- Inculcate habits of creative thinking, critical analysis and rigorousness.
- Make the student appreciate the uniqueness of mathematics as tool of the tools, having the power of generalization and the power of application.
- Equip students with the mathematical techniques and solutions to indigenous problems applied in industrial, business and financial organizations.
- Strengthen academia-professional-world bonding by tailoring the courses and the trainings offered according to needs of the end-user.
- Teaching and research at university/post graduate college level in the departments of Pure Mathematics, Applied Mathematics and Computational Mathematics and so on.
- Research and Development in public and private sector organizations.
- Product development in public and private sector organizations.
- Higher studies and research in Mathematics and the relevant fields.
- MS students will have to pass 24 credit hours courses and 6 credit hours thesis.
- Department can offer any course from the list of approved courses on the availability of resources.
- Summer semester will not be offered.
- Internal assessments include a seminar, quizzes and assignments of every student in each subject. At least one seminar per student per subject is compulsory.
- Number of assessment activities is double to the number of credit hours of each subject.
List of Courses
MA-5001: Commutative Algebra-I
Integral domains, unit, irreducible and prime elements in ring, Types of ideals, Quotient rings, Rings of fractions, Ring homomorphism, Euclidean domains. Construction of formal power series ring R[[X]] and polynomial ring R[X] in one indeterminate. Polynomial extension of Noetherian domains, Quotient ring of Noetherian rings, Ring of fractions of Noetherian rings. Valuation map and Valuation rings.
MA -5002: Homological Algebra-I
Revision of basic concepts of Ring theory and Module Theory, Modules, Homomorphism and exact sequences. Product and co-product of Modules. Comparison of free Modules and Vector Spaces Projective and injective Modules. Hom and Duality Modules over Principal ideal Domain Notherian and Artinian Module and Rings Radical of Rings and Modules Semi-simple Modules.
MA -5003: Commutative Algebra-II
Guass Theorem, Quotient of a UFD, Nagata Theorem. Divisor classes, Divisor class monoid, divisor class group, Divisorial ideals, divisors, Krull rings, Atomic Domains, Domains Satisfying ACCP, Bounded Factorization Domains, Half Factorial Domains, Finite Factorization Domains: Group of divisibility G(D) of a domain D, G(D) and FFD.
MA -5004: Homological Algebra-II
Tensor products of modules, Singular Homology flate Modules. Categories and factors cogenerator. Finitely related (finitely presented) Modules. Pure ideals of a ring pure submodules and pure exact sequences. Hereditary and Semihereditary rings. Ext. and extensions, Axioms Tor and Torsion, universal co-efficient theorems. Hilbert Syzygy theorem, Serre’s theorem, mixed identities.
MA -5005: Banach Algebras
Banach Algebra: Ideals Homomorphisms, Quotient algebra, Wiener’s lemma. Gelfand’s Theory of Commutative Banach Algebras. The notions of Gelfand’s Topology, Radicals, Gelfand’s Transforms. Basic properties of spectra. Gelfand-Mazur Theorem, Symbolic calculus: differentiation, Analytic functions. Integration of A-Valued functions. Normed rings. Gelfand Naimark theorem.
MA -5006: Advanced Complex Analysis-I
Analytic continuation, equicontinuity and uniform boundedness, normal and compact families of analytic functions, external problems, harmonic functions and their properties, Green’s and von Neumann functions and their applications, harmonic measure conformal mapping and the Riemann mapping theorem, the Kernel function, functions of several complex variables.
MA -5007: Advanced Complex Analysis-II
Holomorphic functions, Extension of analytic functions, Levi-convexity: The Levi form, Geometric interpretation of its signature, E.E. Levi’s theorem, Connections with Kahlerian geometry, Elementary properties of plurisub harmonic functions. Cohomology, complex manifolds. The d.operators, the Poincare Lemma and the Dolbeaut Lemma, The Cousin problems, introduction to Sheaf theory.
Vector spaces, Topological vector spaces, product spaces, quotient spaces, bounded and totally bounded sets, convex sets and compact sets in topological vector spaces, closed hyperplanes and separation of convex sets, complete topological vector spaces, mertizable topological vector spaces, normed vector spaces, normable topological vector spaces and finite dimensional spaces. Locally convex spaces: Convex and compact sets in locally convex spaces, bornological spaces, barreled spaces, spaces of continuous functions, spaces of indefinitely differentiable function, the notion of distributions, nuclear spaces, montal spaces, Schwartz spaces, (DF)-spaces and Silva spaces.
MA-5009: Riemannian Geometry
Definition and examples of manifolds. Differential maps. Submanifolds. Tangents. Coordinate vector fields. Tangent spaces. Dual spaces. Multilinear functions. Algebra of tensors. Vector fields. Tensor fields. Integral curves. Flows. Lie derivatives. Brackets. Differential forms. Introduction to integration theory on manifolds. Riemannian and semi Riemannian metrics. Flat spaces. Affine connection. Parallel translations. Covariant differentiation of tensor fields. Curvature and Torsion tensors. Connection of a semi-Riemannian tensor. Killing equation and Killing vector fields. Geodesics. Conformal transformations and the Weyl tensor.
MA -5010: Integral Equations
MA -5011: Inequalities Involving Convex Functions
Jensen’s and related inequalities, Some general inequalities involving convex functions, Hadamard’s inequalities, Inequalities of Hadamard type I, Inequalities of Hadamard type II, Some inequalities iInvolving concave functions, Miscellaneous inequalities.
MA -5012: Harmonic Analysis
Topology. Sets and Topologies. Separation axioms and related theorems. The Stone- Weierstrass theorem. Cartesian products and weak topology. Banach spaces. Normed linear spaces. Bounded linear transformations. Linear functionals. The weak topology for X*.Hilbert space. Involution on ß (H). Integration. The Daniell integral. Equivalence and measurability. The real LP -spaces. The conjugate space of LP. Integration on locally compact Hausdorff spaces. The complex LP –spaces. Banach Algebras. Definition and examples. Function algebras. Maximal ideals. Spectrum, adverse Banach algebras, elementary theory. The maximal ideal space of a commutative Banach algebra. Some basic general theorems
MA -5013: Partial Differential Equations
Cauchy’s problems for linear second order equations in n-independent variables. Cauchy Kowalewski Theorem. Characteristics surfaces. Adjoint operations, Bicharacteristics Spherical and Cylindrical Waves. Heat equation. Wave equation. Laplace equation. Maximum-Minimum Principle, Integral Transforms.
MA -5014: Numerical Solutions of Ordinary Differential Equations
Theory and implementation of numerical methods for initial and boundary value problems in ordinary differential equations. One-step, linear multi-step, Runge-Kutta, and Extrapolation methods; convergence, stability, error estimates, and practical implementation, Study and analysis of shooting, finite difference and projection methods for boundary value problems for ordinary differential equation.
MA -5015: General Relativity
Review of special relativity, tensors and field theory. The principles on which general relativity is based. Einstein’s field equations obtained from geodesic deviation. Vacunmequation. The Schwarzschild exterior solution. Solutions of the Einstein-Maxwall field equations and the Schwarzschild interior solution. Kerr-Newmann solution (without derivation). Foliations. Relativistic corrections to Newtonian gravity Black holes. Kruskal and Penrose diagrams. The field theoretic derivation of Einstein’s equations. Weak field approximations and gravitational waves. Kaluza-Klein theory. Isometrics. Conformal transformations. Problems of “quantum gravity”
MA -5016: Graph Theory
Fundamentals. Definition. Paths cycles and trees. Hamilton cycles and Euler circuits. Planer graphs. Flows, Connectivity and Matching Network flows. Connectivity and Menger’stheorem. External problems paths and Complete Subgraphs. Hamilton path and cycles.Colouring. Vertexcolouring Edge colouring. Graph on surfaces.
MA -5017: Combinatorics
Elementary concepts of several combinatorial structures. Recurrence relations and generating functions. Principle of inclusion and exclusion. Latin squares and SDRs. Steiner systems. A direct construction. A recursive construction. Packing and covering. Linear algebra over finite fields. Gaussian coefficients. The pigeonhole Principle. Some special cases. Ramsey’s theorem. Bounds for Ramsey numbers and applications. Automorphism groups and permutation groups. Enumeration under group action.
MA -5018: Research Methodology
Scientific statements, hypothesis, model, Theory & Law, Types of research, Problem definition, objectives of the research, research design, data collection, data analysis, Interpretation of results, validation of results, Limitation of Science, calibration, Sensitivity, Least count and reproducibility, Stability and objectivity, Difference between accuracy and precision, Literature search, defining problem, Feasibility study, pilot projects / field trials, Formal research proposal, budgeting and funding, Progress report, final technical and fiscal report, Purpose of experiment, good and bad experiments, Inefficient experiments, null and alternative hypothesis, Alpha and beta errors, Relationship of alpha and beta errors to sensitivity and specificity, Designing efficient experiments, Simple random sampling, systematic sampling, Stratified sampling, cluster sampling, Convenience sampling, judgment sampling, quota sampling, snow ball sampling , Identifying variables of interest and their interactions, Operating characteristic curves, power curves, Surveys and field trials, Submission of a paper, role of editor, Peer-review process, importance of citations, impact factor, Plagiarism, protection of your work from misuse, Simulation, need for simulation, types of simulation, Introduction to algorithmic research, algorithmic research problems, types of algorithmic research, problems, types of solution procedure.
MA -5019: Non-Newtonian Fluid Mechanics
Classification of non-Newtonian fluids, Rheological formulae (time-independent fluids, thixotropic fluids and viscoelastic fluids), variable viscosity fluids, cross viscosity fluids, the deformation rate, viscoelastic equation, materials with short memories, time dependent viscosity, the Rivlin-Ericksen fluid, basic equations of motion in rheological models. The linear viscoelastic liquid, Couette flow, Poiseuille flows, the current semi-infinite field. Axial oscillatory tube flow, angular oscillatory motion, periodic transients, basic equations in boundary layer theory, orders of magnitude, truncated solutions for viscoelastic flow, similarity solutions, turbulent boundary layers, stability analysis.
MA-5020: Advanced Analytical Dynamics-I
Equations of dynamic and its various forms, equations of Langrange and Euler, Jacobi’s elliptic functions and the qualitative and quantitative solutions of the problem of Euler and Poisson. The problems of Langrange and Poisson. Dynamical systems. Equations of Hamilton and Appell. Hamilton-Jacobi theorem. Separable systems. Holder’s variational principle and its consequences.
MA -5021: Numerical Solutions of Partial Differential Equations
Boundary and initial conditions, Polynomial approximations in higher dimensions, Finite Element Method: The Galerkin method in one and more dimensions. Error bound on the Galarki method, the method of collocation, error bounds on the collocation method, comparison of efficiency of the finite difference and finite element method. Finite Difference Method: Finite difference approximations. Applications to solutions of linear and non-linear partial differential equations appearing in physical problems.
MA -5022: Functional Analysis
Separation properties. Hahn-Banach theorem. Banach algebras theorem (Introduction). Linear mappings. Finite dimensional spaces. Metrization. Boundedeness and continuity. Seminorms and local convexity. Baire category theorem. The Banach-Steinhaus theorem. The open mapping theorem. The closed graph theorem. Bilinear mappings. The normed dual of normed spaces. Adjoints.
MA -5023: Advance Numerical Analysis
Introduction. Euler’smethod. The improved and modified Euler’s method. Runge-Kutta method. Milne’s method. Hamming’s methods. Initial value problem. The special cases when the first derivative is missing. Boundary value problems. The simultaneous algebraic equations method. Iterative methods for linear equations. Gauss-Siedel method. Relaxation methods. Vector and matrix norms. Sequences and series of matrices. Graph Theory. Directed graph of a matrix. Strongly connected and irreducible matrices. Grerschgoin theorem. Symmetric and positive definite matrices. Cyclic-Consistently ordered matrices. Choice of optimum value for relaxation parameter.
MA -5024: Special topics in Advanced Mathematics-I
MA -5025: Special topics in Advanced Mathematics-II
The course contents should be specified from time to time by the resource person with consultation of the Chairman, Department of Applied Sciences.
M.Sc./BS in Pure Mathematics/Applied Mathematics/Computational Mathematics (minimum 16 year education) degree or its equivalent with a minimum CGPA of 2.0/4.0 in semester system or 60% in annual system/Term system from an HEC recognized institute/ university.
The candidate will have to pass NTU GAT test as.
The applicant must not be already registered as a student in any other academic program in Pakistan or abroad.
- The MS program shall be advertised in the beginning of each academic session.
- An applicant shall apply on a prescribed form within the due date given in the advertisement for admission.
- The completed application form, along with required documents, shall be submitted in the admission office.
- The applications shall be evaluated according to the following criteria for making the merit list.
|BS Mathematics/B.Sc + M.Sc||40/20+20% weightage|
|NTU GAT (General)||40% weightage|
|Certificate Verification Fee||2000||-||-||-|
|Red Crescent Donation||100||-||-||-|
|University Card Fee||300||-||-||-|
|Student Activity Fund||2000||2000||2000||2000|
* There is no Transport Fee for Hostel Resident but they will pay hostel charges