BEGINNING GRADUATE STUDENTS are advised by the Graduate Advisor. 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.

The following describes a core program commonly taken by prospective M.S. or Ph.D. students in mathematics during the first year:

*MATH 7820: Differential Topology (recommended) or MATH 5770: General 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 with advanced preparation or specialized interests may, with permission of the graduate advisor, construct a suitably modified program. Students needing additional preparation are advised to take other courses, such as a 5000-level analysis or algebra sequence, before taking the 7000-level courses:

*MATH 5310: Introduction to Real Analysis*

Basic topology of Euclidean spaces, continuity and differentiation of functions on a single variable, Riemann-Stieltjes integration, and convergence of sequences and series.

*MATH 5330: Advanced Multivariate Calculus*

Differential and integral calculus in Euclidean spaces, implicit and inverse function theorems, differential forms and Stokes' theorem.

*MATH 5651: Advanced Linear Algebra*

This course includes a systematic review of the material usually considered in MATH 3351 such as matrices, determinants, systems of linear equations, vector spaces, and linear operators. However, these concepts will be developed over general fields and more theoretical aspects will be emphasized. The centerpiece of the course is the theory of canonical forms, including the Jordan canonical form and the rational canonical form. Another important topic is general bilinear forms on vector spaces. Time permitting, some applications of linear algebra in differential equations, probability, etc. are considered.

*MATH 5652: Introduction to Abstract Algebra*

Focuses on structural properties of basic algebraic systems such as groups, rings and fields. A special emphasis is made on polynomials in one and several variables, including irreducible polynomials, unique factorization and symmetric polynomials. Time permitting, such topics as group representations or algebras over a field may be included.

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.

Our standard Ph.D. program for a typical student is roughly as follows:

*First academic year:* 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.

*Second academic year:* 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.)