Applied Mathematics

The department offers programs leading to the Master of Science and the Doctor of Philosophy degrees. At the master’s level, students may pursue degrees in mathematics, applied mathematics, or statistics. At the doctoral level, students may pursue degrees in mathematics or applied mathematics.

A student must satisfy all of the general requirements of the graduate school as well as the more specific requirements of the department to earn either a master’s or doctoral degree. Each graduate student is assigned a faculty advisory committee during the first year of study. The committee’s primary responsibility is to help the student plan an appropriate and sufficiently broad program of course work and study that will satisfy both the degree requirements and the special interests of the student. With the aid of the advisory committee, each student must present a study plan indicating how he or she intends to satisfy the requirements for a graduate degree.

The doctorate is conferred not merely upon completion of a stipulated course of study, but rather upon clear demonstration of scholarly attainment and capability of original research work in mathematics. A doctoral student may plan either a traditional program of studies in mathematics (mathematics track) or a program of studies oriented toward applied mathematics (applied mathematics track). In either case, each student must take 36 credit hours of approved courses with a grade average of B or better. For students entering with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, this requirement is reduced to 18 credit hours of approved courses.

Qualifying Exams

Each student will be required to take two written qualifying exams. The exams will be in analysis and algebra for the mathematics track, and in numerical analysis and modeling for the applied mathematics track. Syllabi for the exams are available to students. Exams will be offered twice a year, usually in January and May. Students may attempt each exam up to two times. Under normal circumstances, students are expected to have passed both exams by the end of their fifth semester.

Area Exam

Each student will be required to pass an oral examination showing knowledge of the background and literature in the chosen area of specialization. The exam will be administered by the student’s advising committee, chaired by the principal advisor. The exam should normally take place within one year after final passage of the qualifying examinations and at least one year before the defense takes place. A student may retake the required exam once.

A written syllabus, with a list of the papers for which the student will be responsible, should be prepared and agreed upon by the student and advising committee at least two months before the exam takes place, at which time a specific date and time for the exam should be decided. Both the syllabus and the scheduled date of the exam should then be reported to the graduate committee. Once the syllabus and exam date have been reported to the graduate committee, the student will advance to PhD candidacy.

Yearly Progress Reports

After passing the area exam, students will present yearly progress reports to their advising committees, usually in April. These reports will consist of both a written summary of progress and an oral presentation delivered to the advising committee.

Dissertation, Expository Talk, and Defense

Students are required to produce a written dissertation and present an oral defense. The dissertation is expected to constitute an original contribution to mathematical knowledge. It must be provided to the defense committee (the composition of which is discussed below) at least 10 days prior to the defense. Students are required to give a colloquium-level presentation of their thesis work, open to all students and faculty, followed by an oral defense of the thesis work to the defense committee. The committee consists of at least four faculty members, including the student’s principal advisor and at least one outside faculty member.

Deadlines for the thesis defense and approval of the dissertation are determined by the School of Graduate Studies. It is the student’s responsibility to be aware of deadlines and make sure they are met.

A student in the applied mathematics track must demonstrate knowledge of scientific computing, mathematical modeling, and differential equations. This includes taking qualifying examinations in the areas of computational mathematics and mathematical modeling, and taking certain courses in these three areas, as specified below. 

Qualifying Examination

A doctoral student in the applied mathematics track must take written examinations in numerical analysis and in mathematical modeling, as well as an oral examination in his or her chosen area of specialization. Subjects include but are not restricted to fluid mechanics, statistical mechanics, epidemiology, neuroscience, and more traditional fields of mathematics.

A student with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, must take 18 credit hours of approved courses, which must include at least 6 credit hours of courses offered outside the Department of Mathematics, Applied Mathematics, and Statistics and at least 9 credit hours offered by the Department of Mathematics, Applied Mathematics, and Statistics. The graduate committee will determine which of the specific course requirements stated above have been satisfied by the master’s course work.

Sample study plans for students with concentrations in scientific computing, imaging, mathematical biology, and stochastics follow. The graduate committee will entertain ideas for other serious study plans or qualifying exam subjects in addition to the most common variants.

Scientific Computing Concentration

Imaging Concentration

Life Science Concentration

Stochastics Concentration

PhD students entering with a bachelor’s degree are subject to the same breadth requirements as students pursuing the MS degree in applied mathematics.

Petitions

Any exceptions to departmental regulations or requirements must have the formal approval of the department's graduate committee. Such exceptions are to be sought by a written petition, approved by the student’s advisory committee or thesis advisor, to the graduate committee.

Any exception to university rules and regulations must be approved by the dean of graduate studies. Such exceptions are to be sought by presenting a written petition to the graduate committee for departmental endorsement and approval prior to forwarding the petition to the dean.

Requirements

• Applicants must have a good academic record, e.g., a B-average or rank in the upper third of his or her graduating class at an institution whose status and programs are readily assessed.
• Applicants must meet all of the undergraduate prerequisites for the proposed field of graduate study.
• Statement of Objectives — Some programs have this built into the application. If not, create your own using "Statement of Objectives" as the heading. The statement should be one to two pages and include your purpose in undertaking graduate work as well as an explanation of your study and research interests as they relate to your undergraduate/graduate study and professional goals.
• Academic Transcripts – You are required to identify all post-secondary educational institutions you have attended on your application form, and to submit an official transcript from each. Admission to the School of Graduate Studies is conditional on final certification of degrees awarded. Note to International Students: Documents in languages other than English must be accompanied by certified translations in English.
• Letters of Recommendation - Three letters of recommendation must be submitted from those individuals sufficiently familiar with you, and who can assess your academic preparation, abilities and accomplishments. Letters should be from your most recent instructors in your proposed field of study. If you have been out of school for several years, one letter may be from your employer, supervisor or other person familiar with your most recent activities.
• Test Scores – Your department will identify required standardized tests. When you fill out forms for these use 1105 as the institutional code for Case Western Reserve University. The School of Graduate Studies will forward all official score reports of standardized tests (GRE, TOEFL, etc.) to the program of interest. The department will inform you which standardized tests are required for your program. International students whose first language is not English must demonstrate English proficiency by taking the Test of English as a Foreign Language (TOEFL) and earn a minimum score of 577 if paper-based, or 90 if Internet-based. The International English Language Testing System (IELTS) is also accepted. The minimum acceptable score is 7.0. Some departments may require higher scores. International applicants cannot be admitted without receipt of an acceptable official TOEFL score. The TOEFL is valid for two years after the test date. The GRE is valid for five years after the test date.
• Application Fee - A nonrefundable application fee of $50 is required for each application submitted.

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