This section covers the First Year and Masters Exams, the Language Exam, and both written and oral parts of the Qualiying Exam.

### First Year and Masters Exams

There are eight different semester exams offered corresponding to the following courses: MAS 5311 Introduction to Algebra 1, MAS 5312 Introduction to Algebra 2, MAA 5228 Modern Analysis 1, MAA 5229 Modern Analysis 2, MTG 5316 Introduction to Topology 1, MTG 5317 Introduction to Topology 2, MAD 6406 Numerical Linear Algebra, MAD 6407 Numerical Analysis.

The PhD and MS (exam option) require appropriate passing grades in at least three semester exams with at least one from the Algebra subset (MAS 5311, MAS 5312, MTG 5317, MAD 6406), at least one from the Analysis subset (MAA 5228, MAA 5229, MTG 5316, MAD 6407). There are additional constraints that may be found in the program descriptions.

Each exam is given three times a year, in May, August and January, and lasts for two hours. Possible grades on a single part are High Pass, Pass, Master’s Pass and Fail; you need a Pass or High Pass on three exams in a timely fashion to continue to the PhD, and you have three chances on each exam to do this. However, a student may petition the Graduate Committee to waive one or more of the exams.

Syllabi and some practice problems are available for the first year exams. The Graduate Mathematics Association (GMA) maintains a database of old exams, which you will want to look at. It is a good idea to form study groups with other students, in order to study these exams in detail. Note that exams given before 2103 are 4-hour exams on the entire sequences MAS5311-12 and MAA5228-28, and so are about twice the length of the exams now being given.

### Language Exam

Although much, perhaps most scientific research today is published in English, much of the older literature is written in other European languages, and quite a few mathematicians today write in French, French mathematicians in particular. Accordingly, the Mathematics Department requires a basic reading knowledge of French, German or Russian for completion of the Ph.D. This competence is assessed by an exam in one of these languages, or by the results of exams given by ETS. The language exam must be taken by the end of the third year, and before the Qualifying Oral Exam.

The exam is currently administered by the following faculty:

- French: Dr. Crew, Dr. King
- German: Dr. Rudyak, Dr. Summers
- Russian: Dr. Rudyak

The format of the exam varies depending on the examiner, but it generally involves translation of a short text of between 2 to 5 pages during a fixed period of time. A dictionary is usually allowed. Students should consult the examiner about the details and exact format of the exam.

The choice of language is entirely up to the student, but students should be aware that certain languages may be more appropriate than others for some areas of research. Your advisor may have some useful suggestions.

If you have studied one of these languages as an undergraduate or high school student, perhaps the best way to get started is to check out a book in that language from the library and try to read it. Mathematical prose tends to be relatively simple, and with a few weeks’ work you can probably bring yourself up to speed.

For those starting the study of French or German from scratch, the Department of Languages, Literature and Culture offers from time to time courses for graduate students who need a reading knowledge of French (FRE6060) or German (GER6060). Tutorial arrangements are also possible.

### Written Part of the Qualifying Exam

Written PhD exams are given in Algebra, Analysis, Combinatorics, Logic, Ergodic Theory & Dynamical Systems, Numerical Analysis, Partial Differential Equations and Probability. Each exam lasts for four hours and a student must pass one of them in an area chosen by the thesis advisor to continue to the Qualifying Oral Exam. They are usually offered in the same time period as the First Year Exams. Each exam has a two-semester course sequence for preparation.

Students usually take two sequences leading to PhD exams in their second year as they take steps to secure an advisor. Then they take the appropriate written PhD exam in the May after their second year, and, if necessary, retake the exam in August prior to the start of their third year. The Graduate Mathematics Association (GMA) database of old exams includes written PhD exams.

### Oral Part of the Qualifying Exam

By the end of the third year of graduate study (at the latest, prior to the midpoint of the fall semester of the fourth year), each doctoral student must pass the oral qualifying examination, have a dissertation topic approved and advance to candidacy. Prior to taking the oral exam, all other examination requirements must be completed, including the language exam, the first-year exams, and the written PhD exam. Your advisor will counsel you on which PhD exam to take, in accordance with your chosen research area. Typically, in your second year, you will take the two-semester 6000-level course associated with the exam, and then take the exam in the spring.

Thus before you can advance to candidacy, you must have a research topic. Different students find their topics in different ways. Some come to graduate school with a direction in mind and find an advisor comfortable with the topic. Others try out one or more topics suggested by their advisor or another faculty member in one of their courses or the seminars they attend. Still others hone in on a topic through reading papers, asking questions and having a dialog with their advisor. Some try several topics, sometimes running into deadends or discovering others have mined the field in prior research. A few switch topics even after advancing to candidacy when they become passionate about a different direction.

Once you have a research direction, you may start by working out details in specific examples, exploring their similarities and differences. As you focus in on a larger problem, be sure to figure out how this problem fits with other research in the field. Are there other similar problems, either solved or unsolved? Are there interesting consequences that would flow from solving your problem? Be sure to check that your advisor feels the problem is doable in an amount of time appropriate for the doctorate. Some of the most beautiful problems are easy to state but have proved intractable or taken years and years to solve.

As you prepare a talk on your proposed dissertation topic for your oral exam, please think of your audience. You have been talking with your advisor for some time and share many definitions and background results that others on your committee may find unfamiliar. Your outside member may have little or no background in the area of your research and you should be sure to make part of your talk accessible to everyone. You may also have fellow graduate students attending the first part of the exam cheering you on. Your challenge is to find a way to engage everyone and at the same time showcase your knowledge of the problem and ways to make progress on it.

Check with your advisor about what to expect in the exam, since the format of the oral exam is at the discretion of the supervisory committee. Often it is largely devoted to the candidate’s research proposal and any progress on it, but may include more general questions. The main purpose is for your committee to see that you are capable of shedding light on a research question and outlining one or more avenues by which it might be solved. You must also prepare a written abstract of the material you plan to present at the oral exam, and distribute it to your committee in advance of the exam.

Start early on setting up your oral exam. You need to let the graduate secretary know 10 business days (2 calendar weeks) in advance so that she can send a notice to the departmental faculty. It is your responsability to find a block of two hours’ time when all your committee members can meet, either in person or electronically. When all have agreed on the time, let the graduate secretary know the time, share your proposed dissertation topic with them, and ask for help finding a room.

At the end of a successful examination, the supervisory committee reports an approved dissertation topic to the graduate secretary who records your advancement to candidacy and your dissertation topic.