# Ph.D. Coursework

### First-year Coursework

The typical schedule for a first-year Ph.D. student is as follows:

Fall Semester | Spring Semester |
---|---|

Real Analysis | Complex Analysis |

Algebra I | Algebra II |

Geometry-Topology I | Geometry-Topology II |

Teaching Seminar | Research Seminar |

#### Real Analysis

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.

#### Complex Analysis

Local and global theory of analytic functions of one variable.

#### Algebra I and II

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 algebras.

#### Geometry-Topology I and II

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.

### Advanced Coursework

#### Number Theory I and II

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

#### Geometry-Topology III and IV

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

### Topics Courses

Advanced topics in number theory/representation theory and geometry/topology.

#### Research Seminar

The research seminar is an opportunity for students to present their own research or give lectures on advanced topics. Participation in the research seminar is encouraged by the department. A student may be required by their advisor to participate and/or speak in the research seminar.