**6000 Level - Electives** |

**MTH 6310 - Commutative Rings and Modules** Prerequisite(s): MTH 5311.
Noetherian rings, quotient rings, primary decomposition, integral dependence
and valuations, Dedekind domains, and discrete valuation rings, completions,
dimension theory. |

**MTH 6311 - Non-Commutative Rings and Modules**
Prerequisite(s): MTH 6310. Semi-simple rings and modules, radicals, chain
conditions, decomposition of modules, Goldie's theorem, density and Morita
theory. |

**MTH 6312 - Abelian Group Theory** Prerequisite(s):
MTH 5311. An introduction to the fundamental theory of torsion,
torsion-free, and mixed abelian groups. |

**MTH 6322 - Approximation
Theory** Prerequisite(s): MTH 4322 and 4328. Approximation of
real functions including polynomial and rational interpolation, orthogonal
polynomials, Chebyshev approximation, the fast Fourier transform, splines,
wavelets, and tensor product interpolation. |

**MTH 6325 - Numerical
Solutions of Partial Differential Equations** Prerequisite(s): MTH
4322 and 4328. Finite difference and finite element methods for
elliptic, parabolic, and hyperbolic problems in partial differential
equations. |

**MTH 6340 - Compact Lie Groups** Prerequisite(s):
MTH 5310 and 5340. Compact Lie groups, Lie algebras, representation
theory, orthogonality relations, Peter Weyl theorem, structure theory,
roots, Weyl character formula. |

**MTH 6341 - Lie Algebras**
Prerequisite(s): MTH 5310 and 5316. Lie algebras, semisimple Lie
algebras, root systems, conjugecy theorems, classification theorem,
representation theory, Chevalley algebras. |

**MTH 6342 - Semisimple Lie
Groups** Prerequisite(s): MTH 6340 and 6341. Structure theory
for noncompact groups, induced representations, tempered representations,
Langland's classification of irreducible admissible representations. |

**MTH 6350 - Set and Model Theory** Prerequisite(s): MTH 5311.
Propositional and predicate calculus, Loewenheim-Skolem theorems, properties
of ultraproducts, model completeness, Goedel's completeness/incompleteness
proofs, infinitary language, axioms of set theory, ordinal and cardinal
arithmetic, models of set theory and large cardinals. |

**MTH 6V13 - Advanced Topics in Algebra **1 to 3 sem. hrs. Prerequisite(s):
Consent of instructor. May be repeated for credit with instructor's
consent if under different topic. Maximum 12 sem. hrs. |

**MTH 6V23 - Advanced Topics in Analysis** 1 to 3 sem. hrs. Prerequisite(s):
Consent of instructor. May be repeated for credit with instructor's
consent if under different topic. Maximum 12 sem. hrs. |

**MTH 6V24 - Advanced Topics in Applied Mathematics** 1 to 3 sem. hrs.
Prerequisite(s): Consent of instructor. May be repeated for credit with
instructor's consent if under different topic. Maximum 12 sem. hrs. |

**MTH 6V28 - Advanced Topics in Numerical Analysis** 1 to 3 sem. hrs.
Prerequisite(s): Consent of instructor. May be repeated for credit with
instructor's consent if under different topic. Maximum 12 sem. hrs. |

**MTH 6V30 - Advanced Topics in Topology**
Prerequisite(s): Consent of instructor. Topology is the study of abstract mathematical spaces with the ultimate goal of finding invariants which are preserved under continuous transformation. Along with algebra and analysis, topology is one of the main areas of modern mathematics and as such every doctoral program in mathematics should have a course designed to cover the more advanced aspects of topology. This course would be taken primarily by doctoral candidates with a strong interest in topology. |

**MTH 6V43 - Advanced Topics in Representation Theory** 1 to 3 sem. hrs.
Prerequisite(s): Consent of instructor. May be repeated for credit with
instructor's consent if under different topic. Maximum 12 sem. hrs. |