# Postgraduate Courses

- MAED 5021Mathematical Analysis and Its Applications I[3-0-0:3]DescriptionThis course is an introduction to analysis and its applications. Countable and uncountable sets, metrics, norms, open sets, closed sets, continuity, homeomorphisms, connectedness, complete metric spaces, totally bounded sets, fixed points, compact metric spaces, uniform continuity, category, the Baire Category Theorem, sequences of functions, pointwise and uniform convergence, the space of bounded functions.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Analyze if a given set is countable or uncountable, construct bijections between two sets, and analyze the cardinality of the Cantor set.
- 2.Analyze if a function defined on a set is a metric or a norm, analyze if a sequence in a metric or norm space converges, determine whether or not two metrics or two norms are equivalent, analyze if a metric space is of the second category, and apply the Baire Category Theorem in different contexts.
- 3.Analyze if a given subset of a metric space is open or closed, compact, connected, totally bounded, and construct the relative metric.
- 4.Analyze if a given mapping from a metric space into another metric space is continuous and uniformly continuous by using different continuity characterizations, and determine if a given map is a homeomorphism.
- 5.Analyze if a metric space is complete by using nested set theorem and Balzono-Weierstrass Theorem.
- 6.Apply fixed point theorem in the context of differential equations.

- MAED 5031Mathematical Analysis and Its Applications II[3-0-0:3]Prerequisite(s)MAED 5021DescriptionThis course is a continuation of MAED 5021. It covers the space of continuous functions, the Stone-Weierstrass Theorem, functions of bounded variation, the Riemann-Stieltjes Integrals, Fourier Series, Lebesgue measure, measurable functions, Lebesgue integrals, Convergence in measure, Lp spaces, differentiation.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Apply the Weierstrass Theorem to different approximation problems, and analyze if a subset of C(X) is equicontinuous, compact or/and dense in C(X).
- 2.Construct continuous linear functionals on C([a,b]) by using functions of bounded variation, and apply the Riesz Representation Theorem.
- 3.Construct the Riemann-Stieltjes integrals.
- 4.Analyze some elementary properties of Fourier Series.
- 5.Apply Lebesgue Measure Theory to analyze a subset in R is measurable, construct the Lebesgue integrals on R and analyze if a function is absolutely continuous on R.
- 6.Apply Lebesgue Differentiation Theory in R.

- MAED 5032Applications of Geometry and Analysis[3-0-0:3]Prerequisite(s)MAED 5021DescriptionThis course will show how mathematical analysis can be used to investigate a variety of physical phenomena. Kepler's laws of planetary motion will be derived from Newton's law gravitation. Further topics will be the use of differential geometry, in particular Gaussian curvature, in describing space curves and surfaces. Additional topics will be other historically important uses of differential equations, Fourier series and harmonic analysis.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Formulate the quantities that describe space curves.
- 2.Present standard examples of curves that are interesting in high school education.
- 3.Formulate a curve as the motion of a point particle.
- 4.Formulate the quantities that describe surfaces.
- 5.Present standard examples of surfaces that are interesting in high school education.
- 6.Formulate arc-length in the language of Riemannian metrics.
- 7.Formulate the use of normal vectors and curvature in the description of surfaces.
- 8.Apply the theory of curves, and surfaces to physical problems.

- MAED 5111Classical and Abstract Algebra[3-0-0:3]DescriptionThis is a course which develops the rigorous definition of real and complex numbers, and various aspects of polynomials.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Use Peano axioms to define natural numbers and their arithmetic rigorously.
- 2.Use the method of Dedekind cuts to rigorously define the field of real numbers.
- 3.Define complex numbers and apply them to algebraic and geometric problems.
- 4.Describe the connection between polynomials and geometric objects defined by them.
- 5.Work with various special polynomials and show a variety of their applications.

- MAED 5121Algebra and Its Applications I[3-0-0:3]DescriptionThis course reviews the notion of groups, rings and homomorphisms and quotient constructions. Applications are then described for finite fields and number theory, and coding theory.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Formulate precisely the mathematical axioms which a group (and ring) must satisfy.
- 2.Derive certain more advanced properties of groups and rings from the axioms.
- 3.Identify when a group or ring occurs in a physical or mathematical situation, and infer useful information based on properties satisfied by groups and rings.
- 4.Apply algebraic concepts to organize and solve problems.
- 5.Explain how abstract algebra is used to solve certain classical questions of geometry and algebra.
- 6.Explain how abstract algebra is used to prove certain number theory results, and how these results are used for encryption purposes in modern day communications.

- MAED 5131Algebra and Its Applications II[3-0-0:3]Prerequisite(s)MAED 5121DescriptionThis course develops group theory and presents applications. Among the applications covered are the use of group theory to show the impossibility of expressing the roots of a general polynomial in terms of radicals, and the use of finite fields in random number generation, construction of orthogonal Latin squares, and the finite Fourier transform. Other applications are connections of group theory with Rubik’s Cube and uses of Lie groups.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Work abstractly with the mathematical objects called groups and rings.
- 2.Identify when a group occurs in a physical or a mathematical setting, and then exploit group and ring theory to better analyze the situation.
- 3.Use normal and quotient groups in the setting of Galois theory to explain the impossibility of expressing the roots of a general polynomial in terms of radical.
- 4.Apply groups to investigate problems of symmetry: for example, enumeration of sets by Polya counting, applications to Rubik's cube, and continuous motions.

- MAED 5211Classical and Modern Geometry[3-0-0:3]DescriptionThis course is about important ideas in the development of geometry. Topics include logic and axiomatic system, projective geometry, Hilbert’s axioms, hyperbolic models, geometric transformations, hyperbolic trigonometry.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Explain the axiomatic nature of Euclid’s Elements and its shortcomings and show how Hilbert's axioms provide a rigorous foundation for Euclidean geometry.
- 2.Formulate the parallel postulate and explain its role in the development of hyperbolic geometry.
- 3.Use geometric transformations to characterize various geometries.
- 4.Use hyperbolic trigonometry to study various properties of hyperbolic geometry.

- MAED 5321Combinatorics[3-0-0:3]DescriptionTopics in combinatorics including counting combinations and counting permutations, generating functions, and enumerative combinatorics. Inclusion-exclusion principle, Polya’s theory of counting, graph theory and applications.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Explain the reasons behind combinatorial principles and applications.
- 2.Design combinatorial problems using the principles taught in class.
- 3.Present projects related to certain theorems or problems in combinatorics.
- 4.Construct problems to show the usefulness of the principles for teaching purposes.

- MAED 5421Probability and Statistics[3-0-0:3]DescriptionThis course illustrates how to use statistical and probabilistic techniques with an open source statistics package R to analyze data, especially in education.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Identify the core ideas on analyzing data.
- 2.Communicate effectively the statistical results to both lay and expert audiences utilizing appropriate information and suitable technology.
- 3.Apply rigorous, analytic and highly numerate methods to analyze and solve problems in daily life and at work.
- 4.Carry out objective analysis and prediction of quantitative information with independent judgement.

- MAED 5731Problem Solving Strategies[3-0-0:3]DescriptionThis course covers ways to teach strategies and techniques to students to solve mathematical problems as they appeared in mathematical competitions. It covers many techniques, including transformation methods for solving geometry problems, and graph theoretical methods for solving combinatorial problems.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Recognize various techniques to solve certain types of math problems.
- 2.Present solutions and compare the strength and weakness of the techniques.
- 3.Modify problems to create new problems.
- 4.Identify conditions in solving problems to investigate further applications.

- MAED 5821Topics in Mathematics[3-0-0:3]DescriptionThis course introduces a selection of topics from interdisciplinary mathematics, which may include mathematical physics, mathematical biology, image processing, etc.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Apply theoretical knowledge, principles and techniques to mathematical physics, mathematical biology and traffic flows.
- 2.Recognize roles of mathematics in solving some mathematical physics, mathematical biology and traffic flows.
- 3.Apply various theorems to study ideal mathematical models, such as ideal 2-dimensional fluid flows, ideal electrostatic force fields, steady state temperature, etc.
- 4.Present solutions of some physical problems in a systematic way.

- MAED 5851Scientific Computation[3-0-0:3]DescriptionThis course introduces various case studies drawn from different areas of science to illustrate the use of computers as a problem-solving tool. Each integrates physical principles and mathematical models, as well as numerical techniques and computer implementations, into a coherent perspective.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Apply theoretical knowledge, principles and techniques to physical problems.
- 2.Recognize roles of mathematics in scientific computation.
- 3.Implement various numerical algorithms on computers and apply them to real life problems.
- 4.Present numerical output from a computer code in a systematic way.

- MAED 6980MSc Project[2 credits]DescriptionIn this course, the student will do a project in an area of mathematics under the guidance of a faculty member. The project either surveys a topic or describes an investigation completed by the student.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Explain advanced mathematical theories, concepts and principles using precise mathematical language.
- 2.Apply independent judgment to investigative mathematical work.
- 3.Apply a rigorous logical and analytic approach to execute tasks and solve mathematical problems.
- 4.Communicate mathematical concepts and methods effectively to a range of audiences, both orally and in writing.

- MAED 6981Independent Study[3 credits]DescriptionIndependent study of selected mathematical topics under the supervision of a faculty member. The course may be repeated for credit if different topics are studied. Consent of instructor is required.Intended Learning Outcomes
On successful completion of the course, students will be able to:

- 1.Explain and apply mathematical knowledge of the selected topics of current interest which may not be covered by existing courses.
- 2.Apply independent judgment to investigative mathematical work.
- 3.Present findings and solutions in a systematic way.