### Level – 3M

#### Course units effective from academic year 2016/2017 to date

 Course Code MMT301M3 Course Title Advanced Algebra I Credit Value 03 Hourly Breakdown Theory Practical Independen Independent Learning 45 — 105 Objectives: Introduce the elements of commutative algebra through a study of commutative rings and modules over such rings Intended Learning Outcomes: Prove factorization of homomorphisms, Correspondence theorem, the classical isomorphism theorems and their consequences Define certain special types of ideals Discuss various aspects of certain special types of ideals, by means of proofs and examples Prove module isomorphism theorems and other results related to submodules, quotient modules, direct sum and direct product of modules Prove certain results related to finitely generated modules, notably Nakayama’s lemma. Recall various results regarding exact sequences Define Unique Factorization Domains (U.F.D), Principal Ideal Domains(P.I.D) and Euclidean Domains (E.D)  Discuss various results regarding U.F.D, P.I.D and E.D, by  means of proofs and examples Explain the processes of constructing the rings and modules of fractions  Prove certain results related to primary decomposition, notably 1st and 2nd uniqueness theorems ·Prove certain theorems regarding Noetherian and Artinian modules  Discuss various results regarding Noetherian and Artin rings, in particular the Hilbert basis theorem Contents: Ring homomorphisms, factorization theorems, isomorphism theorems , nil and nilpotent ideals, prime and maximal ideals, primary ideals, nilradical and Jacobson radicals, operations on ideals, integral domains, principal ideal domain(P.I.D), unique factorization domain(U.F.D), Euclidean domain (E.D), modules, module homomorphisms, submodules and quotient modules, module isomorphism theorems, direct sum and direct product of modules, finitely generated modules, exact sequences, rings and modules of fractions, localization, primary decomposition , chain conditions, Noetherian rings, Artin rings. Teaching Methods: Lectures,  Tutorials,  Problem solving, Use of e-resources and Handouts Assessment/ Evaluation Details: In-course Assessments 30% End-of-course Examination 70% Recommended Readings: Atiyah, M.G. and MacDonald, G., AnIntroduction to Commutative Algebra, Addison-Wesley,1969 Dummit, D. S. and Foote, R. M., Abstract Algebra, Wiley, 2006 Sharp, R. Y., Steps in commutative Algebra, Cambridge University Press, 2001 Rotman, J.J., Advanced Modern Algebra, JAMS, 2010.
MMT302M2: Topology I
 Course Code MMT302M2 Course Title Topology I Credit Value 02 Hourly Breakdown Theory Practical Independen   Independent Learning 30 – 70 Objectives: Provide an introduction to the general topology through a selection of topics Intended Learning Outcomes: Define certain notions associated with the topological spaces Recall the proofs of certain facts related to bases, subbases , subspaces and quotient spaces Prove certain results concerned with continuous functions and homeomorphisms Compare the product and box topologies Prove various results regarding convergence in the topological spaces Prove the Urysohn’s lemma, Urysohn’smetrization theorem, and the Tietze extension theorem Reproduce certain fundamental results regarding the compact spaces Discuss various aspects of connected, path connected and locally connected spaces Course Contents: Topological Spaces and Continuous functions: Definition of topology, bases and subbases, closed sets and limit points, subspaces, continuous functions,  homeomorphisms, the product topology, the weak topology, quotient spaces, Convergence: Sequences, nets, ultranets filters, ultra filters. Compactness and Connectedness: Compact spaces, finite intersection property, limit point compactness, compactness in the real line, connected spaces, connectedness in the real line, path connectedness, components and local connectedness The Separation Axioms and Countability Axioms: space, space,  Hausdorff space, regular space, completely regular space, Tychnoff space, normal space, perfectly normal space, completely normal space,  the countability axioms, Urysohn’s Lemma,Urysohn’smetrizationtheorem, the Tietze Extension theorem. Teaching Methods: Lectures, Tutorials, Problem solving, Use of e-resources and Handouts. Assessment/ Evaluation Details: In-course Assessments 30% End-of-course Examination 70% Recommended Readings: James R. Munkres, Topology, 2nd Edition, Prentice-Hall, 2000. James Dugundji, Topology, University Book stall, 1972. Stephen Willard, General Topology, Dove publications, 2004. C.Wayne Patty, Foundations of Topology, PWS-Kent publishing Company, 2009.
MMT303M2: Functional Analysis I
 Course Code MMT303M2 Course Title Functional Analysis I Credit Value 02 Prerequisites PMM201G3, PMM203G3 Hourly Breakdown Theory Practical IndependentLearning 30 —– 70 Objectives: Introduce the fundamental concepts of normed linear space and Banach space Provide a clear notion of concepts of linear functional and dual space Understand the concepts of inner product space and Hilbert space Intended Learning Outcomes: Explain the fundamental concepts of normed linear space and Banach space Discuss equivalence of norms and its properties Examine the convergence and absolute convergence of a series in Banach space Recall the concepts of a bounded linear functional and its properties Analyze the properties and applications of dual spaces Prove Hahn Banach theorems in real and complex normed linear spaces Define inner product space and Hilbert space Prove Riesz  representation theorem, Bessel’s inequality and Parseval identity Make use of separability of a Hilbert space for the existence of orthonormal basis Course Contents: Normed linear spaces and Banach spaces: Normed linear spaces, Equivalent norms, Riesz lemma, Completeness and Banach spaces, Convergence and absolutely convergence of series in Banach space , Schauder basis, Separable  normed linear spaces. Linear functionals and Dual spaces: Linear operators and linear functionals, Bounded linear functional, Isometry, Isomorphism, Dual spaces and its applications, Hahn- Banach theorems Inner product spaces and Hilbert spaces: Inner products, Hilbert spaces,  Closed subspaces and orthogonal projection, Riesz Representation theorem, Orthonormal system and othonormalization, Bessel’s inequality and Parseval identity, Existence   of orthonormal basis in a separable Hilbert space Teaching Methods: Lectures,  Tutorial discussion,  use of e-resources and Handouts Assessment/ Evaluation Details: In-course assessment                                              30% End of course Examination                                    70% Recommended Readings: John B. Conway, A Course in Functional Analysis, Springer, 1997 Bryan P. Rynne , Martin A. Youngson, Linear Functional Analysis, Springer, 2008 Walter Rudin, Functional Analysis, McGraw – Hill, 1991 Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley1989 FrigyesRiesz and BelaSz.-Nagy, Functional Analysis, Dover Publications, 1990
MMT304M3: Numerical Linear Algebra
 Course Code MMT304M3 Course Title Numerical Linear Algebra Credit Value 03 Hourly Breakdown Theory Practical IndependentLearning 45 — 105 Objectives: Understand the numerical methods for solving large systems of linear equations Recognize the underlying mathematical concepts of computer aided numerical algorithms Understand the iterative methods for computing eigenvalues of large matrices Intended Learning Outcomes: Outline the fundamental concepts in numerical linear algebra Apply the matrix factorization algorithms to solve system of linear equations Determine bounds for relative error in the solution of a system of linear equations Examine the convergence of iterative methods for solving system of linear equations Apply  iterative methods to solve a system of linear equations Examine the convergence of iterative methods for computing the eigenvalues of matrix Apply  iterative methods to compute eigenvalues of a matrix Apply the Grahm-Schmidt orthogonalization process to a matrix Solve the linear systems by using readily available software Course Contents: Direct Methods: Linear algebra Review, Elementary triangular matrices and Gauss elimination, Elementary permutation matrices and pivoting, Elementary Hermitian Matrices and matrix factorization, iterative refinement Matrix Analysis: Canonical forms and positive definite matrices, Vector and Matrix norms, Spectral radius, Condition of problems and scaling Norm ReducingMethods: Iterative methods and error bounds, convergence results for special matrices, choice of relaxation parameter, sparce matrix technique, Conjugate gradient method Similarity Reduction methods:House holders’ method, Eigensystems of Hessenberg matrices and tridiagonal matrices, Jacobi method, Given’s method, LR method and QR method Power Methods:  Direct power method, Raleigh quotients, Deflation process, Shift of the origin, Inverse iteration Teaching Methods: Lectures,  Tutorials,  Problem solving, Use of e-resources and Handouts Assessment/ Evaluation Details: In-course Assessments           30% End-of-course Examination 70% Recommended Readings: Trefethen, N. andBau, D., Numerical Linear Algebra, SIAM, 1997. Golub, G. and Charles, V. L., Matrix Computations, John Hopkins University Press, 1996. Beilina, L., Karchevskii, E. and Karchevskii, M., Numerical Linear Algebra: Theory and Applications, Springer, 2017.
MMT305M3: Mathematical Modeling and Programming
 Course Code MMT305M3 Course Title Mathematical Modeling and Programming Credit Value 03 Hourly Breakdown Theory Practical Independent Learning 45 — 105 Objectives: Provide knowledge and skills to build mathematical models of real-world problems, analyze them and make predictions about behavior of problems taken from physics, biology, chemistry, economics and other fields. Enable to investigate and apply standard mathematical programming problems. Intended Learning Outcomes: Formulate  mathematical models for solving given word problems Sketch the qualitative solution of the formulated model problems involving Differential equations Modify  simple models for the change of environment Solve single species population models Discuss interacting two species population models Solve deterministic programming problems using dynamic programming algorithm Apply solution methods of unconstrained nonlinear extremum problems Make use of Lagrangian MultipliersmethodandKarush-Kuhn-Tucker(KKT) conditions to locate local minimizers ApplyWolfe’s algorithm for solving quadratic programming problems Apply separable programming algorithm to solve nonlinear programming problems Course Contents: Fundamentals of modelling and word problems:Basic approach of Mathematical Modeling, its needs, types of models and limitations, setting up mathematical models for simple problems in words. Qualitative solution sketching for first order differential equations: Direction field, solution sketch, convexity phase portrait, equilibrium solutions, stability. Population models for Single and interacting species: Basic concepts, Exponential growth model, formulation, solution, interpretation and limitations, Compensation and dispensation, Logistic growth model, formulation, solution, interpretation and limitations.  Types of interaction between two species. Lotka-Volterra prey-predator model, formulation, solution, interpretation and limitations. Lotka-Volterra model of two competing species, formulation, solution, interpretation and limitations. Dynamic programming models: Stage, State, Recursive equation, Developing optimal decision policy, Bellman’s principal of optimality, characteristics of deterministic dynamic programming, solving linear programming problem using dynamic programming approach. Nonlinear programming fundamentals: Fundamentals of optimization, types of problems, Optimality conditions for unconstrained and constrained problems, Lagrangian Multipliers method and Karush-Kuhn-Tucker conditions. Quadratic and Separable programmings: Quadratic programming, Wolfe’s algorithm, Wolfe’s modified simplex method, separable programming, separable function, piece wise linear approximation of separable nonlinear programming problem, separable programming algorithm. Teaching Methods: Lectures,  Tutorials, Problem solving, Use of e-resources and Handouts Assessment/ Evaluation Details: In-course Assessments 30% End-of-course Examination 70% Recommended Readings: Winston, W. L., Introduction of Mathematical Programming Applications and Algorithms, Duxbury press, California, 1995. Hillier, F. S. and Lieberman, G.J., Introduction to Operations Research, 7th edition, McGrawHill, New York, 2001. Taha, H. A., Operations Research an Introduction, 8th edition, Pearson Prentice Hall, New Jersey, 2007. Braun, M. ,Coleman, C. S. and Drew,D.A., Vol. 1, Vol. 2 and Vol.3 – Differential Equation Models,Springer-Verlag, New York, 1983. Meyer,W.,  Concepts of Mathematical Modeling, McGraw Hill, New York, 1994.
MMT306M3: Number Theory and Combinatorics
 Course Code MMT306M3 Course Title Number Theory and Combinatorics Academic Credits 03 Hourly Breakdown Theory Practical Independent Learning 45 — 105 Objectives : Provide in depth knowledge of classical number theory through a study of selection of topics. Develop an understanding of the core ideas and concepts of fundamental and advanced counting techniques Recognize the power of abstraction and generalization, and apply logical reasoning to investigate some combinatorial problems Intended Learning Outcomes : Recall certain theorems in the theory of congruences Relate various number theoretic functions through  Möbius inversion formula Discuss various properties of Euler  function Prove various results regarding primitive roots of prime numbers and composite numbers Recall Euler’s criterion and its consequences Discuss various properties of Legendre symbol, notably quadratic reciprocity law Solve various quadratic congruences Solve certain non-linear Diophantine equations Prove various results regarding certain numbers of special form Solve common counting problems using elementary counting techniques involving the multiplication rule, permutations, and combinations Apply the Principle of inclusion and exclusion to solve variety of counting problems Apply the recurrence relations to model a wide variety of counting problems and solve them Make use of generating functions to solve many type of counting problems subject to variety of constraints, and solve recurrence relations Identify the best counting technique  for given counting problem Contents: Number Theory: Congruences, The Chinese remainder theorem, Wilson’s theorem, Fermat’s little theorem, Euler φ function, Euler’s theorem, Number theoretic functions, Möbius inversion formula, Primitive roots, Quadratic residues, Euler’s Criterion, Quadratic reciprocity, Quadratic Congruences, Legendre symbol, Perfect numbers, Mersenne primes and Amicable numbers, Fermat’s numbers, Certain nonlinear Diophantine equations Basic Counting Techniques: Fundamental principles of counting:Sum, product, subtraction, and division rules. Permutations and  combinations, the Binomial Theorem, Pascal’s identity and Triangle, Vandermonde’s Identity, permutations and combinations with limited and unlimited repetition, multinomial theorem and its applications, generating permutations and combinations,The Pigeonhole Principle and some of its interesting applications. Advanced Counting Techniques: Inclusion-Exclusion: The principle of inclusion-exclusion, an alternative form of inclusion-exclusion, the Sieve of Eratosthenes, derangementsRecurrence Relations:Modeling with recurrence relations,solving linear homogeneous recurrence relations with constant coefficients, linear non-homogeneous recurrence relations with constant coefficients, divide-and-conquer algorithms and recurrence relations.Generating Functions: Extended binomial theorem, counting problems and generating Functions, using generating functions to solve recurrence relations. Exponential generating functions and its applications on solving some permutation problems. Teaching Methods : Lectures,  demonstration, tutorial discussion, problem solving, use of e-resources and handouts Assessment / Evaluation Details : In course Assessments 30% End of course Examination  70% Recommended Readings : Kenneth H. Rosen, Discrete Mathematics and its Applications, 7th edition, McGraw Hill, 2012. Bernard Kolman, Robert C. Busby, S. C. Ross, Discrete Mathematical Structures, 4th edition, Prentice Hall, 2001 Grimaldi R.P, Discrete and combinatorial mathematics: an applied introduction, 5th edition, Pearson education Inc, 2004 Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill (2007).
MMT307M2: Topology II
 Course Code MMT307M2 Course Title Topology II Credit Value 02 Hourly Breakdown Theory Practical Independen   Independent Learning 30 — 70 Objectives: Provide an in depth understanding of compactness in the topological spaces Introduce the basic concepts of algebraic topology through a selection of topics Intended Learning Outcomes: Discuss the processes of one point and  Stone–Cechcompactifications Examine the relationship between various types of compactness in the matric space Prove the Tychonoff theorem Compare the topology of pointwise convergence, topology of compact convergence and compact-open topology in a function space Prove various  results concerned with homotopicity of maps Identify covering maps and covering spaces for certain topological spaces Recall certain results and their proofs about fundamental groups. Identifythe fundamental group of Prove the  Fundamental theorem of Algebra and theBorsuk-Ulam theorem Prove certain facts of the singular homology theory Classify various types of knots Course Contents: Compactness: compactness in metric spaces, sequential compactness, Heine-Borel Property, total boundedness, complete metric spaces, the TychonoffTheorem,local compactness, one point and Stone-Cechcompactification, pointwise and compact convergence, compact-open topology The Fundamental group and Covering Spaces: homotopy of paths, the fundamental group, covering spaces, fundamental group of the circle, the fundamental theorem of algebra, antipodes, the Borsuk-Ulam theorem, retractions and fixed points, deformation  retracts and homotopy type Homology Theory:  standard – simplex, boundary operator, – cycles, homology groups, induced homomorphism. Introduction to Knots: types of knots, preliminary results. Teaching Methods: Lectures, Tutorials, Problem solving, Use of e-resources and Handouts. Assessment/ Evaluation Details: In-course Assessments 30% End-of-course Examination 70% Recommended Readings: James R. Munkres, Topology, 2nd Edition, Prentice-Hall, 2000.  B.K.Lahiri, A First course in Algebraic Topology, Narosa, 2000. C.Kosniowski,  A First course in Algebraic Topology, Cambridge University press, Cambridge , 2008 A.Hatcher, Algebraic topology, Cambridge University press, Cambridge, 2002. SashoKalajdzievski,  Topology and Homotopy,  CRC press, 2015.

### Level – 4M

#### Course units effective from academic year 2016/2017 to date

MMT401M4: Measure Theory
 Course Code MMT401M4 Course Title Measure Theory Credit Value 04 Hourly Breakdown Theory Practical Independent Learning 60 — 140 Objectives: Introduce the fundamental concepts of Lebesgue measure spaces and abstract measure spaces Develop clear ideas on the concept of Lebesgue measurable functions, integrals, and their convergence properties Discussthe fundamental connection between differentiation, and integration Intended Learning Outcomes: Construct Lebesgue measures on the real line Define abstract measure space Illustrate the properties of abstract measure space Discuss the properties of measurable functions and the convergence of sequence of measurable functions Explain the simple function approximation of measurable functions Formulate integrals in a measure space Discuss the convergence of integrals Extend the measures from algebras/semialgebra to σ-algebras Formulateproduct measures Prove Fubini’s theorem, and Tonelli’s theorem Discuss the fundamental connection between differentiation, and integration Course Contents: Measure Spaces:Preliminaries:Algebra and σ-algebras of sets,Borel sets; Lebesgue measure: Outer measure, Measurable sets, and Lebesgue measure,Properties,Example of a non-measurable set, Borel measures; General measure: Definition of measure, Measure space, Complete measure space, Examples, Properties. Measurable Functions:  Basic properties of measurable functions, Examples, Borel measurable functions, Approximation Theorem; Littlewoods’s three principles: Egoroff’s theorem. Integration:Integral of nonnegative functions, Integrability of a nonnegative function, Fatou’s Lemma, Monotone convergence theorem, Lebesgue Convergence Theorem, Generalized Convergence Theorem. Extension of Measure:Measure on an algebra, Extension of measures from algebras to σ-algebras, Carathéodory’s theorem, and Lebesgue-Stieltjes integral. Product Measure:Measurable rectangle, Semialgebra, Construction of product measures, Fubini’s theorem, and Tonelli’s theorem Differentiation and Integration:Differentiation of monotone functions: Vitali’s lemma, Functions of bounded variations; Differentiation of an integral: Indefinite integral, and Absolutely continuous functions. Teaching Methods: Lectures, Tutorials, Handouts, Problem solving, Use of e-resources Assessment/ Evaluation Details: In-course Assessments         30% End-of-course Examination 70% Recommended Readings: Halsey Royden, Patrick Fitzpatrick, Real Analysis,4th Edition, 2010 Walter Rudin, Real and Complex Analysis, 3rd Edition, 1986 G De Barra, Measure theory and Integration, 2nd Edition, 2003 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd Edition, 2007
 Course Code MMT402M3 Course Title Advanced Algebra II Credit Value 03 Hourly Breakdown Theory Practical Independent Learning 45 — 105 Objectives: Provide an in-depth knowledge in the theory of groups through a selection of advanced topics. Intended Learning Outcomes: Prove the orbit-stabilizer theorem, Cauchy’s theorem, the class equation and other related results Define Sylow-p-subgroup Prove the Sylow-theorems Examine the simplicity of groups by means of Sylow theorems Prove the fundamental theorem of finite abelian groups Discuss the simplicity of Alternative Group An Construct a group that is free on a given non-empty set Analyse the presentations of certain groups, in particular the dihedral groups and the quaternion groups Prove the Zassenhaus lemma, theorems of Schreier and Jordan-Holder and certain other results related to the chain conditions Discuss various properties of the Frattini subgroup of a given group Define nilpotent group, soluble group, supersoluble group and residually nilpotent group Examine various properties of nilpotent groups, soluble groups, supersoluble groups and residually nilpotent groups Course Contents: Direct products, group acting on sets, p-groups, the Sylow theorems, finite abelian groups, symmetric groups, simplicity of An,  free groups, presentations of groups, series of subgroups, composition series, chain conditions, Frattini subgroup, nilpotent groups, soluble groups, super soluble groups, residually nilpotent groups. Teaching Methods: Lectures,  Tutorials, Handouts, Problem solving, Use of e-resources Assessment/ Evaluation Details: n-course Assessments 30% End-of-course Examination 70% Recommended Readings: Abstract algebra, Joha A Beachy William D Blair, 4th edition, Waveland, 2019. Contemporary Abstract Algebra Joseph.A.Gallian, 9th edition, Cengage, 2016.  Abstract Algebra, David S Dummit,Richard M. Foote,4th edition, Wiley, 2018.  A course in the theory of groups – Derek.J.S.Robinson 2nd edition, Springer, 1995.
MMT403M4: Functional Analysis II