## Section 4 The first year of graduate study

In broad strokes, as a first year student, you will take nine courses, help TA for courses in two of the Fall, Winter, and Spring terms, and prepare for a preliminary exam in three core areas covered by your first-year courses. Exams are typically offered just before the start of summer term in such a way that no student takes two on one day.

The department offers approximately sixteen graduate courses each year, seven of which are aimed at core topics covered by the end of year preliminary exam. Core graduate courses fall into two categories, some with a fixed syllabus and others whose syllabus alternates from year to year, making it possible to take the second offerings in later years to add to the breadth of your mathematical knowledge. Advanced courses are typically topics courses based in part on student interest.

### Subsection 4.1 Advisors

In general, an individual first-year advisor is assigned to each graduate student to help them transition to graduate school and serve as a general advisor until the student has an advancement/thesis committee more closely aligned with their research pursuits.

These advisors help students work out academic plans (which are supposed to be written and filed with the department), and be a sounding board and neutral third party to help resolve issues surrounding work as a graduate student. If a graduate student is not compatible with his or her assigned advisor, they can ask for a different one by contacting the Graduate Representative (listed in the first lines of the web page Information for Current Graduate Students).

In addition to a student's personal advisor, there is an Advisor to Graduate Students and an Graduate Representative whose roles are described in the Appendix.

### Subsection 4.2 Pure or Applied — Choosing among research areas

The first thing to note here is that, while there is a clear difference between traditional pure mathematics and traditional applied mathematics, reflected in two different paths for your first two years of graduate study, the two are part of a multi-dimensional continuum. *You will not actually choose between pure and applied math. You will choose a thesis area that falls somewhere within this continuum.*

During your first year, you may be interested in some areas that are more pure and some that are more applied. You do not have to choose an area of concentration right away. You can keep your options open between pure and applied. To do this, look at the *courses and advice* sections for both pure and applied math, and discuss with your advisor what you should be doing in the way of attending seminars and searching out possible research projects. When you choose courses, keep in mind that your thesis committee in your eventual area of research will probably require you to fill in any gaps in your course preparation, and do your best to leave as few gaps as possible.

*The one choice you must make during your first year is whether to take the “pure math” preliminary exam or the “applied math” preliminary exam.* If you are really keeping your options open, you should take enough core courses to cover the material in both exams. Being interested in both pure and applied areas does not mean you can pick an assortment of preliminary exams sampling from both. Passing preliminary exams in applied math, for example, will only reinforce the breadth of background of a student who ultimately writes a thesis in pure math. If that student has also taken the core courses in pure math and done well, their thesis committee is unlikely to object to their having taken the “wrong” exam.

During your second year, you must begin to narrow your options somewhat. Before the summer that begins your second year, you must decide whether to arrange the summer research project described in the advancement procedure for applied math. Again, doing a summer project in mathematical biology does not rule out eventually writing a thesis in signal processing, or in algebraic combinatorics. However, not doing any summer research project may affect whether applied mathematics faculty members are willing to accept you as a thesis student. You should be talking to prospective thesis advisors and thesis committee members about this.

Note that if you are not following the applied mathematics path by doing a summer research project, then you must follow the pure mathematics path by taking Math 117. The guidelines for pure mathematics call for you to assemble an advancement committee, including a tentative advisor. You should be working on this during the summer that begins your second year, at the latest. This does not mean you have to choose your thesis area that summer. It is quite possible to include more than one tentative advisor, who may work in quite different areas, on your committee. The committee will help you design a second-year program, and an advancement examination (which may include different options), that will allow you to make a choice closer to the end of your second year. Taking this option does not rule out writing a thesis in an applied area, provided your committee includes a possible advisor in that area who has agreed to your plan.

Once you enter your third year, you should begin work in a specific thesis area. It is difficult, although not impossible, to change areas after your third year, and still have time to complete a quality thesis. The choice of research area is not a lifetime commitment. Many Dartmouth faculty members work now in areas quite different from their thesis research.

### Subsection 4.3 Courses and Advice — Applied Math

Students interested in doing research in applied mathematics, specifically in the areas of computational mathematics, data analysis and assimilation, imaging, or uncertainty quantification, must have a working knowledge of material taught in certain core courses. Courses appearing in traditional applied mathematics programs include numerical analysis, partial differential equations, and methods in applied mathematics. In addition, courses like stochastic processes and uncertainty quantification are more germane to the research interests of current Dartmouth faculty, although they too are becoming more common in applied math programs.

#### Subsubsection 4.3.1 Introductory Courses

Among the first-year core applied math courses a student should take are:

- Math 106: Stochastic Processes and Uncertainty Quantification
- Math 116: Numerical Analysis
- Math 126: Partial Differential Equations
- Math 136: Methods in Applied Mathematics

Note that the content of Math 116 and Math 106 alternates in even and odd years, so students should take each of these courses twice, completing the core course requirement in two years.

In addition to the courses listed above, there may also be courses of interest in pure mathematics or offered in other departments that should be strongly considered, especially in those topics related to their research application domain. Such course offerings will vary from year to year, so it is a good idea to discuss opportunities with your first-year advisor (or potential thesis advisory committee) each term before registering for classes.

In all, as a first-year student, you will take nine courses, six of which must be classroom courses. Beyond those six, a student may choose other classroom courses of interest, or sign up for supervised reading courses.

#### Subsubsection 4.3.2 Follow-on courses

All students in applied math need to take the alternate topic offerings of Math 106 and Math 116 (in their second year). But even in the first year, there may be courses of interest offered in other departments that should be strongly considered, especially in those topics related to their research application domain. Such course offerings will vary from year to year, it is a good idea to discuss opportunities with your first-year advisor (or potential thesis advisory committee) each term before registering for classes.

#### Subsubsection 4.3.3 Seminars

Applied mathematics students are required to attend the weekly applied mathematics seminar. They are also strongly encouraged to participate in Math 150, which is a student-run applied math topics seminar course.

#### Subsubsection 4.3.4 Research Projects

Students interested in doing research in applied mathematics should seek out opportunities *early* — in their first year — to get involved in ongoing research projects, and should start working on their own (often related) research projects even if ultimately they do something else for their dissertations. The best way to do this is to attend group meetings and learn about various research investigations within the group. Note that applied math faculty are often interested in collaborative projects, and many group meetings are held with two or more professors, students, and postdocs. The best way to find out about group meetings is to talk with more advanced students and find out what they are working on. Advanced students and postdoctoral fellows are available and encouraged to support first and second-year students as they begin their own investigations. All students have the opportunity to present their research findings at group meetings and at the applied math seminar.

#### Subsubsection 4.3.5 Advanced Planning for the Summer Research/Internship

Part of the advancement procedures for applied students involves a significant summer research experience/internship, and while the actual research will formally occur at the start of the second academic year (summer), plans need to made and approved before the end of the spring term of the first year.

While the research project may be supervised by any professor in mathematics, students are strongly encouraged to seek out interdisciplinary opportunities, especially in research labs. *It is imperative that students who wish to do an internship discuss this with either their first year advisor or some potential Advancement Committee member by February 1 of their first year (sooner is better)*. The applications for internships are often due by mid February.

The GPC needs to approve each proposed project. Students must submit to the GPC a one-to-two page written research plan, which has been signed and approved by a prospective Advancement Committee (two professors, or by the prospective advisor and the external supervisor). The plan should be submitted to the GPC no later than two weeks before the end of spring term.

Requirements for documenting the outcome of the research experience/internship as well as information concerning how this experience is integrated into the Advancement Exam can be found in the section on the second-year summer research/internship.

### Subsection 4.4 Courses and Advice — Pure Math

While the spectrum of research areas in pure mathematics is far broader than the contents of the core first-year courses in algebra, analysis and topology, these six courses form an essential core on which further study builds.

#### Subsubsection 4.4.1 Basic Courses

The basic courses in algebra, analysis, and topology are:

- Math 101: Algebra
- Math 103: Analysis
- Math 104: Topology

The subject matter for these courses is aimed at advanced undergraduates and beginning graduate students.

#### Subsubsection 4.4.2 Follow-on Courses

Sequels to the basic courses are:

- Math 111: Topics in Algebra
- Math 113: Topics in Analysis
- Math 114: Topics in Topology

The content of these courses will vary from year to year.

These six courses can be supplemented both by approved upper-level undergraduate courses as well as other graduate courses divided between introductory and advanced topic courses. There is also flexibility to add depth by taking supervised reading courses highly tailored to student interest (see Math 127, 137). A full course load for a graduate student is nine courses per year; there is also a breadth requirement for all students which requires a minimum number of classroom courses each year; that number for first-year students is six.

Choosing courses for the year should be done in consultation with the first-year advisor. He or she can get a sense of the student's background, how prepared they are to take the standard courses, and suggest appropriate courses that get their career off to a solid start while also helping the student look through the options for the entire first year.

#### Subsubsection 4.4.3 Seminars

Students are strongly encouraged to sit in on various research seminars offered by the department. This will give a sense of the kind of research in which various faculty are involved, and what might be areas of interest to pursue. This will also aid in the student's efforts to constitute an Advancement Committee which shepherd the student through the second-year advancement process.

### Subsection 4.5 TA and grading responsibilities

First and second-year graduate students have two terms of responsibility in which they assist with undergraduate courses. As a TA, the responsibilities generally include staffing walk-in tutorials for two hours three times per week and helping grade all exams. Preparation for each tutorial session is an assumed part of your responsibility. TAs may also have the opportunity to host review sessions prior to midterms. Total weekly time commitment averages ten hours per week.

While the general model is to TA twice in each year, in certain circumstances a student may negotiate to replace one term of TAing for a term of grading homework for an upper-level undergraduate or beginning graduate course. This negotiation is between the student, the course instructor, and the student's first/second-year advisor. In the case that the instructor is the advisor, the Advisor to Graduate Students will serve as the third member. The idea is that all three must agree that this is a mutually beneficial arrangement.

Note that in the term you TA for the second time (each year), you must register for Math 107. This is the official way in which the department tracks that you have completed this degree requirement. Your grade (pass/fail) in Math 107 is based on the assessment of the instructors for whom you have TAed, and on end-of-term evaluations filled out by the students with whom you have interacted.

Please also see Reference Sheet for TAs which includes a great deal of information about resources and regulations.

### Subsection 4.6 The preliminary exam

All students are required to take and expected to pass a preliminary exam in pure or applied mathematics. This exam is typically offered just before the start of the summer term, and if necessary, again just before the start of the fall term. The exams differ in content (see below), and taking this exam is a necessary step before a student may constitute their advancement committee. The sections below provide more detail.

#### Subsubsection 4.6.1 The preliminary exam — Applied Math

The preliminary exam in applied mathematics consists of topics from three of the four core first-year courses: Math 106, 116, 126, and 136. The current format is a three-hour written exam with two questions each from the chosen areas. It is offered in a window of time just before the start of summer term.

#### Subsubsection 4.6.2 The preliminary exam — Pure Math

First-year students in pure mathematics will be required to take a written preliminary exam covering basic algebra, analysis, and topology, limited to topics normally covered in Math 101, 103, and 104, together with ancillary courses such as Math 71, 63, and 54. The exam will be offered at the just before the start of the summer term, and again just before the beginning of the fall term.

The exam will be set and graded by an ad hoc Pure Math Preliminary Exam Committee, to include the instructors in Math 101, 103, and 104, and a representative of the GPC. This committee will report the results of the exam to the GPC, who will decide which students have passed the exam. The exam will be written, though its precise format is still under consideration.

#### Subsubsection 4.6.3 What if a student fails the preliminary exam?

If a student does not pass the preliminary exam just prior to the start of summer term, the GPC will look at the student's work in the program so far, including their performance on the exam. Most likely, the GPC will say nothing more than that they should try again in the fall. In that case, the student should continue their efforts just as if they passed the exam. Talk to your advisor. Pursue your summer research project or take Math 117; begin working on forming an Advancement Committee; talk to prospective thesis advisors, and take their advice about what to work on during the summer. Of course, the student should also spend some time over the summer preparing to retake whatever part of the exam they did not pass. In exceptional cases the GPC may have additional requirements or expectations of the student, or agree to a different course of action.

If a student retakes the exam just prior to fall and still does not pass, the GPC will determine whether it is in everyone's best interest for the student to continue in the program past the end of fall term. In order to continue, the student will need at least two things. One is an Advancement Committee, including at least one person who agrees to supervise the thesis if the student passes the Advancement Examination. The other is a plan for addressing the deficits shown by the exam performance, which must be approved by both the Advancement Committee and the GPC.