### The Masters Preliminary Exam

The Masters Preliminary Exam is taken by all incoming AMCS graduate students at the University of Pennsylvania, just prior to the start of the fall semester (generally in late August). It plays three roles:

- It serves as a placement exam, to help determine whether students should begin with 500-level courses or with 600-level courses (or with a mixture).
- It is a requirement for each of the graduate degrees in applied mathematics, in order to ensure that those who receive graduate degrees have a solid mathematical foundation.
- It provides an incentive for incoming grad students to review basic material, which will then help them in their beginning graduate classes.

Students who do not pass the exam the first time will have a second chance to pass it at the end of the spring semester (generally in late April or early May). Those students take the Proseminar (MATH 504, 505) or other 400/500-level math courses during their first year, to strengthen their problem solving ability, their background in mathematics, and their familiarity with material on the prelim.

The preliminary exam focuses on the key material from an undergraduate mathematics program that is most important to those entering a applied mathematics graduate program. The first half of the exam is given in the morning, and the second half in the afternoon. Each of these two parts consists of six problems, and students are given two and a half hours for each part.

The exam consists of problems in linear algebra, advanced calculus, basic complex analysis and probability. Some problems are computational, some ask for proofs, and some ask for examples or counterexamples. Each part of the exam (morning and afternoon) constains a mixture of types of problems, and a mixture of analysis and algebra problems.

The key to success on the preliminary exams is practice! Here are some practice problems and previous years' exams:

- Prelim practice problems.
- Fall 2008 part 1, Fall 2008 part 2.
- Fall 2009 part 1, Fall 2009 part 2.
- Fall 2010 part 1, Fall 2010 part 2.
- Fall 2011 part 1, Fall 2011 part 2.
- Fall 2012 part 1, Fall 2012 part 2.
- Spring 2013 part 1, Spring 2013 part 2.

The following list of topics gives a general idea of the material that is covered on the exam:

- I. Analysis
- Continuity, uniform continuity, properties of real numbers, intermediate value theorem, metric spaces, topological spaces, compactness, epsilon-delta proofs.
- Differentiable functions of one variable: differentiation, Riemann integration, fundamental theorem of calculus, mean-value theorem, Taylor's theorem
- Sequences and series of numbers and functions, uniform convergence, equicontinuity, interchange of limit operations, continuity of limiting functions.
- Ordinary differential equations (separable, exact, first order linear, second order linear with constant coefficients), applications such as orthogonal trajectories.
- Multivariable calculus: partial derivatives, multiple integrals, integrals in various coordinate systems, vector fields in Euclidean space (divergence, curl, conservative fields), line and surface integrals, vector calculus (Green's theorem, divergence theorem and Stokes' theorem), inverse and implicit function theorems, Lagrange multipliers.
- Power series and contour integration.
- Basics of Fourier series.

- II. Linear Algebra
- Linear Algebra:
- Vector spaces over
**R**,**C**, and other fields: subspaces, linear independence, basis and dimension. - Linear transformations and matrices: constructing matrices of abstract linear transformations, similarity, change of basis, trace, determinants, kernel, image, dimension theorems, rank; application to systems of linear equations.
- Eigenvalues and eigenvectors: computation, diagonalization, characteristic and minimal polynomials, invariance of trace and determinant.
- Inner product spaces: real and Hermitian inner products, orthonormal bases, Gram-Schmidt orthogonalization, orthogonal and unitary transformations, symmetric and Hermitian matrices, quadratic forms.
- Positive definite matrices and the variational characterization of eigenvalues and eigenspaces.

- Vector spaces over
- Numerical linear algebra
- Basic algorithms for solving linear systems of equations
- Notions of stability and conditioning
- Basic algorithms for finding eigenvalues and eigenvectors

- Linear Algebra:
- III. Probability and statistics
- The basic notions of events and probability, simple discrete distributions
- Independence of events
- Random variables
- Moments, the characteristic function
- Simple examples of estimators, and the notion of bias
- IV. Complex analysis
- Definitions of analytic functions
- Cauchy theorem and integral formula
- Power series
- Residue calculations
- Elementary conformal maps

### Rules and Regulations for the AMCS PhD Oral Exam and Thesis Committee

The purpose of the oral exam is to assess a student's readiness to transition into full-time research and eventually write his or her dissertation. It is something of a hybrid between the subject-oriented oral exam administered by the Math department and the thesis proposal used in many fields of science and engineering.

The AMCS oral exam will normally be taken by the end of a student's fifth semester of study, and in any case no later than the summer after the sixth semester. Exceptions may only be granted by the AMCS Graduate Group Chair. Students are allowed to take further course-work after passing the oral exam.

Preparation for the oral exam provides the student with the opportunity to get a strong grasp on the background needed for his/her research program, and to formulate a clear plan for his/her overall research. The student, and his/her thesis advisor should agree upon an outline for the background material to be mastered as part of the oral exam process. This should be done approximately six months before the exam (usually by the start of the second summer), and a copy of the outline signed by the student and his/her advisor must be provided to the Graduate Group Chair at the time it is completed.

As part of this process the student will select a Thesis Committee. The Thesis Committee will consist of at least two AMCS-affiliated faculty members, including

the student's primary supervisor, as well as, at least one other member, who is not required to be a member of the AMCS affiliated faculty. The primary supervisor may not serve as the Chair of the committee. The composition of the Thesis Committee must be approved by the Graduate Group Chair.

The exam itself affords the student with a chance to demonstrate his/her knowledge of the subjects in which they plan to work, and explain to the committee their overall research plans. The committee should provide constructive suggestions regarding these plans. If a student fails to demonstrate adequate working knowledge of the field in which they plan to work, or lacks a novel problem, with significant mathematical content for their research, then the student risks failing the exam. A student who fails will be given a chance to retake the exam, within six months, and will be asked to leave the program if his or her performance on this second exam is not satisfactory.

### Detailed descriptions of various aspects of the process:

Scheduling the exam:

Once the thesis committee has been assembled, the student should schedule his/her exam. The Graduate Chair shall then be advised of the time and date of the Thesis Committee meeting and oral examination. Ideally, the exam should take place by the end of the Fall semester of the third year, but in any case, no later than the summer after the third year. One week prior to the meeting, students will submit their thesis proposal to the members of their Thesis Committee, along with an outline of the topics that he or she has studied in the process of preparing his/her thesis proposal.

The thesis proposal should:

- begin with a 250 word abstract emphasizing the main objectives of the research, its broad significance and its relationship to Mathematics.
- provide an introduction to the general area of the research project, including a thorough literature review on the topic of study. This section will help the committee assess if student has sufficient background in the math and the subject-specific topics.
- describe the overall objectives of the proposed research.
- contain preliminary results and a general research plan for achieving the research objectives.
- not exceed 20 double-spaced typed pages.

Conduct and outcome of the exam:

At the time of the Oral Exam, the Thesis Committee will meet with the student and his/her advisor individually for a few minutes prior to the student’s presentation. After presentation of the research project, the Thesis Committee will orally examine the student on the background material and their proposed research. The committee may "pass" the student, "conditionally pass" the student, with a written list of conditions to be fulfilled for the grade to be converted to a "pass," or "fail" the student. The student's thesis advisor may participate in the oral exam, but shall not participate in the evaluation of the student, except when requested by the Chair of the Oral Exam Committee. The advisor's participation is limited to attendance only, and he/she shall have no vote, except in case of a tie.

The Thesis Committee Chair should report the result of the Oral exam to the Graduate Chair, in writing, as soon as possible. An Oral Exam Thesis Committee Report Form will be provided to the Committee for this purpose. The written evaluation by Thesis Committee will be given to student for their review and will become part of the student's record. If the project is approved, the student has passed the oral examination. Form 150 will then be submitted to the Graduate Division of the School of Arts and Sciences for notation on the student's transcript.

Following a successful oral exam, the student must meet with his or her Thesis Committee at least once a year. These committee meetings are designed to facilitate communication and provide guidance during the research process. The Thesis Committee may require meetings every six months. The Graduate Chair will remind the student and thesis advisor when a meeting should be scheduled. It is important to note that it takes some time to organize these meetings, thus it is wise to start the planning process early.

The Chair of the Thesis Committee will submit a written evaluation of each meeting to the Graduate Chair and the student for his/her review. This report will become part of the student's record. At these meetings, students must show satisfactory progress towards completion of their dissertation research as judged by the thesis committee. If not, the thesis committee can recommend dismissal from the program.

Consequence of two failures:

If a student twice fails the oral exam, then they will not be permitted to continue in the AMCS PhD program. If they have fulfilled all the requirements, and have not already received the degree, then they may be awarded a Masters degree.