# Mathematics Course Descriptions

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MATH 800 - Comprehensive Review of a Selected Topic in Mathematics**

The student will conduct an in-depth literature review of a selected area in Mathematics and prepare a report pertaining to the selected topic. The topic will be chosen in consultation with the supervisor and the Department Head. A final examinaton (written, oral or both) will be conducted by a committe of the Department.

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MATH 802 - Major Essay in Mathematics**

Essay on a selected topic for students in the course-based MSc program in Mathematics.

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MATH 803 - Approved Summer School**

This course is available to full-time Mathematics graduate students in good standing. Students will participate in a summer school offered by an approved institute. The school and credit award must be approved by the Graduate Coordinator for Mathematics and Statistics (or designee).
***Prerequisite: Approval of Department Head.***
*Note: Students may only take MATH 803 once in their program.*

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MATH 810 - Measure & Integration**

Integration and measure theory, spaces of continuous functions, and Lp spaces.

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MATH 812 - Complex Analysis**

Riemann mapping theorem, analytic continuation, Riemann surfaces.

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MATH 813 - Functional Analysis**

Banach spaces, Banach algebras, and operator theory.

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MATH 814 - Operator Algebras**

C*-algebras and von Neumann algebras.

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MATH 816 - Introduction to Quantum Information Theory**

A first course in the mathematics of quantum information theory. Topics include information measures, quantum states and observables, qubits, entanglement, quantum channels, entropy, and measurements.

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MATH 818 - Intro to Lie Algebras and Representation Theory**

The course is an introduction to the structure of finite dimensional complex semisimple Lie algebras, via root systems, as well as their finite dimensional irreducible representations, through highest weight modules.

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MATH 819 - Topics in Analysis**

Advanced study of selected areas of analysis.

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MATH 820 - Introduction to Commutative Algebra**

A first graduate course in commutative algebra. Topics include prime and maximal ideals, radicals, Nakayama's Lemma, exact sequences, tensor products, localization, Noetherian and Artinian rings and selected additional topics.

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MATH 821 - Number Theory**

Topics from analytic and algebraic number theory.

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MATH 822 - Linear Algebra**

Vector spaces, linear transformations and matrices, canonical forms, multilinear algebra.

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MATH 823 - Algebra**

Advanced study of group theory, Galois theory, and ring and module theory.

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MATH 824 - Topics in Algebra**

Advanced study of selected areas of algebra.

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MATH 825 - Matrix Analysis**

Matrix canonical forms, norms, spectral theory, perturbation theory, special classes of matrices.

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MATH 826 - Combinatorial Matrix Theory**

Amtrices arising from directed and undirected graphs, and related connections between matrix theory and combinational mathematics.

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MATH 827 - Graph Theory**

Advanced study of selected areas of graph theory.

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MATH 828 - Combinatorics**

Advanced study of selected areas of combinatorics.

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MATH 831 - Differential Geometry**

Differentiable manifolds, the tangent bundle, differential forms, and the general Stokes' theorem.

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MATH 832 - Topics in Differential Geometry and Topology**

Advanced study of selected areas of differential geometry and topology.

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MATH 837 - Intro to Algebraic Number Theory**

A course on rings of integers of algebraic number fields, Dedekind rings, and Ideal Class Groups.

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MATH 838 - Associative Algebras, Groups, and Representation Theory**

An introductory course on the fundamental results concerning associative algebras, groups, and the representation theory of groups and algebras.

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MATH 841 - General Topology**

Separability of spaces, paracompactness, metrization theorems, function spaces.

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MATH 842 - Algebraic Topology**

Introduction to homotopy groups, and to the homology and cohomology of topological spaces.

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MATH 843 - Homological Algebra**

A first graduate course in homological algebra. Topics include modules over rings, chain complexes, homology, projective and injective resolutions, derived functors, abelian categories, derived categories, and selected additional topics.

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MATH 849 - Topics in Topology**

Advanced study of selected areas of topology.

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MATH 869 - Numerical Analysis**

Advanced study of selected areas of numerical analysis.

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MATH 881 - Partial Differential Equations**

The Cauchy problem, the Fredholm alternaive in Banach space, the potential equation, the Dirichlet problem, the heat equation, Green's functions, and the separation of variables.

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MATH 882 - Topics in Applied Mathematics**

Advanced study of selected topics in applied mathematics.

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MATH 890AG - Topics in Combinatorics**

This course will include transitivity in graphs, eigenvalues of graphs, homomorphisms of graphs, and some results from extremal set theory, particularly the Erdos-Ko-Rado theorem that can be proven using algebraic graph theory.

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MATH 890AH - Topics in Complex Manifold Theory**

definition, examples incl. projective spaces, tori, type decompositions as (1, 0), (0, 1), holomorphic functions, holomorphic forms, sheaves, sheaf cohomology, Dolbeault cohomology, divisors, fiber bundles incl. line bundles and vector bundles, almost complex manifolds, Hermitian metrics, Kaehler metrics, connections

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MATH 890AI - Lie Groups**

This course is an introduction to the main fundamental results of Lie Group theory through an extensive study of the classical groups.

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MATH 890AK - Extremal combinatorics**

An introduction to extremal combinatorics and extremal set theory.

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MATH 890AL - Permutation Groups**

A course in the theory of permutation groups, with an emphasis on actions of finite permutation groups on combinatorial structures, such as graphs, designs and geometries.

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MATH 890AM - Topics In Analysis II**

Advanced study of selected areas of analysis and operator algebras.

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MATH 890AN - Advanced Topics in Functional Analysis**

Locally convex topologies, geometry of Banach spaces, bounded and
unbounded operators on Banach spaces, spectral theory.

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MATH 890AP - Quiver representations of algebras**

The algebra of a quiver; Auslander-Reiten quivers; classification of finite dimensional algebras and their representation theory in terms of quivers.

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MATH 890AQ - Matrix Analysis and Entrywise Positivity Preservers**

This course is an advanced course in matrix analysis and will concentrate on the topic of entrywise positivity preservers. Preservers are functions that operate on the individual entries of matrices and preserve the cone of positive semidefinite matrices.

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MATH 890AT - Design Theory**

This course will be an introduction to design theory. This course will include block designs, symmetric designs, Hadamard matrices and orthogonal arrays. We also study distance regular graphs, projective and affine space. We will look at focus on constructions and bounds of designs as well as connections to other areas of math.

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MATH 890AU - Combinatorial Association Scheme**

This class will be on Association Schemes with a combinatorial perspective. The course will look at specific association scheme arising in graph theory, such as distance regular graphs, strongly regular graphs and the Johnson scheme. Including a focus on the symmetric group and how it applies to Schurian association schemes.

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MATH 890AV - Continuum Mechanics**

Tensor analysis, fundamentals of continuum mechanics, Navier-Stokes equations.

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MATH 900 - Seminar**

Preparation and presentation of a one-hour lecture to graduate students and faculty.

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MATH 901 - Research**

Thesis research.

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MATH 902 - Research Tools in Mathematics**

This course teaches students about the computing and library resources available in the Mathematics and Statistics department. This course also includes an introduction to using LaTeX for preparing papers, writing research proposals, and giving academic presentations.

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MATH 903 - Comprehensive Exam 1**

Students must complete a comprehensive exam in one of the following topics: Matrix Theory and Linear Algebra, Commutative Algebra, Abstract Algebra, or Combinatorics and Graph Theory. It is evaluated on a pass/fail basis.

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MATH 904 - Comprehensive Exam 2**

Students must complete a comprehensive exam in one of the following topics: Topology, Algebraic Topology, Functional Analysis, Measure and Integration, Differential Geometry, or Probability Theory. It is evaluated on a pass/fail basis.

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MATH 905 - Research Proposal**

Students are required to submit a written research proposal for their PhD thesis research project during its early stages. The candidate will give a seminar before the department to defend their proposal. The topic must be approved by the research supervisor and the candidate's PhD committee. It is evaluated on a pass/fail basis. This course is required of all PhD students in Mathematics, and will usually be completed following the completion of MATH 903 and 904.