Thoughts on my oral exam

In Stony Brook’s maths department, before embarking on any serious research, a graduate student must complete an oral qualifying examination. I took mine in December 1st 2020. I was assessed on Holomorphic Dynamics (major topic) and Several Complex Variables (minor topic) by Dzmitry (Dima) Dudko, my major advisor, Eric Bedford, my minor advisor, and Mikhail (Misha) Lyubich. Here are my thoughts on how it went and some tips on how you can ace your orals.

What is the format?

The purpose of the oral exam is to assess the ability of a PhD student to learn advanced maths and to prepare them for mathematical research. In the US, it is usually completed in the second/third year of their PhD program. The oral exam covers two topics: a major topic and a minor topic. The student has to firstly find a major advisor with whom they can take a reading course on the selected major topic for about two semesters. The minor topic does not have to be complementary to the major one; it can be a completely different second interest the student wishes to pursue for about a semester.

In preparation for the orals, the student must create a syllabus based on their major and minor topics. During the exam, the orals committee members would ask a variety of mathematical questions based on the syllabus. The exam typically lasts for about 1.5-2 hours. Upon completion, the student is expected to leave the room and wait for the orals committee to discuss the student’s performance and decide whether the student has passed. To a number of people, the orals is a formality, and a pass is almost guaranteed if you are deemed ready to pursue mathematical research by the committee based on how the orals went (and likely also based on the student’s past interactions with the advisors). A fail means that you might have not done enough and would have to repeat the exam again another time.

My experience

Choosing the topics and the advisors was a fairly quick process to me because I’ve known from the beginning that I would want to focus on complex analysis and dynamics for research. Throughout the entire year, I have read, with the help of Dima and Eric, the following books.

1. L. Ahlfors. Lectures on Quasiconformal Mappings. Van Nostrand, 1966.
2. C. McMullen. Complex Dynamics and Renormalization. Princeton University Press, 1994.
3. M. Lyubich. Conformal Geometry and Dynamics of Quadratic Polynomials, Vol I-II. Book in preparation, 2020.
4. L. Bers, Introduction to Several Complex Variables, New York Univ. Press, 1964.

I wrote my syllabus based on 2-4. I created a summary to help me remember things I need to know by heart. I also did mock orals with both Dima and Eric, as well as more senior graduate students specialising in dynamics and/or several complex variables. All of these certainly helped me prepare myself for the orals.

I didn’t sleep well the night before and I was anxious and afraid that I would mess up somehow. I skimmed through my summary an hour before the orals, making sure that I did not forget anything important. The time went by rather quickly and it was finally time. Covid-19 is still around and so we arranged a zoom meeting for this. I joined the meeting and everyone was on time.

The fact that I know everyone in the orals committee helped us maintain a rather relaxed and friendly environment throughout. I was able to maintain my composure most of the time. I tackled the questions on definitions of things (invariant line fields, Lattès map, laminations, domains of holomorphy, cohomology, etc) fairly easily. To my surprise, I was also able to recall sketch some proofs of a number of theorems (invariant line fields induces queer components, Oka extension theorem).

The main obstacles were the problems which require more thought and impromptu computation. These include locating all the periodic cycles of period 3 of $f(z)=z^2-1$ using external rays, drawing geodesic laminations associated to a Julia set, computing the domain of convergence of the Taylor series of $(1-z-w^2)^{-1}$ about $(0,0)$, and calculating the Cech cohomology $H^1(\mathbb{C}^2 \backslash {(0,0)})$. I certainly forgot some details and made occasional mistakes in some of the examples I came up with and whenever that occurred, the committee members would give me some hints of what went wrong and what I could do about it. The way I reacted to these hints varied; there were times when I certainly noticed the mistake and corrected myself immediately, but there were other times when I just did not know what went wrong and either they told me the answer or we decided to move on.

It had been going for 75 minutes when every committee member had their chance to pose questions, but Misha thought we could possibly go on a little longer. He skimmed through my syllabus one more time and said, “Can you tell me about the Oka extension theorem? It sounds quite interesting.” I stated the theorem and I was rather surprised by myself that I managed to give a sketch of the proof to him. He then asked me a more philosophical question on this theorem. Eric was quick to notice I couldn’t come up with a decent answer and he helped me fill in the details.

We finally finished and I was sent to the waiting room to let them discuss how it went. I walked around the house, drank some water and ate a banana whilst waiting and thinking about the questions they asked me previously. The orals was tiring, but it was also certainly thought provoking. Some of the challenging problems were actually interesting in their own right and I couldn’t even get them out of my head for days. For that, I am grateful that I actually learned something new from it.

After about 4-5 minutes, they summoned me back to the main room. Eric cheekily said, “Since you mentioned the queer components during orals earlier, I guess you can tell me whether or not queer components exist in the coming years because from now on you can do research!” (The existence is actually a HUGE conjecture in holomorphic dynamics, lol.) From their faces, I could tell that all the committee members were pleased of how it went. I guessed that I did well enough to pass!

Some tips

1. Establish good relationship with your advisors from the beginning. Be respectful and friendly. Have at least one mathematical question to ask in every meeting – this can be from one of the exercises you struggled completing, on any doubts you have pertaining to a particular theorem, or on a particular topic you randomly encounter in a seminar you recently attended. Also, don’t hesitate to have a little non-mathematical chit chat.
2. Make sure that you know all the definitions and examples relevant to everything you write in your syllabus. Knowing how to use important theorems is more practical than knowing the proofs line by line. This also includes knowing counterexamples to these theorems, e.g. what would go wrong if this particular assumption does not hold in this theorem? I made a summary for myself and I found this particularly helpful!
3. Make sure you send out your syllabus to your orals committee members early! If your exam is online, email them beforehand in advance. If your oral exam is in person, print it out and bring some copies. It is essential that they ask you questions relevant to what you (at least, claim to) have studied and prepared for.
4. Have plenty of rest the night before your orals. Do not stress out and overwork yourself. During the orals, make sure you are in a comfortable room and have a glass of water next to you.
5. Upon being asked a rather involved question, you should take your time to think and not blurt out random answers without proper logical thought. Do not panic! Feel free to ask for a minute or two to think. Plan on how you should structure your thoughts and explain your answer in a coherent way.
6. Chances are there will be a number of questions which you do not immediately know how to answer, and it is alright! What not to do is to do nothing and say “I don’t know” without looking like you are trying. Let them know if you would like to do any rough working on the board to help you come up with an answer. When you are unsure of the answer, say out loud any slightest ideas you have. Often the committee members will give you a hint or two to help you out. Remember that in the end, they are rooting for you!