Last week, some members of Mathematics had a meeting with our PhD students to talk about their course work, their inputs about the graduate programme etc. A sombre mood prevailed at the meeting because these students are now preparing for their comprehensive examinations (also called qualifiers in many other places) in the last week of July. Clearing the comprehensives is necessary for them to continue in the programme. Over here, this means that they will write exams in three foundational topics, like analysis, algebra, topology or discrete mathematics, for which they have been prepared through course work over the last year. They are allowed two attempts in each subject within a limited time duration, but for almost all of them, the July exam will be the first attempt.
The students were worried about the extensive syllabus for the exams (they were not as worried when this syllabus had been broken down into different courses) and expressed concern over the passing mark etc. To allay their apprehensions, one of my colleagues told them that the exam was a way of testing their grasp of fundamental topics. While it is very important that they focus exclusively on preparing for these exams for the remaining part of the summer, they should not feel stressed and take small breaks if they are feeling very exhausted with the preparation. All the faculty members present shared memories of their own comprehensive exams very candidly- when and how they took it and how they chose their PhD topics after the exams.
These exams, when thought of by themselves, do seem monstrous. But, it helps a lot if one sees them as a phase of the PhD program, after which the students can start independent study and pick up topics that they really enjoy. Although we hope that the meeting made these students feel better, the pressure will of course persist until these exams are cleared and left behind.
This meeting brought back some memories about my comprehensive exams, way back in 2004! The programme at my grad school was structured differently from the Mathematics programme at IISER Pune. Students were assigned a supervisor upon joining and were expected to pursue course work and their research projects simultaneously. The comprehensive examination was usually conducted in the second year of the program, but could also be postponed till the third year. For this exam, the student had to submit a research proposal, give a talk on it and then face an interview in which they were quizzed about the proposal. The interview was not restricted to what was written on the proposal - it could have questions about any fundamental topics which the committee felt the student should know before embarking upon the proposed project.
Students would usually take their comprehensive exams during the summer to avoid clashes with their course work. However, I really wanted to go to India during the summer. So, my advisor (Prof. RM) suggested that I take the exam during term time. I wrote up the proposal during Christmas vacation and after running it by Prof. RM, submitted it some time in January. The exam was scheduled exactly after a month. Meanwhile, in that semester, I was taking three courses and also teaching a first year course to a large class (this was my first teaching experience).
The real fun started after the submission of the proposal. Other than my advisor, there were three members in the committee, out of which one was the meeting coordinator and two (Member 1 and Member 2) were subject experts. I submitted the proposals to them in person and asked them to give me some feedback before the exam. Member 1 read the proposal within two days of submission and gave me some feedback about what parts needed clarification, details etc.
Member 2 was a formidable senior number theorist, of whom most grad students were very scared. The courses offered by him (almost all of which I had taken) were very challenging. I have spent a lot of sleepless nights working through his assignments. When I went to his office to give him the write-up, he sternly glanced at it for a few minutes and mentioned a book that he expected me to master before the exam. I almost fell off the chair [1]. He said that he did not believe in students taking exams unless they were thoroughly prepared and went on to describe some students whom he had examined before and who had failed the comps.
Struck by lightning and dazed, I went straight to Prof. RM's office, told him about the conversation with Member 2 and requested that the exam be postponed. I was ready to sacrifice the India trip. The conversation that took place afterwards is a great example of how to supervise a student!
He explained to me the basic objective of the exam: to test whether the student understands fundamental concepts and techniques that would be needed for his/her research project. He then went to the board, mentioned a problem in elementary number theory [2] and we started discussing how one solves it. He then mentioned that the purpose of the comprehensive exam is to make sure, for example, that I can apply fundamental summation techniques to such questions before I embark on a research project. He then took out the book mentioned by Member 2 from his bookshelf and we went through the table of contents. I realized that I already knew many topics from this book. He pointed out the other sections that were relevant to my proposal and convinced me that by working for two hours a day on this book, I will master these topics and be ready to face the comps within a month.
When I again expressed my inability to do this, he started talking about the good things that would happen if I could clear the comprehensive exam within a month. "You can visit your family, relax [3] and come back with a fresh mind to work on your research problem. You can finish by early 2006 and also invite your father for the convocation ceremony in the summer." I got convinced that adhering to the above timelines would be great and decided not to postpone my exam.
I came out of the office feeling relieved and much more confident. I woke up two hours earlier every morning and worked through several sections of the book.
My exam lasted for one and a half hours. Members started by asking me basic definitions, examples, non-examples related to modular forms. After the warm up, I was asked to prove some lemmas mentioned in the proposal and was asked if it was possible to generalize them to other cases [4]. Member 2 came prepared with a long list of questions, as expected. As we ran out of time, we could not cover all his questions. So, he graciously gave me a xerox copy of his questionnaire. The exam was not half as dreadful as I had anticipated and I was declared passed.
It was lunch time when the interview ended. After lunch, I attended a class taught by Member 1 and taught my first year course. Later in the evening, I had dinner at my favourite Indian restaurant and saw a Hindi movie with a Japanese friend.
In the next teaching semester, Member 2 also offered a course on the topics that he had asked me to learn before the exam. I took this course. Later, he gave me valuable inputs and feedback for my thesis and also wrote reference letters for my job applications.
To all students taking comprehensives this summer, all the very best. Study well, take it easy and think of all the good things that will follow after you are done with them!
[1] For those who know what I am writing about, he asked me to master Serge Lang's "Introduction to Modular Forms" and a proof of the Eichler-Selberg trace formula (it wasn't enough to just know the proof for the level 1 case). He was particularly insistent upon Atkin-Lehner theory.
[2] Average order of the divisor function (counting the number of divisors of $n$).
[3] As an added incentive, he recommended some very nice non-math books to read in the summer.
[4] Again, for those who know what I am talking about, I was asked to write out the terms of the Eichler-Selberg trace formula and evaluate them for specific cases. A lot of the questions were about class numbers of binary quadratic forms and about Dirichlet characters.
The students were worried about the extensive syllabus for the exams (they were not as worried when this syllabus had been broken down into different courses) and expressed concern over the passing mark etc. To allay their apprehensions, one of my colleagues told them that the exam was a way of testing their grasp of fundamental topics. While it is very important that they focus exclusively on preparing for these exams for the remaining part of the summer, they should not feel stressed and take small breaks if they are feeling very exhausted with the preparation. All the faculty members present shared memories of their own comprehensive exams very candidly- when and how they took it and how they chose their PhD topics after the exams.
These exams, when thought of by themselves, do seem monstrous. But, it helps a lot if one sees them as a phase of the PhD program, after which the students can start independent study and pick up topics that they really enjoy. Although we hope that the meeting made these students feel better, the pressure will of course persist until these exams are cleared and left behind.
This meeting brought back some memories about my comprehensive exams, way back in 2004! The programme at my grad school was structured differently from the Mathematics programme at IISER Pune. Students were assigned a supervisor upon joining and were expected to pursue course work and their research projects simultaneously. The comprehensive examination was usually conducted in the second year of the program, but could also be postponed till the third year. For this exam, the student had to submit a research proposal, give a talk on it and then face an interview in which they were quizzed about the proposal. The interview was not restricted to what was written on the proposal - it could have questions about any fundamental topics which the committee felt the student should know before embarking upon the proposed project.
Students would usually take their comprehensive exams during the summer to avoid clashes with their course work. However, I really wanted to go to India during the summer. So, my advisor (Prof. RM) suggested that I take the exam during term time. I wrote up the proposal during Christmas vacation and after running it by Prof. RM, submitted it some time in January. The exam was scheduled exactly after a month. Meanwhile, in that semester, I was taking three courses and also teaching a first year course to a large class (this was my first teaching experience).
The real fun started after the submission of the proposal. Other than my advisor, there were three members in the committee, out of which one was the meeting coordinator and two (Member 1 and Member 2) were subject experts. I submitted the proposals to them in person and asked them to give me some feedback before the exam. Member 1 read the proposal within two days of submission and gave me some feedback about what parts needed clarification, details etc.
Member 2 was a formidable senior number theorist, of whom most grad students were very scared. The courses offered by him (almost all of which I had taken) were very challenging. I have spent a lot of sleepless nights working through his assignments. When I went to his office to give him the write-up, he sternly glanced at it for a few minutes and mentioned a book that he expected me to master before the exam. I almost fell off the chair [1]. He said that he did not believe in students taking exams unless they were thoroughly prepared and went on to describe some students whom he had examined before and who had failed the comps.
Struck by lightning and dazed, I went straight to Prof. RM's office, told him about the conversation with Member 2 and requested that the exam be postponed. I was ready to sacrifice the India trip. The conversation that took place afterwards is a great example of how to supervise a student!
He explained to me the basic objective of the exam: to test whether the student understands fundamental concepts and techniques that would be needed for his/her research project. He then went to the board, mentioned a problem in elementary number theory [2] and we started discussing how one solves it. He then mentioned that the purpose of the comprehensive exam is to make sure, for example, that I can apply fundamental summation techniques to such questions before I embark on a research project. He then took out the book mentioned by Member 2 from his bookshelf and we went through the table of contents. I realized that I already knew many topics from this book. He pointed out the other sections that were relevant to my proposal and convinced me that by working for two hours a day on this book, I will master these topics and be ready to face the comps within a month.
When I again expressed my inability to do this, he started talking about the good things that would happen if I could clear the comprehensive exam within a month. "You can visit your family, relax [3] and come back with a fresh mind to work on your research problem. You can finish by early 2006 and also invite your father for the convocation ceremony in the summer." I got convinced that adhering to the above timelines would be great and decided not to postpone my exam.
I came out of the office feeling relieved and much more confident. I woke up two hours earlier every morning and worked through several sections of the book.
My exam lasted for one and a half hours. Members started by asking me basic definitions, examples, non-examples related to modular forms. After the warm up, I was asked to prove some lemmas mentioned in the proposal and was asked if it was possible to generalize them to other cases [4]. Member 2 came prepared with a long list of questions, as expected. As we ran out of time, we could not cover all his questions. So, he graciously gave me a xerox copy of his questionnaire. The exam was not half as dreadful as I had anticipated and I was declared passed.
It was lunch time when the interview ended. After lunch, I attended a class taught by Member 1 and taught my first year course. Later in the evening, I had dinner at my favourite Indian restaurant and saw a Hindi movie with a Japanese friend.
In the next teaching semester, Member 2 also offered a course on the topics that he had asked me to learn before the exam. I took this course. Later, he gave me valuable inputs and feedback for my thesis and also wrote reference letters for my job applications.
To all students taking comprehensives this summer, all the very best. Study well, take it easy and think of all the good things that will follow after you are done with them!
[1] For those who know what I am writing about, he asked me to master Serge Lang's "Introduction to Modular Forms" and a proof of the Eichler-Selberg trace formula (it wasn't enough to just know the proof for the level 1 case). He was particularly insistent upon Atkin-Lehner theory.
[2] Average order of the divisor function (counting the number of divisors of $n$).
[3] As an added incentive, he recommended some very nice non-math books to read in the summer.
[4] Again, for those who know what I am talking about, I was asked to write out the terms of the Eichler-Selberg trace formula and evaluate them for specific cases. A lot of the questions were about class numbers of binary quadratic forms and about Dirichlet characters.
