Beginning graduate students are advised by the Director of Graduate Studies (see contacts). Usually in the second year students acquire a major professor who does all subsequent advising. The responsibility rests with the student to contact a prospective major professor. The advisor approves course selections, monitors progress, and generally oversees the student's program of study. Satisfactory progress is usually measured by a grade of at least B in all courses.

While supported by a teaching assistantship (e.g., running two hours per week of discussion sections of calculus), students take core courses in algebra, analysis, and topology, providing the foundation for all further graduate work.

The following describes a core program commonly taken Ph.D. students in mathematics during the first year:

*MATH 7820: Differential Topology*

Topological spaces and continuous functions; product and quotient topologies; compactness and connectedness; separation and metrization.

*MATH 7340: Complex Analysis I*

Fundamental theorems of analytic function theory.

*MATH 7751: Algebra I*

Detailed study of groups, rings, fields, modules, and multilinear algebra.

*MATH 7000: Seminar on College Teaching*

Discussion of issues related to the practice of teaching, pedagogical concerns in college level mathematics, and aspects of the responsibilities of a professional mathematician.

*MATH 7310: Real Analysis and Linear Spaces I*

Introduction to measure and integration theory.

*MATH 7752: Algebra II*

Further topics in groups, rings, fields, and multilinear algebra.

*MATH 7800: Algebraic Topology I*

The fundamental group and covering spaces, Van Kampen theorem, and applications to group theory. Simplicial, cellular, and singular homology; Eilenberg-Steenrod axioms; categories and functors.

*MATH 7010: Seminar on Research in Mathematics*

This seminar discusses the issues related to research in Mathematics.

Students who pass general exams upon arrival can take more advanced courses in the first year.

In the second year and beyond, students choose from more specialized courses. Ph.D. students past the third year are mainly involved in seminars and independent research. In seminars, students have the opportunity to lecture on published work or their own research, gaining experience in exposition of advanced mathematical topics. For descriptions of the remaining graduate courses, see the Graduate School's catalogue.

Teaching responsibility usually involves four contact hours per week, typically running one discussion hour and meeting one's own class of a 1000-level math course for three hours. Students continue to take a range of basic courses, but chosen with potential areas of specialization in mind. General Examinations preferably are passed either just before this year starts, or as soon as possible thereafter, and the Second-Year Proficiency Examination is taken at the end of the second academic year.

*Third academic year:* Students should be integrating themselves into the research life of the department through advanced courses and participation in seminars. Often students are doing independent reading toward acquiring the specialized background needed for doing research and, guided by a potential thesis advisor, making the transition into dissertation research.

*Fourth academic year and beyond:* Students should be reading the literature related to a thesis topic, and making progress on original research. (We view five or six years as the normal time needed to complete graduate work.)