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AIM Ph.D. Coursework Requirements

Students in the AIM Ph.D. program take both mathematics courses and courses from other departments. In addition, to facilitate the choice of a partner discipline and to illustrate the breadth of interdisciplinary research, the program requires students to enroll in the AIM Student Seminar (Math 501) for the first three semesters.

Basic Requirements

The AIM Ph.D. degree requires a minimum of 11 regular courses (3 credits each) and 3 credits of the AIM Student Seminar (Math 501). Remaining coursework should be composed of Candidate Research (Math 995) or other approved graduate courses. The AIM coursework requirements exceed those of the regular mathematics Ph.D., but the needs of interdisciplinary research mandate a more extensive type of graduate training. The 11 regular courses should be composed of the following:

  • a total of at least seven mathematics courses, which must include at least four AIM core courses (see below) and at least an additional two math courses at or above the 600 level.
  • at least four graduate level courses from the chosen partner field, or another field outside of mathematics.

All courses must be at the graduate level (400 level and above), and with the exception of Math 501 must be graded courses completed with a grade of at least B, with an overall average of B+. The AIM Program Director will supervise coursework selection in the first two years. Coursework in subsequent years is normally supervised by the mathematics and partner discipline co-advisors.

AIM Core Courses

The following courses are fundamental to many areas of applied mathematics and related application areas. These courses provide the basic mathematical framework on which the AIM graduate experience is built.

  • Math 525: Probability Theory
  • Math 526: Stochastic Processes
  • Math 555: Complex Variables
  • Math 556: Methods of Applied Math I: Applied Functional Analysis
  • Math 557: Methods of Applied Math II: Asymptotic Analysis
  • Math 558: Applied Nonlinear Dynamics
  • Math 565: Combinatorics and Graph Theory
  • Math 566: Combinatorial Theory
  • Math 571: Numerical Methods for Scientific Computing I: Numerical Linear Algebra
  • Math 572: Numerical Methods for Scientific Computing II : Numerical Methods for Differential Equations
  • Math 656: Introduction to Partial Differential Equations
  • Math 657: Nonlinear Partial Differential Equations

Well prepared students may, with permission, substitute alternate mathematics courses for some AIM core courses.

The AIM Student Seminar Course (MATH 501)

The AIM Student Seminar course is a three-semester sequence taken by all AIM Ph.D. students (AIM M.S. students will join the course for the first two semesters of study). It is specifically designed to assist with many of the unique challenges confronting AIM graduate students. For Ph.D. students, one of these challenges is the choice of a dissertation committee that includes two different co-advisors, one from mathematics and one from another partner discipline. Another challenge common to the interests of both M.S. and Ph.D. students is the development of a sound understanding of the way that mathematics plays a role in diverse application areas.

Math 501 is a course only for AIM graduate students. It is coordinated with the departmental Applied and Interdisciplinary Mathematics Research Seminar, and students are required to participate in the research seminar on a weekly basis as a course requirement. Students will therefore meet once each week during the regular course time and again during the time of the research seminar. Currently these two meetings are scheduled for 12-1 PM and 3-4 PM respectively on Fridays. Attendance of the AIM research seminar as part of Math 501 is vital to sound interdisciplinary training. In this way, students will be exposed to a wide range of research topics.

The specific content of the Math 501 course as delivered during the regular course meetings is highly variable, but the purpose of the course is clear: to address specific issues related to the process of successfully completing a graduate degree in the AIM program and becoming an active member of the research community. The weekly meetings of the class are generally partitioned among three types of sessions:

  1. “Focus on. . ." presentations: These are short presentations on various subjects pertinent to the pursuit of an AIM degree as well as other aspects of professional development. In prior semesters some of the presentations have been:
    • Focus on . . . how to find your advisor and co-advisor
    • Focus on . . . how to use the library at the University of Michigan
    • Focus on . . . applying for summer research programs
    • Focus on . . . using LATEX for scholarly writing
       
  2. AIM Faculty Portraits: These are short presentations by faculty members from mathematics and other departments, providing a direct channel for students to discover what research is being done in various application areas, and to see what kind of preparation is required for participating in such research. Any faculty member giving an AIM Faculty Portrait is a potential thesis co-advisor for AIM Ph.D. students.

  3. AIM Research Seminar “Warm-up talks": These are presentations by particularly dynamic speakers slated to speak in the AIM Research Seminar (or surrogates thereof) as a way to provide background material with the goal of making the AIM Research Seminar lecture more valuable for the AIM students. The warm-up talks will (i) present the background to the research to be discussed at a more advanced level in the subsequent Research Seminar talk, (ii) put the work in context and discuss the importance of the results, and (iii) generally provide an introduction to topic of the Research Seminar. This enables students to derive greater benefit from the AIM Research Seminar series and to gain meaningful exposure to a broad range of problems.

Suggested Mathematics and Partner Discipline Courses

Here we list some courses that might be appropriate for AIM graduate students. Reading courses may also be arranged between students and willing faculty from mathematics or other departments. These can provide opportunities for personalized instruction going beyond what is available in regular courses, and can count for course credit.

Example Course Tracks

Here are some sample course tracks in various application areas:

Research in Electrical Engineering (Image Processing)

  Fall Semester Winter Semester
Year 1
  • MATH 501 (AIM Student Seminar)
  • MATH 556 (Methods of Applied Mathematics I)
  • MATH 561 (Linear Programming I)
  • MATH 571 Numerical Scientific Computing I)
  • MATH 501 (AIM Student Seminar)
  • MATH 525 (Probability)
  • MATH 557 (Methods of Applied Mathematics II)
  • MATH 563 (Advanced Mathematical Biology)
Year 2
  • MATH 501 (AIM Student Seminar)
  • IOE 612 (Network Flows)
  • MATH 558 (Ordinary Differential Equations)
  • MATH 651 (Topics in Applied Mathematics I)
  • EECS 586 (Algorithms)
  • IOE 611 (Nonlinear Programming)
  • MATH 526 (Discrete Stochastic Processes)
Year 3
  • EECS 587 (Parallel Computing)
  • MATH 656 (Introduction to Partial Differential Equations)
  • PHYSICS 406 (Statistical and Thermal Physics)

Research in Mathematical Biology (Cancer Modeling)

  Fall Semester Winter Semester
Year 1
  • MATH 501 (AIM Student Seminar)
  • MATH 463 (Mathematical Modeling in Biology)
  • MATH 525 (Probability)
  • MATH 555 (Complex Variables)
  • MATH 501 (AIM Student Seminar)
  • MATH 513 (Introduction to Linear Algebra)
  • MATH 558 (Ordinary Differential Equations)
  • MATH 563 (Advanced Mathematical Biology)
Year 2
  • MATH 501 (AIM Student Seminar)
  • EPID 511 (Introduction to Public Health Genetics)
  • MATH 556 (Methods of Applied Mathematics I)
  • MICROBIOL 553 (Cancer Biology)
  • MATH 567 (Introduction to Coding Theory)
  • MATH 572 (Numerical Scientific Computing II)
Year 3
  • EHS 513 (Pathology and the Basis of Disease)
  • MATH 656 (Introduction to Partial Differential Equations)
  • MATH 671 (Numerical Methods I)
  • MICROBIOL 440 (Immunology)

Research in Computer Science (Compressive Sampling)

  Fall Semester Winter Semester
Year 1
  • MATH 501 (AIM Student Seminar)
  • MATH 525 (Probability)
  • MATH 555 (Complex Variables)
  • MATH 591 (General and Differential Topology)
  • MATH 501 (AIM Student Seminar)
  • MATH 526 (Discrete Stochastic Processes)
  • MATH 566 (Combinatorial Theory)
  • MATH 567 (Introduction to Coding Theory)
Year 2
  • MATH 501 (AIM Student Seminar)
  • ECON 617 (Game Theory)
  • EECS 574 (Computational Complexity)
  • MATH 556 (Methods of Applied Mathematics I)
  • EECS 684 (Current Topics in Databases)
  • MATH 597 (Real Analysis and Measure Theory)
  • STATS 701 (Topics in Appl. Statistics II - Large Data Sets)
Year 3
  • MATH 547 (Probabilistic Modeling in Bioinformatics)
  • MATH 654 (Introduction to Fluid Mechanics)
  • MATH 650 (Fourier Analysis)
  • STATS 711 (Topics in Theoretical Statistics II)

Research in Fluid Mechanics (Geophysical Fluid Flow)

  Fall Semester Winter Semester
Year 1
  • MATH 501 (AIM Student Seminar)
  • MATH 525 (Probability)
  • MATH 602 (Real Analysis II / Functional Analysis)
  • PHYSICS 401 (Mechanics)
  • MATH 501 (AIM Student Seminar)
  • MATH 557 (Methods of Applied Mathematics II)
  • MATH 572 (Numerical Scientific Computing II)
  • MATH 710 (Modern Analysis II)
Year 2
  • MATH 501 (AIM Student Seminar)
  • AOSS 420 (Environmental and Ocean Dynamics)
  • MATH 558 (Ordinary Differential Equations)
  • MATH 654 (Introduction to Fluid Mechanics)
  • MATH 656 (Introduction to Partial Differential Equations)
  • AOSS 605 (Current Topics / Physical Climate)
  • MATH 526 (Discrete Stochastic Processes)
  • PHYSICS 406 (Statistical and Thermal Physics)`
Year 3
  • AOSS 605 (Current Topics / Wave-Mean Flow Interaction)
  • MATH 756 (Advanced Partial Differential Equations)
  • AOSS 422 (Boundary Layer Methods)
  • GEOSCI 500 (Computer Programming for Geoscience)

Research in Mathematical Finance

Note: This example illustrates the possibility of an exceptionally well-prepared student substituting a higher level course (MATH 626) for a corresponding AIM Core course (in this case, Math 526).

  Fall Semester Winter Semester
Year 1
  • MATH 501 (AIM Student Seminar)
  • MATH 525 (Probability)
  • MATH 565 (Combinatorics and Graph Theory)
  • STATS 620 (Applied Probability and Stochastic Modeling)
  • MATH 501 (AIM Student Seminar)
  • MATH 572 (Numerical Scientific Computing II)
  • MATH 650 (Fourier Analysis)
  • STATS 625 (Statistics of Random Processes)
Year 2
  • MATH 501 (AIM Student Seminar)
  • FIN 580 (Options, Futures, and Corporate Decision Making)
  • MATH 656 (Introduction to Partial Differential Equations)
  • EECS 498 (Special Topics / C++ Programming)
  • IOE 552 (Financial Engineering I)
  • MATH 625 (Probability and Random Processes I)
Year 3
  • ECON 407 (Finance Econometrics)
  • IOE 553 (Financial Engineering II)
  • STATS 610 (Statistical Inference)
  • MATH 623 (Computational Finance)
  • MATH 626 (Probability and Random Processes II)