Course Descriptions

CREDIT HOURS: 3

This course is to be offered by and completed at an AARMS Summer School hosted at an Atlantic University. To register you must have permission from the Graduate Coordinator.

CROSS-LISTING: MATH 4001

CREDIT HOURS: 3

This course is to be offered by and completed at an AARMS Summer School hosted at an Atlantic University. To register you must have permission from the Graduate Coordinator.

CROSS-LISTING: MATH 4002

CREDIT HOURS: 3

Lebesgue's theory of measure and integration. The topics include: sigma-algebras, measures, construction of measures, Lebesgue measure on the real line, measurable functions, the Lebesgue integral and convergence theorems, Lp spaces, signed and complex measures, decomposition of measures and the Lebesgue-Radon-Nikodym theorem, product measures and the Fubini-Tonelli theorem.

EXCLUSIONS: MATH 4010.03

CREDIT HOURS: 3

Topics include: review of analytic complex functions including topological properties of the plane, Mobius mappings, exponential, logarithmic, trigonometric and related functions, integration and the Cauchy theorem. Cauchy's integral formula, residues, harmonic functions, analytic continuation, entire and meromorphic functions, some results of conformal mapping, including the Riemann mapping theorem.

CROSS-LISTING: MATH 4020.03

CREDIT HOURS: 3

This introduction to commutative algebra includes a selection of the following topics: prime and maximal ideals, primary decomposition, Noetherian rings, Hilbert's Basis Theorem and the Nullstellensatz.

CROSS-LISTING: MATH 4025.03

CREDIT HOURS: 3

This course covers special topics in Commutative Algebra.

CREDIT HOURS: 3

Introduction to module theory: modules, submodules, quotient modules, module homomorphisms, generators for modules, direct sums, free modules, tensor products, exact sequences, projective modules, injective modules and flat modules. Modules over principal ideal domains. Additional topics may include homological algebra, Ext and Tor functors, symmetric and exterior algebras.

EXCLUSIONS: MATH 4045.03

CREDIT HOURS: 3

Field theory, field extensions, Galois theory and applications.

CROSS-LISTING: MATH 4055.03

CREDIT HOURS: 3

This is a first course in algebraic geometry and will introduce students to the basic properties of affine and projective varieties. Topics covered will include a selection from: local properties of plane curves, elliptic curves, Bezoutâ€™s Theorem, Riemann-Roch Theorem.

CROSS-LISTING: MATH 4065.03

CREDIT HOURS: 3

This course, together with STAT 5067.03 provides a solid basis in the theory of statistical inference. After a review of some probability and distribution theory, the Bayesian and classical theories of estimation and testing are introduced.

NOTES: Please see course description for STAT 5066 for more details.

CROSS-LISTING: STAT 5066.03.

EXCLUSIONS: MATH 4066.03, STAT 4066.03/

CREDIT HOURS: 3

An introduction to algebraic number theory, with special emphasis on quadratic and cyclotomic fields. A more general study of rings of integers of algebraic number fields focuses on divisibility properties. Other topics include Dedekind domains, ideals and their factorization into prime ideals, and class groups and class numbers.

CROSS-LISTING: MATH 4070.03

CREDIT HOURS: 3

Categories, functors, natural transformations and adjointness are introduced with emphasis on examples drawn from undergraduate Mathematics and theoretical Computer Science. The calculus of diagram chasing, limits, colimits and Kan extensions is explored in detail.

CROSS-LISTING: MATH 4135.03

CREDIT HOURS: 3

Topics of current interest in category theory will be discussed with an emphasis on open problems. No previous knowledge of category theory is required. The necessary concepts will be discussed in the context of their applications. However, a certain familiarity with the basic concepts of modern mathematics such as found in courses on algebra and topology would be an asset.

CROSS-LISTING: MATH 4136.03

CREDIT HOURS: 3

An introduction to the basic principles of functional analysis including the following topics: infinite dimensional vector spaces, normed spaces, inner-product spaces, Banach and Hilbert spaces, linear and continuous linear functionals, the Hahn-Banach Theorem, the principle of uniform boundedness, dual spaces, weak* topology, and the Alaoglu theorem, the open mapping and closed graph theorems, and consequences and applications.

CROSS-LISTING: MATH 4140.03

CREDIT HOURS: 3

Complex variables and applications including solutions to Laplace equation, ideal fluid flow and Joukowski airfoil. Fourier series and generalizations; separation of variables; completeness. Green's functions in one and two dimensions. Asymptotic evaluation of integrals and special functions. Plus some additional topics in mathematical physics.

CROSS-LISTING: PHYS 5160.03

EXCLUSIONS: MATH 4165.03, PHYS 4160.03

CREDIT HOURS: 3

An introduction to topological spaces that includes the following topics: classification in terms of cardinality of bases, separation, etc., product spaces, Tychonoff theorem, compactness, compactifications, Tychonoff spaces, metrization.

CROSS-LISTING: MATH 4170.03

CREDIT HOURS: 3

An introduction to algebraic topology including the following topics: the definitions, properties and methods of computation of the fundamental group of a topological space; simplicial, singular and cellular homology groups; basic properties and methods of computation of homology groups; a selection of application such as the classification of surfaces and fixed point theorems.

CROSS-LISTING: MATH 4180.03

CREDIT HOURS: 3

A graduate-level introduction to ordinary differential equations. Topics covered include flows, existence and uniqueness theorems, continuity of solutions, coordinate transformations, symmetry methods and reductions, linearization of dynamical systems, and ODEs on manifolds.

CROSS-LISTING: MATH 4190.03

CREDIT HOURS: 3

Qualitative theory is concerned with determining the behaviour of solutions of differential equations without finding explicit solutions. Topics are selected from Liapunov stability theory, stable and unstable manifolds of singular points and periodic solutions, classification of plane singular points, structural stability and Hamiltonian systems. Other topics at the instructorâ€™s discretion.

EXCLUSIONS: MATH 4200.03

CREDIT HOURS: 3

This course is a basic introduction to the theory of partial differential equations. Topics covered include: modelling physical systems, method of characteristics, Laplace, wave and heat equations, separation of variables, eigenfunction expansions, integral transforms, maximum principles and Ritz Raleigh theory.

CROSS-LISTING: MATH 4220.03

**MATH 5001 AARMS Summer Course I**CREDIT HOURS: 3

This course is to be offered by and completed at an AARMS Summer School hosted at an Atlantic University. To register you must have permission from the Graduate Coordinator.

CROSS-LISTING: MATH 4001

**MATH 5002 AARMS Summer Course II**CREDIT HOURS: 3

This course is to be offered by and completed at an AARMS Summer School hosted at an Atlantic University. To register you must have permission from the Graduate Coordinator.

CROSS-LISTING: MATH 4002

**MATH 5010 Introduction to Measure Theory and Integration**CREDIT HOURS: 3

Lebesgue's theory of measure and integration. The topics include: sigma-algebras, measures, construction of measures, Lebesgue measure on the real line, measurable functions, the Lebesgue integral and convergence theorems, Lp spaces, signed and complex measures, decomposition of measures and the Lebesgue-Radon-Nikodym theorem, product measures and the Fubini-Tonelli theorem.

EXCLUSIONS: MATH 4010.03

**MATH 5020 Analytic Function Theory**CREDIT HOURS: 3

Topics include: review of analytic complex functions including topological properties of the plane, Mobius mappings, exponential, logarithmic, trigonometric and related functions, integration and the Cauchy theorem. Cauchy's integral formula, residues, harmonic functions, analytic continuation, entire and meromorphic functions, some results of conformal mapping, including the Riemann mapping theorem.

CROSS-LISTING: MATH 4020.03

**MATH 5025 Commutative Algebra**CREDIT HOURS: 3

This introduction to commutative algebra includes a selection of the following topics: prime and maximal ideals, primary decomposition, Noetherian rings, Hilbert's Basis Theorem and the Nullstellensatz.

CROSS-LISTING: MATH 4025.03

**MATH 5035 Topics in Commutative Algebra**CREDIT HOURS: 3

This course covers special topics in Commutative Algebra.

**MATH 5045 Advanced Algebra I**CREDIT HOURS: 3

Introduction to module theory: modules, submodules, quotient modules, module homomorphisms, generators for modules, direct sums, free modules, tensor products, exact sequences, projective modules, injective modules and flat modules. Modules over principal ideal domains. Additional topics may include homological algebra, Ext and Tor functors, symmetric and exterior algebras.

EXCLUSIONS: MATH 4045.03

**MATH 5055 Advanced Algebra II**CREDIT HOURS: 3

Field theory, field extensions, Galois theory and applications.

CROSS-LISTING: MATH 4055.03

**MATH 5065 Algebraic Geometry**CREDIT HOURS: 3

This is a first course in algebraic geometry and will introduce students to the basic properties of affine and projective varieties. Topics covered will include a selection from: local properties of plane curves, elliptic curves, Bezoutâ€™s Theorem, Riemann-Roch Theorem.

CROSS-LISTING: MATH 4065.03

**MATH 5066 Advanced Statistical Theory I**CREDIT HOURS: 3

This course, together with STAT 5067.03 provides a solid basis in the theory of statistical inference. After a review of some probability and distribution theory, the Bayesian and classical theories of estimation and testing are introduced.

NOTES: Please see course description for STAT 5066 for more details.

CROSS-LISTING: STAT 5066.03.

EXCLUSIONS: MATH 4066.03, STAT 4066.03/

**MATH 5070 Algebraic Number Theory**CREDIT HOURS: 3

An introduction to algebraic number theory, with special emphasis on quadratic and cyclotomic fields. A more general study of rings of integers of algebraic number fields focuses on divisibility properties. Other topics include Dedekind domains, ideals and their factorization into prime ideals, and class groups and class numbers.

CROSS-LISTING: MATH 4070.03

**MATH 5135 Introduction to Category Theory**CREDIT HOURS: 3

Categories, functors, natural transformations and adjointness are introduced with emphasis on examples drawn from undergraduate Mathematics and theoretical Computer Science. The calculus of diagram chasing, limits, colimits and Kan extensions is explored in detail.

CROSS-LISTING: MATH 4135.03

**MATH 5136 Topics in Category Theory**CREDIT HOURS: 3

Topics of current interest in category theory will be discussed with an emphasis on open problems. No previous knowledge of category theory is required. The necessary concepts will be discussed in the context of their applications. However, a certain familiarity with the basic concepts of modern mathematics such as found in courses on algebra and topology would be an asset.

CROSS-LISTING: MATH 4136.03

**MATH 5140 Introduction to Functional Analysis**CREDIT HOURS: 3

An introduction to the basic principles of functional analysis including the following topics: infinite dimensional vector spaces, normed spaces, inner-product spaces, Banach and Hilbert spaces, linear and continuous linear functionals, the Hahn-Banach Theorem, the principle of uniform boundedness, dual spaces, weak* topology, and the Alaoglu theorem, the open mapping and closed graph theorems, and consequences and applications.

CROSS-LISTING: MATH 4140.03

**MATH 5165 Mathematical Methods in Physics**CREDIT HOURS: 3

Complex variables and applications including solutions to Laplace equation, ideal fluid flow and Joukowski airfoil. Fourier series and generalizations; separation of variables; completeness. Green's functions in one and two dimensions. Asymptotic evaluation of integrals and special functions. Plus some additional topics in mathematical physics.

CROSS-LISTING: PHYS 5160.03

EXCLUSIONS: MATH 4165.03, PHYS 4160.03

**MATH 5170 General Topology**CREDIT HOURS: 3

An introduction to topological spaces that includes the following topics: classification in terms of cardinality of bases, separation, etc., product spaces, Tychonoff theorem, compactness, compactifications, Tychonoff spaces, metrization.

CROSS-LISTING: MATH 4170.03

**MATH 5180 Introduction to Algebraic Topology**CREDIT HOURS: 3

An introduction to algebraic topology including the following topics: the definitions, properties and methods of computation of the fundamental group of a topological space; simplicial, singular and cellular homology groups; basic properties and methods of computation of homology groups; a selection of application such as the classification of surfaces and fixed point theorems.

CROSS-LISTING: MATH 4180.03

**MATH 5190 Ordinary Differential Equations**CREDIT HOURS: 3

A graduate-level introduction to ordinary differential equations. Topics covered include flows, existence and uniqueness theorems, continuity of solutions, coordinate transformations, symmetry methods and reductions, linearization of dynamical systems, and ODEs on manifolds.

CROSS-LISTING: MATH 4190.03

**MATH 5200 Ordinary Differential Equations - Qualitative Theory**CREDIT HOURS: 3

Qualitative theory is concerned with determining the behaviour of solutions of differential equations without finding explicit solutions. Topics are selected from Liapunov stability theory, stable and unstable manifolds of singular points and periodic solutions, classification of plane singular points, structural stability and Hamiltonian systems. Other topics at the instructorâ€™s discretion.

EXCLUSIONS: MATH 4200.03

**MATH 5220 Introduction to Partial Differential Equations**CREDIT HOURS: 3

This course is a basic introduction to the theory of partial differential equations. Topics covered include: modelling physical systems, method of characteristics, Laplace, wave and heat equations, separation of variables, eigenfunction expansions, integral transforms, maximum principles and Ritz Raleigh theory.

CROSS-LISTING: MATH 4220.03