We were asked to gather in a classroom at 14:30. Then they asked us to choose one from two committees for interview.

1st committee consist of analysis(real and complex), ODE/PDE, and few more

2nd consist of Algebra(Linear and Abstract), Topology and discrete maths(I am not sure about last one)

I was interested in Topology and set theory so I choose 2nd committee. Interviews were going to last for 2 days. Many students were having interview at IISER Pune on next day and they were requesting to arrange their interview on same day. Around 80% of them got same day. I was in no hurry so I choose 2nd day.

My interview started around 4pm. There were around 12 faculties in panel.

I entered in classroom.

There were smiling faces looking friendly. One faculty asked to introduce my self.

Me: My self Ravi. I did my MSc from NIT Warangal. ( I stoped. He said finised.? I replied yes by moving my head. 🙂 )

Then one aged professor asked about my primary Interest.

Me: I like topology and Set theory.

Intr: Do you mean point set topology?

Me: Yes sir. (they got surprised. May be I was so specific)

Intr: We have not much people working in this field. What is your 2nd preference.?

Me: (I was taking so-much time to answer. My facial expression was saying that I don’t want other field)

Intr: We dont have people working on this.

Me: I like self study. (They laughed. They said you need some guide during your PhD.)

Intr: So what if we give you number theory or algebraic topology? would you like to work?

Me: (I was taking time)

Intr: We will not ask you questions from number theory. Just tell us if you can work in other field if you get.

Me: (after taking some time) No. ( I was thinking wtf I just said)

Intr: Why? (One senior faculty was not happy with this question. Maybe he was thinking that this question will make me uncomfortable or nervous in interview)

Me: (after taking some time)I like many subjects in maths like Probability, Analysis, graph theory etc… But I will be spending 4 years in this particular subject. Thus I should go to the subject I find most interesting. Okay, My second preference is algebraic topology.

Intr: So which theorem in topology you like most?

Me: (Taking so-much time. Not coming one single theorem in mind)

Intr:(With smiling face) Just tell us one theorem that you find interesting…

Me:(After taking a little time) Convergence class theorem (This one is given in *General Topology* by *John Kelly. I was good at all the terms that appear in this theorem** like NET, DIRECTED SET*)

Intr: What is this theorem.

Me:(stated it but not precisely. Below pic consisting definition is from General Topology by John Kelly)

Intr: what is Net

Me: (defined)

Intr: What is directed set.

Me: (defined)

Intr: What is relation between Continuous functions and NET.

Me: (I stated something else. After I explained they asked same question again)

Me: (I answered this question. I learned this time that We should listen them carefully)

Intr: Which topics you are good in Topology?

Me: Connectedness , Compactness etc…

Intr: Give an example of a set which is connected but not path connected.

Me: (I gave example of *Topologists SINE curve)*

Intr: Is it locally connected?

Me: (I was not having Idea of what is locally connected. I was taking time)

Intr: How a neighborhood looks at (0,0) (They asked to draw ).

Me: (answred)

Intr: Now you can answer. (I was taking time) Do you know what is Locally compact set?

Me: No sir.

Intr: is this set compact?

Me: (I was trying to prove by definition)

Intr: is it closed?

me: Since we are adding all limit points of graph (x,sin(1/x)), this set is closed.

Intr: Now answer previous question.

Me: Yes. It is also bounded (They asked reason. I gave) Thus it is compact since R^2 is euclidean space.

Intr: Here is one last question. Are you comfortable with metric spaces? ( I said yes) So Let X is a connected metric space with at-least two points. Prove that it is uncountable.

Me: (I was trying. I was doing rough work on board. I was trying to prove that there will be a point for each real no. on some interval. I was trying and trying. I was getting nervous. I knew that I was trying in right direction but since i was not getting answer , I changed the the approach and started trying to prove that I will have a perfect subset. they asked what is perfect set. I answered. They stopped me as time was over for interview.)

I left the room. This last question was very easy. I was near to the answer and I changed my strategy and this could reduce my chance of selection. I was 50% sure about getting selected.

**IISER PUNE**

Infrastructure of IISER-P is beautiful and academically also this institute looked nice. I reached here early morning around 6am. Booked a room in hostel. Did breakfast in mess. Then written test started around 8:30 am. A good point about IISER-P is that all students were not asked on same day unlike IIT-B and IIT-M. Students were distributed over 3 days. I was invited on the last day. Here we were not shortlisted unlike IIT-B and IIT-M. We all were having interview on the same day. There were three parallel panel for interview. My was in 1st panel after lunch. Interviews were running from 30 minutes to 70 minutes. Maths dept. of this institute has a beautiful waiting room. Here is a pic of a painting I captured.

Books appearing on right side of pic are great classical books. Do read them if you get time. Here is pics of that side.

My interview started around 4pm. When I went to that classroom, one of the three panel member were outside of class. Let we call him Mr. A. He said that other panel members went outside and coming in two minutes. He was really very very friendly with smiley face. He asked about myself and my degree. I said I am from Gujarat. I said I had completed my MSc this year and result of 4th sem announced just two days back. I said I got 7.55. He asked how mush is 7.55 in your institute NIT W. I said not much good. There are students with 9 point also. He asked if i had appeared for interview anywhere and how was it. I said I had interview at IIT Madras and it was somewhat good. He asked about my interest. I said I want to work in any of *set theory, mathematical logic *and* topology*. He looked surprised. He asked if I had read *Naive Set Theory by Halmos* for axiomatic set theory.* *I said this book is soo much descriptive and I didn’t like this book. I liked appendix of *General topology by John Kelly.* He felt good. Meanwhile two other member came and we all four went into classroom. That sir introduced my self to two other members with mentioning that I am interested in Set Theory.

Mr. A: So do you know godel’s incompleteness theorems? ( I said no)

Mr. A: Do you no ZFC? ( I said Yes sir. He said ‘Ohhh every one interested in set theory knows this.’ He didn’t asked any question further in set theory or logic)

Intr: So since you are interested in topology… can you give an example of continuous function f from X to Y such that f is bijective, f inverse of each element is connected set and Y is connected and X is not connected.

Me: Let X is [-2, -1) union [0,1]. Y is [-1,1]. f is identity on [0,1]. f(x)=x+1 on [-2,-1).

Intr: What is topology on X and Y.

Me: Subspace topology of R^2.

Intr.: Yes. You are right. Here X is not connected. what condition we should add such that X becomes connected (She didn’t exactly said this but trying to say this).

Me: (I was not getting question properly. I said extra condition is f inverse is also continuous. I should not answer this because then f becomes homeomorphism and there is no fun)

Intr.: (She was not satified. She knew that I am not getting question. She tried a lot.) Do you know quotient sapce? (I said yes). Define it.

Me: (Defined it nicely with two different approach. One with partition of a topological space X and one with given function f from topological space X to set Y)

Intr.: prove or disprove that if f is a quotient map from X to Y such that f inverse(y) is connected for each y and Y is connected than X is also connected.

Me: (Did Some rough work. Tried for few minutes. Proved. I was surprised that I proved it. She was asking her doubts about proof. Mr. A. was explaining my proof to her)

Intr.: Which other subject you are comfortable with?

Me: Algebra.

Mr.A.: (He asked something related to conjugacy classes. I said I don’t know about conjugacy classes) He asked what is group of rotaions of CUBE.

Me: (I tried to rotate vertically, Horizontally, Diagonally but I was not getting answer. Mr.A. gave me a lot of hints. but i had never spent time on this topic. I was not able to answer)

Mr. A.: How many members are there in this group? (I was not getting answer. )

Mr.A.: Can you give a lower bound ?

Me: 12. since there is an element of order 3 and an element of order 4.

Intr.: Prove or disprove: If V is a vector space then that there exist finite number of proper subspaces of V such that Union over them is V.

Me: (I was trying and trying to prove that it is false. looked obvious in R^1, R^2 and R^3. I was trying to solve using induction on something I don’t exactly remember now. She said use induction on dim(V). I tried. Was not able to solve. Time finished. Total duration was around 70minutes)

I left Pune on next day morning.

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I didn’t in IISER Pune. I got selected in IIT Madras. I will be joining IIT-Madras from 10th July.