Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor. For more detailed information visit the Math 643R Wiki page.

Students should gain familiarity with a particular area of analysis selected by the instructor.

Examples of topics that may be offered include:

1. Metric number theory.
2. The theory of L-functions.
3. Operations research.