These exams should be thought of as stepping stones towards your Ph.D. Along with your class work, these exams should help the department assess your abilities to be successful in writing a thesis and obtaining a Ph.D. But most importantly, they give the student a way to check his/her progress. As you move from year to year and pass these different exams, you will have a concrete set of successes you can look back on that, along with your class work and research, build a solid doctoral education.

Information regarding Exams & Presentations.

Preliminary Exam

The purpose of the examination is to assess the qualifications of students in critical areas of undergraduate mathematics. The preliminary exam (prelim) consists of 12 questions from single variable calculus, multivariable calculus, and linear algebra. Students choose 10 questions to be graded. The duration of the test is four hours. The level of difficulty of questions is similar to the GRE mathematics subject test, however the emphasis is different. The linear algebra and calculus questions from GRE preparation books can serve as good practice questions for the test. The format of the test is different from the GRE test in that there are no multiple choice questions.  It may be helpful for you to keep the following tips in mind.

  • You may get copies of sample prelims from previous years either online or from the Graduate Student Coordinator.
  • Mathematics GRE subject test study guides can be purchased at the GRE web site.
  • Relevant textbooks in Calculus, Linear Algebra, and Differential Equations are available for loan from the Graduate Student Coordinator.
  • Once you have decided when you will take the prelim you must sign up with the Graduate Student Coordinator. You have the option to not take it even if you sign up. However, you should let the Graduate Student Coordinator know if you change your mind.
  • Remember to get a good night's sleep, eat well before the exam, review the material for several weeks prior to the exam, find a study partner, etc....
  • You may ask the Graduate Student Coordinator which faculty are on the preliminary exam writing #committee. Feel free to contact faculty for helpful studying suggestions.
  • Students who earn an 80% or higher on the GRE Math Subject Test are exempt from taking the prelim.

Qualifying Exam

As you take graduate classes, you should keep in mind that you will have to pass a qualifying exam covering three 6000 level classes of your choosing. Your selection might be based upon your aptitude for the subject matter, your relationship with the professors of the classes, and the course's relationship with your tentative research topic. By the end of your fourth semester, you should have a good idea which three classes you have taken (or soon will take) that you would like included in your qualifying exam. You should discuss the tentative course list with your academic advisor.

After choosing your three classes, you should set up a date and time that works for the examining #committee and which meets the exam time frame requirement. Please note: you will want to confirm that the professors you took the courses from will be on your examining #committee. There are rules that pertain to the situation when one of your chosen faculty members is unable to attend the examination.

In any case, once you know the subjects and the examiners, you will want to begin stage two of your preparation for this exam. (Stage one is to take the courses in the first place.) You should review your class notes, as well as find more references on the subjects, acquire books you didn't use in class, etc., to ensure that you have a very good understanding of the material. Then talk to the people who will be on your examination #committee, ask them for the things they think are the most important. If possible, set up more than one meeting with each person, so that you get more than one impression. Ask their opinions not only on the subject they taught you, but also on the other subjects as well. Don't forget, they will all be in the room with you and it is in your best interest to know as much as possible about what will be asked before you walk in. Similarly, it is in your best interest for these people to be confident of your knowledge before your exam.

Note that it is NOT essential to have found a thesis advisor before this exam, but at least you should be well acquainted with your academic advisor by this time.  You should bring a copy of the qualifying exam form to the faculty member who will be chairing your examination #committee.

Candidacy Presentation

After you have found a thesis advisor and have begun looking into a research topic, together you will form a thesis committee.  The Institute requires at least three full-time tenure-track Rensselaer faculty members, and at least one scientist or professor from outside the Math Sciences Department. Once things have solidified, you and your advisor will decide when it is time for your candidacy presentation. This must occur before you complete 75 credit hours (an Institute requirement). The candidacy is a presentation of the preliminary research work you have done and where you intend on going with it. It is a time to gather feedback from your entire committee, since it may be the only time they are all in the same room until your thesis defense. It should be viewed as a presentation and request for feedback, but you can expect penetrating questions from your committee.

Writing Your Thesis

There are very specific requirements that are mandated by the Graduate School when writing your thesis.  Most mathematics students choose to write their thesis using LaTeX. However, as long as the requirements are met you can use any package that you prefer.

There is a online guide called Preparing a Thesis With LaTex.

The Writing Center has a document called Thesis Writing.

There is also a Thesis Writing Manual available from the Graduate School.


When your thesis is complete you must defend it in a public exam conducted by your #committee.  Here are a few other things to keep in mind.

  • Schedule the defense when all #committee member can be present.  It is also important to keep in mind the last dates that the Graduate School accepts thesis (depending in whether you plan to graduate in December, May, or August).
  • You will need to reserve a room and notify the Graduate Student Coordinator.
  • The Record of Thesis/Project Examination form must be filled out after you have successfully completed your defense.
  • You will need to pay a binding fee (about $80) to the Registrar (keep a receipt).
  • There are also Copyright forms and an Earned Ph.D. Student Survey that you can pick up from the Graduate School.  If you choose to submit the Copyright forms, you will also need a money order for $45.
  • You should bring several (at least three) copies of the thesis cover sheet on bond paper to your defense and get original signatures on each.  It is also a good idea to bring a copy of your thesis in case of questions.
  • The Graduate School has strict requirements for submitting your thesis.  Be sure that your work meets all the requirements stated in the Doctoral Candidate Thesis Checklist.
  • The Math Sciences Department should receive one hard-bound copy of your thesis.  Also, you should make copies of your thesis for your #committee members.  Three are several local companies that can make spiral bound copies, and the RPI bookstore can make hard-bound copies.

Fairness & Appeals

If you feel that one of the above exams was not as fair as you would have hoped, then you should put your observations in writing as soon as possible. Discuss these observations with your advisor. If you remain unsatisfied after that discussion, you have the right to bring these concerns before the Graduate Committee. A formal appeal in writing should be submitted to the Graduate Committee as soon as possible. This process is in place to provide checks and balances to the system. You should make every effort to discuss things with your advisor first.  However, you should not be afraid to use the system.

Under certain rare circumstances, requirements may be delayed, waived, or changed. All appeals should be made in writing to the Graduate Committee. Prior discussion with your academic advisor is assumed. Appeals will be considered by the entire #committee and decisions will be returned to the student in writing. Keep in mind these requirements were set up not to hinder you from gaining a good education, but to ensure that you obtain a quality degree in a timely fashion.