# Graduate Course Descriptions

**MATH 8806 Algebra I (Fall: 3)
MATH 8807 Algebra II (Spring: 3)**

The Math 8806-8807 sequence will cover the following topics: Group Theory (Group actions, Sylow, Nilpotent/Solvable, simple groups, Jordan-Holder series, presentations); commutative algebra (uniqueness of factorization, Jordan decomposition, Dedekind rings, class groups, local rings, Spec); finite fields; algebraic numbers; Galois theory; Homological algebra; Semisimple algebra.

**MATH 8808 Geometry/Topology I (Fall: 3)
MATH 8809 Geometry/Topology II (Spring: 3)**

The MATH 8808-8809 sequence will cover the following topics: Point-set topology, fundamental group and covering spaces, smooth manifolds, smooth maps, partitions of unity, tangent and general vector bundles, (co)homology, tensors, differential forms, integration and Stokes' theorem, de Rham cohomology.

**MATH 8810 Real Analysis (Fall: 3)**

Topics include: Measure Theory, Hilbert Space and Fourier Theory. Possible topics from: Lebesgue measure starting on R, convergence and Fubini theorems, generalizing to locally compact spaces and groups.

**MATH 8811 Complex Analysis (Spring: 3)**

Topics include: Local and global theory of analytic functions of one variable.

**MATH 8820 Introduction to Representation Theory (Fall: 3)
**Introduction of a broad range of representation theory, including representations of finite and compact Lie groups, and finite dimensional representations of complex semisimple Lie groups and Lie algebras, and quantum groups.

**MATH 8821 Number Theory I (Fall: 3)
MATH 8822 Number Theory II (Spring: 3)**

Possible topics include: Factorization of ideals, local fields, local-vs-global Galois theory, Brauer group, adèles and idèles, class field theory, Dirichlet L-functions, Chebotarev density theorem, class number formula, Tate's thesis.

**MATH 8831 Geometry/Topology III (Fall: 3)
MATH 8832 Geometry/Topology IV (Spring: 3)**

Possible topics include: differential geometry, hyperbolic geometry, three-dimensional manifolds, knot theory.

**MATH 8845 Topics in Algebra and Number Theory (Fall: 3)
**Selected topics in Algebra and Number Theory.

**MATH 8855 Topics in Geometry and Topology (Fall: 3)
**Selected topics in Geometry and Topology.

**MATH 8890 Graduate Teaching Seminar I**

This seminar is directed at building the teaching skills of beginning graduate students, who will typically be serving as Teaching Assistants. Topics include: use of blackboard, explaining problems, assigning points in grading exams and homework, classroom management. The seminar will include both practice and case-studies based discussion.

**MATH 8891 Graduate Teaching Seminar II**

This seminar is directed at building the teaching skills of advanced graduate students, who are teaching their own classes for the first time. Topics include: preparation of a syllabus, preparation of an examination, assigning course grades, lecture organization and preparation, lecture delivery, classroom management. The seminar will include both practice and case-studies based discussion.

**MATH 8892 Graduate Research Seminar**

This seminar is directed at building the research skills of graduate students.

**MATH 8899 Readings and Research (Offered by arrangement with individual faculty members)**

Department permission is required.

This is an independent study course, taken under the supervision of a Mathematics Department faculty member. Interested students should see the Assistant Chair for Graduates.