# PhD prelims in some Applied Mathematics programs

This is an informal survey and a collection of links for PhD preliminary examinations in some other Applied Mathematics programs. The programs are selected not based on the UCD administration's notion of "peer institutions", but rather as the institutions in our professional community.

## UC Denver PhD Program in Applied Mathematics

All must take applied analysis and applied linear algebra. Two classes lead to exams. requirements past exams

- Applied analysis: advanced calculus
- Applied linear algebra

## Brown University

No prelims such as at UCD. "Prelims" are oral, total four hours, two in one major area and two hours in two minor areas. They are more like the comprehensive exams at UCD, but taken early in the program. Division of Applied Mathematics handbook

## Colorado State University: Mathematics

## Colorado State University: Statistics

**Doctoral Candidacy Requirements:** (Similar to UCD preliminary exams)

The DCR consists of two components: two mandatory examinations described in (a) and (b) below; and a particular demonstrated competency described in (c) below.

*(a)*An examination in probability and mathematical statistics based on the course content of STAT 520 and STAT 530.*(b)*An examination in statistical methods and linear models based on the course content of STAT 540 and STAT 640.*(c)*Competence in a choice of two subjects from sampling, time series, and stochastic processes (based on the course content of STAT 605, STAT 525, and STAT 521, respectively) must be demonstrated. This may be accomplished by obtaining at least an A minus (A-) grade in the corresponding course or, if the student should get less than an A minus (A-) in such course, the student will receive one and only one chance to pass a Candidacy Examination on the same subject within thirteen (13) months from the end of the course. A student who does not take the relevant course at Colorado State University can demonstrate competency by taking the Candidacy examination and will be granted a second chance if he/she should not pass the first taking of the exam.

**PhD Preliminary Examination** (Similar to UCD Comprehensive Exam) The Ph.D. preliminary examination is an oral exam administered by the student's graduate advisory committee after having passed STAT 720 and STAT 730. The Graduate School requires that this examination be administered at least two terms before the final dissertation defense.

## Colorado School of Mines

**Qualifying Exams: **Each student must pass three written exams, in the areas of Pure Mathematics, Applied Mathematics, and Computational Mathematics. Each exam will cover one course from the corresponding list below. Each student will choose which courses from each list he or she will be examined on. Students are not required to take the courses in order to take the exams.

*Pure Mathematics:*Math 500-Linear Vector Spaces, Math 502-Real and Abstract Analysis or Math 503-Funtional Analysis.

*Applied Mathematics:*MATH 510-Ordinary Differential Equations and Dynamical Systems or Math 514-Applied Mathematics I

*Computational Mathematics:*Math 550-Numerical Solutions of Partial Differential Equations or Math 551-Computational Linear Algebra.

Students should complete the qualifying exam within two years after enrolling in the PhD program.

**Thesis Proposal:** The thesis proposal consists of a written description of the research topic. The proposal is presented to the thesis committee. This should be done at least one year before the Thesis Defense. Typically, students defend the proposal in an oral presentation to the thesis commitee.

## Illinois Institute of Technology

**Qualifying Exams:** Fulfilled by achieving better than a 3.5/4.0 GPA on the courses taken in three of the core areas of study and passing an oral examination within the first five semesters of study (within the first three semesters for students entering with a Master's degree). The oral exam covers three of the four core areas, which can be chosen by the student.

**Core Courses**

- Applied Analysis I & II

- Stochastic Analysis & Stochastic Dynamics

- Discrete Applied Mathematics I & II

- Computational Mathematics I & II

**Comprehensive Exam:** Consists of an oral examination based on the student's research proposal. The exam aims to ensure that the student has the background to carry out successful research in his/her chosen area and the proposed research has sufficient scholarly merit.

## Purdue

**Qualifying Examinations:** The student must pass four written examinations. The exams are based on material that is covered in the courses listed and on material from undergraduate prerequisites. Qualifying exam areas/courses are as follows:

- Complex Analysis (MA 530)
- Real Analysis (MA 544)
- Abstract Algebra (MA 553)
- Linear Algebra (MA 554)
- Numerical Analysis (MA 514)
- Probability (MA 519)
- Partial Differential Equations (MA 523)
- Differential Geometry (MA 562)
- Topology (MA 571)
- Mathematical Logic (MA 585)

The Qualifier Deadline for students who enter the program with a master's degree is the January Qualifier Exam Session of their second year. The Qualifier Deadline for students without a master's degree is the January Qualifier Exam Session of their third year. Students who have not passed the four exams on or before the session of their Qualifier Deadline will have their privileges to continue in the Mathematics PhD program terminated.

**Advanced Topics Examinations:** After passing the Qualifying Examinations, a student must find a faculty member willing to serve as the Advanced Topics Examination Coordinator.

The Advanced Topics Proposal form lists the Coordinator, two courses beyond the qualifying level on which the student is to be examined (or a body of mathematics roughly equivalent to this), and one other faculty member who, with the Coordinator, administers the Advanced Topics Examination. At the discretion of the Coordinator, the examination may also cover a third subject, possibly with a third examiner. The examinations may be oral or written, and may be given separately or together. To pass the examination requires agreement of all members of the committee and the consent of one to serve as the student's thesis advisor.

A student may take the Advanced Topics Examination at most twice; however, the examination should be passed within one and one half years of passing the Qualifying Examinations. In special cases the Graduate Committee may grant an extension of this time limit. Each time the examination is taken, a new Advanced Topics Examination Proposal Form must be filed in the Graduate Office.

## University of Alabama-Birmingham

**Qualifying Exam:** This is an exam in Real Analysis and Applied Linear Algebra. A student admitted directly into the Ph.D. program is expected to take this exam by the end of the first year at the latest.

**Comprehensive Exam:** Consists of a written and an oral part.

## University of Colorado Boulder; Applied Math

Numerical analysis and applied analysis prelims and either one of the other two preliminary exams, partial differential equations, and probability/statistics. Classes lead to exams. PhD degree requirements syllabi and previous prelims

- Applied analysis: advanced calculus, measure theory, functional analysis
- Computational math: numerical analysis, linear algebra, ODEs, finite differences for PDEs
- PDEs: Green's function, separation of variables, Fourier, characteristics
- Probability and statistics: basic probability, estimation, hypothesis testing

## University of Illinois

- comprehensive requirements: based on five core courses; or getting A or A-
- prelim more similar to our comprehensive: oral, directed towards thesis research
- Computational Science and Engineering Option is stipend based on thesis topic and advisor with the CS&E program

## University of Missouri-Kansas City

**Comprehensive Examination** has two components, one written and one oral. The written component consists of 2 written examinations. The oral component consists of one oral examination.

*Written Exam* For Mathematics Emphasis the 2 written exams are based on two of the following three doctoral core courses

- Math 5519 (General Algebra II)
- Math 5523 (Real Variables II)
- Math 5542 (Advanced Numerical Analysis II).

For Statistics Emphasis the 2 written exams are based on two of the following three doctoral core courses

- Stat 5576 (Probability)
- Stat 5578 (Advanced Mathematical Statistics)
- Stat 5588 (Theory of General Linear Models)

The 2 written exams are given within two consecutive weeks in October and February of each year. "Passing" the written component of the Comprehensive Examination means passing both written exams. The passing score is 70% for each of the 2 written exams.

Each of the 2 written exams can be taken at most twice.

*Oral Exam* A one-hour oral examination will be given after a student passes both written exams making up the written comprehensive component. The purpose of the oral examination is for your committee to confirm that you have grasped the general knowledge of the mathematical sciences required for a successful IPhD candidate.

The oral examination covers fundamental concepts deemed to be necessary for a doctoral student to know, including but not limited to the problems that already appeared in either of the 2 written exams which were part of your written comprehensive component.

If you fail at the first attempt the oral examination, you may make a second attempt at a time agreed upon by your Ph.D. Supervisory Committee. If you fail at the second attempt the oral examination, your Ph.D. Supervisory Committee will vote whether you will be passed or failed for the entire "Comprehensive Examination" (written and oral components together). If the vote is "failing", you will be terminated from the Ph.D. program in Mathematics.

## University of Nebraska-Lincoln

**Course Requirements:** Each student must complete, with grades of B or better, at least three two-course sequences. At least one of the sequences must be from Group A below and at least one must be from
Group B below.

*Group A*

- Math 825-826 (analysis)
- Math 830-831 (differential equations)
- Math 842-843 (applied mathematics)

*Group B*

- Math 817-818 (algebra)
- Math 850-852 (discrete mathematics)
- Math 871-872 (topology)

**Qualifying Exam:** Each student must pass, at the qualifying (Q) level, two subject exams; together these two exams constitute the Ph.D. Qualifying Exam. Subject exams are offered in each of the six areas listed in part I above. One of the two exams must be either Math 817-818 (algebra) or Math 825-826 (analysis). The other exam may be chosen from any of the remaining five options.

**Comprehensive Exam:**The student's Ph.D. Supervisory Committee will determine the timing and the content of the Ph.D. Comprehensive Exam. In particular, the Comprehensive Exam must include a written portion and may, at the discretion of the Supervisory Committee, include an oral portion.

Typical Comprehensive Exams in the Mathematics Department consist of two parts: a four- hour written exam on a standard two-course sequence at the 900-level (often Algebra 901- 902 or Analysis 921-922) and a second exam, either oral or written, that is more specialized toward the student's research area. The decision as to whether the student has passed the Comprehensive Exam, and if not, which part(s) of the exam must be repeated, rests with the Supervisory Committee.

## University of Pennsylvania

- Ph.D. students take four courses each semester that they are on fellowship
- stipend of at least $21,630 for ten months, in addition to a full-tuition scholarship and free health insurance (for up to six years, when fully funded)
- Ph.D. students are expected in their first year to complete the requirements for the masters degree, and to achieve a distinguished pass on the Masters Preliminary Exam. (By doing this and by doing well in their classes, students satisfy the Qualifications Evaluation.) Ph.D. students are expected to take their oral Ph.D. Preliminary Exam by the end of the second year, unless given an extension by the Graduate Group Chair
- Masters Preliminary Exam single exam from
- Analysis, up to differential equations and multivariate; but not metric spaces
- Linear algebra, Abstract algebra

## Penn State: Mathematics

- Prelims must be passed within 18 months, offered twice a year
- Algebra: A. linear algebra (vector spaces, linear transformations), and either B. abstract algebra (theory of groups, rings) or C. applied algebra (numerical linear algebra)
- Analysis: A. real analysis (Lebesgue measure theory), and either B. complex analysis or C. applied analysis (functional analysis with applications to linear differential equations)

## University of Washington; Mathematics

**Preliminary Exams:** Pass three preliminary exams by September of the beginning of the third year. These exams, four hours in length, currently cover

- Algebra
- Real Analysis
- Complex Analysis
- Topology and Geometry of Manifolds
- Linear analysis

A student may substitute completion of a full three-quarter sequence of a designated core course, in which grades of 3.8 or above are received each quarter, for the passing of the corresponding preliminary exam. Only one such exam can be replaced in this manner.

**General Examination:** An oral examination on a special area of intended research, given by a committee after the student has passed the preliminary exams and the language exams. This exam can be given only after two years of graduate study. Normally, it should be taken by the middle of the student's fourth year. In addition, the student will prepare a written General Paper and distribute it to the committee at least two weeks before the date of the General Exam. The content of this paper is decided upon in consultation with the committee. For example, this might be a 10-20 page expository account of the student's research area, culminating in a problem or list of problems to be studied, together with a discussion of some of the relevant literature.

## University of Washington; Applied Mathematics

**Qualifying Examination:** In order to sit for the Qualifying Examination, students have to provide the Examination Committee (consisting of three members of the Applied Mathematics faculty) with a written summary of the research accomplished with their research advisor over the summer of their first year. This research summary is to be handed in on the first day of the Qualifying Examination.

The bulk of the Qualifying Examination consists of three exams of three hours each, spread over three days. The exams cover a core of undergraduate and graduate material necessary for successful completion of the Ph.D. program. These core areas are the material covered by:

- AMATH 567 Applied Analysis
- AMATH 568 Advanced Methods for Ordinary Differential Equations
- AMATH 569 Advanced Methods for Partial Differential Equations
- AMATH 581 Scientific Computing or AMATH 584 Applied Linear Algebra and Introductory Numerical Analysis
- AMATH 585 Numerical Analysis of Boundary Value Problems
- AMATH 586 Numerical Analysis of Time Dependent Problems

as well as knowledge from:

Calculus, ODE, Linear Algebra and Advanced Calculus.

**General Examination:** The general examination has three parts, and is generally taken by the end of the third year:

1. A written thesis proposal. The student is expected to outline a research topic which has the potential of leading to a PhD dissertation in applied mathematics. This written proposal should contain an outline of a thesis project, as well as an overview of the existing relevant literature. The written proposal should be made available to the supervisory committee one week before the general exam.

2. A thesis proposal (open to the public) consisting of a thirty minute (maximum) presentation on the content of the written proposal. During and after this public proposal the student will take questions from the general audience.

3. The thesis proposal is followed by a rigorous oral examination (not open to the public) where the supervisory committee verifies that the student is ready to undertake the proposed work.

## University of Wyoming

**Qualifying Exams** in 3 areas: (1) Algebra, Combinatorics & Number Theory; (2) Analysis; and (3) Applied & Computational Mathematics. The exams are offered at the beginning of the fall and spring semesters (twice a year) and are based on the relevant core courses.

- Doctoral students must pass 2 qualifying exams.

- Students may attempt a particular category of exam (algebra, analysis, applied) 2 times.

- Any failed attempts made during students' first two semesters do not count toward the tally. Successes made in the first two semesters count as passes; failures are not recorded.

- Students who have not attempted exams in their first two semesters must take at least one exam at the beginning of their third semester.

- Students must complete their qualifying exams by the end of their 6th semester.

**Preliminary Exam** consists of two parts: a written exam on two topics related to the student's proposed dissertation, and an oral exam. The oral exam is usually preceded by a public presentation in which the student describes the proposed dissertation and partial results.

Before taking their prelims, students must pass 2 qualifiers, attain proficiency in the professional tool prescribed by their committee, file a program of study, and complete at least 30 hours of coursework.