Graduate Study

Students must complete at least eight (8) graduate-level courses, at least two (2) with Honors grades.  Four (4) courses must meet core requirements listed below.  Courses counted toward the 8-course minimum must be full-credit graduate courses with clear focus that are related to Applied & Computational Mathematics in the judgment of the DGS and/or Research Advisor.  Courses such as “Special Investigation,” “Dissertation Research,” “Master’s thesis,” or “Seminar” do not count towards the 8-course requirement.  

Additionally, first year students will be required to complete one (1) course on the responsible conduct of research (i.e. MATH 991) and the Seminar in Applied Mathematics (AMTH 525).  These courses do not count toward the 8-course minimum requirement.

Students will take a mix of core courses and electives.  The detailed program of study for each student will be determined by the DGS and the Research Advisor (after the first year).  Students are expected to take 2-3 courses in each term of their first year and all coursework must be completed by the end of their second year.  See Appendix A for a sample pathway through the program.   

Core classes will be selected from the four broad areas listed below and students are required to take a minimum of one class from each area: 

1. Analysis (real, complex, and functional) 

2. Probability theory and statistics 

3. Computing (either numerical analysis OR algorithms and complexity theory) 

4. Application area (discrete mathematics, science/engineering, data science, ODEs/PDE) 

Elective courses may be chosen from the following areas: 

• Artificial Intelligence 

• Biomedical, Chemical, or Electrical Engineering 

• Computer Vision 

• Machine Learning 

• Materials Science 

• Mathematical Physics 

• Theoretical Computer Science and Discrete Mathematics 

Students are expected to engage in research beginning in the second semester of the first year. The first-year advisor will assist in identifying potential research opportunities for each student with an eye toward helping students select a research program for their dissertation research. 

Students are required to complete two semesters of teaching, typically in their second year, as a Teaching Fellow (TF-10).  Teaching fellow assignments will be coordinated in the summer after the first year of study by the DGS in coordination with the DUS.  

Students will select an advisor by the end of the first year in the program and, once selected and approved by the DGS, students will meet with the advisor to formalize the goals and expectations for each year of study in the program.

Co-advising is strongly encouraged.  Given the emphasis both mathematics and application, we encourage students to work with two faculty, one who has a more theoretical approach and one who is a domain expert.

The qualifying exam will test the depth and breadth of the student’s applied and computational mathematical knowledge and capacity for critical examination and inquiry, ensuring they are well suited to complete doctoral level research in Applied and Computational Mathematics.  The exam is meant to ensure students are prepared for the Ph.D. program, not as a means to filter students from the program. The qualifying exam should be taken in the fall semester of the second year and may be retaken once in the spring semester if needed. Students may be asked by their research advisor to take additional coursework in the spring semester of the second year if the exam reveals that their preparation is lacking.  If additional coursework is required beyond the 8-course minimum, the student will receive written notification from the DGS outlining the additional requirement(s).   

The qualifying exam will consist of either an oral presentation to the qualifying committee followed by questions from the committee or a series of written examinations on topics chosen by the advisor.  The content of the exam will be tailored to the research interests of the student,  emphasizing broad foundational knowledge in mathematics, applied and computational mathematics, and their area of interest, as well as specific topics and current developments, in the chosen area.  

Students will be expected to pass their qualifying exam by March 15th of their second year. If the student fails to pass the second time, they are normally considered no longer in good standing and asked to leave the program. They will be awarded a terminal Master’s degree if all course requirements have been satisfied. 

To be admitted to candidacy, students must submit a written research prospectus and pass an oral area examination. The purpose of this examination is to ensure the student is prepared to do independent research and has acquired the expected breadth and depth of knowledge.     

The thesis prospectus should summarize the first significant research project undertaken with the advisor and demonstrate familiarity with relevant literature.  Further, it should outline plans for the thesis research, contextualize the work, and provide a timeline for completion. The student is not formally committed to the plans outlined in the prospectus, but those plans should represent the student’s current thinking regarding their dissertation research plans. 

The corresponding oral presentation will be to the thesis committee.  At the discretion of the Research Advisor, the presentation may be public, and questions may be allowed by members of the audience. A closed session will follow which is restricted to questioning of the student by the committee members. The questioning focuses primarily on topics broadly related to the student’s prospectus, with the goal of determining if the student is prepared for the next stages of graduate study. 

A student who does not pass the Area Examination the first time is given a second opportunity to take it. The second exam must be taken and the results reported to the DGS and Graduate Registrar’s Office by March 15th of the third year. If the student fails to pass the second time, they are normally considered no longer in good standing and asked to leave the program. They will be awarded a terminal Master’s degree if all course requirements have been satisfied. 

At the latest, the student must be admitted to candidacy in the Graduate School by the end of the third year. A student is not permitted to register for a fourth year unless the Admission to Candidacy has been granted. 

Upon admission to candidacy, students must submit an annual written progress report by May 1 of each year.  Students are expected to meet regularly with their research advisor and are required to meet at least once annually with their thesis committee to discuss progress, plans for the upcoming academic year, and career development opportunities, which may include conference participation, networking opportunities, and summer internships.

The dissertation must report original research and demonstrate creative thought and scholarly achievement.  The exact dissertation components will be determined in conversation with the research advisor and DGS based on the student's research program. However, at minimum, the dissertation will consist of written output equivalent to three first-authored papers with an introduction and conclusion, representing a cohesive body of work. 

Upon submission of a final draft of the dissertation, the student will present their research publicly. The examination will be announced by the Graduate Registrar at least one week before it is held, and it is open to all who wish to attend. It consists of a summary by the student of their research activity, with questions by the Committee and members of the audience. The examination typically ends with a closed session for only the Committee and the student, during which the Committee can ask additional questions. 

Students must be in residence for at least three (3) years.  The residence requirement must normally be met within the first four (4) years of study.  Any exception to the residency requirement must e approved by the DGS, the department chair, and the appropriate associate dean in the GSAS.