Suggested Basic Courses
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Graduate students normally take four basic courses each semester in the first year. Many students will take further basic courses in later years. The basic courses have two distinguishing features:
(1) They do not have pre-requisites, except that the second semester of a two-semester sequence requires the first semester.
(2) They have regular homework and tests. Students do a significant amount of written work and get feedback on it.
The basic courses, listed below, are designed to help prepare students for research, and to provide a foundation on which to build a broad understanding of mathematics. An asterisk indicating that the course is being offered in fall, 2004. Syllabi are found at the end of the doctoral regulations.
601*, 602: Algebra I, II
603*, 604: Real Analysis I, II
605*, 606: Complex Analysis I, II
607*, 608: Topology I, II
609*, 610: Logic I, II
611: Nonlinear Dynamical Systems
612: Discrete Mathematics
613*: Optimization
614: Basic PDE (Applied Analysis)
617*: Numerical Analysis I
618: Numerical Methods in PDE (requires Numerical Analysis I)
625*: Differentiable Manifolds
633*: Nonlinear Analysis
637: Linear Control
643: Probability
644: Stochastic Analysis (requires Probability)
Many students consider several different research areas before settling on one, and it is important to choose basic courses wisely. Below several faculty members indicate which basic courses are most important for their research area, which should be taken in the first year, and which could be postponed.
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Katrina Barron:
601*, 602: Algebra I, II
603*, 604: Analysis I, II
605*, 606: Complex Analysis I, II
607*, 608: Topology I, II
625*: Differentiable Manifolds
Richard Hind:
601*, 602: Algebra I, II
603*, 604: Analysis I, II
605*, 606: Complex Analysis I, II
607*, 608: Topology I, II
625*: Differentiable Manifolds
Julia Knight: My students should take logic (609-610) and algebra (601-602) in the first year, plus one of the analysis sequences (603-604, or 605-606). Either in the first year or later, they should take topology (607-608), and at least one semester of the other analysis sequence. I think it would also be good to sample some applied courses, such as numerical analysis (617), probability (643), or discrete mathematics (612).
Juan Migliore: The basic courses are listed below, with an asterisk beside those being offered this fall. The syllabi are given on our web site, and in the current version of our doctoral regulations.
601*, 602: Algebra I, II --------> important first year
603*, 604: Analysis I, II
605*, 606: Complex Analysis I, II ---------> important first year
607*, 608: Topology I, II ------------> important first year
609*, 610: Logic I, II
611: Nonlinear Dynamical Systems
612: Discrete Mathematics ----------> useful, can wait til second year
613*: Optimization
614: Basic PDE (Applied Analysis)
617*: Numerical Analysis I -----> useful, can wait til second year or later
618: Numerical Methods in PDE (requires Numerical Analysis I)
625*: Differentiable Manifolds
633*: Nonlinear Analysis
637: Linear Control
643: Probability
644: Stochastic Analysis (requires Probability)
Israel Michael Sigal:
For the first year
633*: Nonlinear Analysis
614: Basic PDE (Applied Analysis)
603*, 604: Analysis I, II
For later:
605*, 606: Complex Analysis I, II
607*, 608: Topology I, II
611: Nonlinear Dynamical Systems
617*: Numerical Analysis I
618: Numerical Methods in PDE (requires Numerical Analysis I)
625*: Differentiable Manifolds
637: Linear Control
643: Probability
644: Stochastic Analysis (requires Probability)
Andrew Sommese: I want my students to take most first year courses before they graduate. Absolutely essential in the first year are: Numerical Analysis (one semester) followed by Algebra (two semester). If possible also: probability (one semester) followed by complex analysis (two semesters). In the second semester, I would like them to take discrete mathematics if it is offered. To the extent possible Optimization, Numer. Anal. II, Prob. II, and logic: these can be replaced by many courses if a student has a strong interest in the other courses.
In terms of numbers
Ideal:
1st Semester:
Math 617 followed by Math 601, Math 643, Math 605 in that order
2nd semester
Math 602, 612, 606, 618, 644 in that order
Next best group:
Math 617, 613, Math 609, 610
Nancy Stanton: The most important basic courses for a graduate student who wants to work with me are:
601, 602: Algebra I, II
603, 604: Analysis I, II
605, 606: Complex Analysis I, II
607, 608: Topology I, II
625: Differentiable Manifolds
and also possibly 614: Basic PDE (Applied Analysis)
Of these, the four two-semester sequences should be taken in the first year.
Bruce Williams:
601*, 602: Algebra I, II
603*, 604: Analysis I, II
605*, 606: Complex Analysis I, II
607*, 608: Topology I, II
625*: Differentiable Manifolds
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Other faculty members are encouraged to send their suggestions to Judy Hygema (hygema.1@nd.edu)
