## Level 3

**PM 3050: Group Theory (60L, 4C)**

Dependencies: None

Syllabus:Group Theory: Definition of a group, examples, basic properties; Subgroups; Order of an element and its properties; Cyclic groups and cyclic subgroups; Cosets; Lagrange’s theorem; Product of subgroups; Normal subgroups; Quotient groups; Homomorphisms; Isomorphism theorems; Permutation groups and Cayley’s theorem; Automorphisms and inner automorphisms; Action of a group on a set; Conjugacy classes; Cauchy’s and Sylow’s theorems; Commutator subgroup; Nilpotent and solvable groups; Direct product of groups; Finite abelian groups.

Assessment: End of semester examination.

**PM 3052: Real Analysis II (45L, 3C)**

Dependencies: PM 3056

Syllabus:Riemann Integral: Darboux definition for a bounded function on a closed bounded interval; Necessary and sufficient condition for integrability; Integrability of monotone functions and continuous functions; Linearity of the integral; Additivity of the integral over the interval of integration; Monotonicity of the integral; Integrability of composite; Integrability of the modulus and the product; Integral as a limit of a sum; Integration and differentiation; Fundamental theorem of calculus; Integration by parts; Integration by substitution; Mean value theorems; Interchanging derivative and integrals.Functions of bounded variation and rectifiable curves: Definition of bounded variation; Total variation; Additive property of total variation; Total variation on [a,x] as a function of x; Functions of bounded variations expressed as the difference of increasing functions; Continuous functions of bounded variation; Total variation as the integral of the absolute value of the derivative.Improper Riemann Integrals: Tests for convergence of improper integrals (analogues to Cauchy condition, absolute convergence, comparison test, ratio limit test, Dirichelet’s and Abel’s tests); Cauchy’s intergral test; Euler’s constant; Derivation of Stirling’s formula; Gamma and Beta functions.Riemann-Stieltjes integral: Definition of Riemann-Stieltjes integral; Cauchy criterion for integrability; Some results on existence of integral; Linearity of integral; Additivity of integral; Integrability of monotone functions and continuous functions; Integral as a limit of a sum; Change of variable; Mean value theorem.

Assessment: End of semester examination.

**PM 3053: Complex Analysis (60L, 4C)**

Dependencies: None

Syllabus:The origin of complex numbers; The algebra, geometry and topology of complex numbers.Functions of a complex variable: Limits, continuity, analyticity and Cauchy-Riemann equations; Harmonic functions; Julia and Mandelbrot sets.Elementary functions: Exponential, trigonometric, hyperbolic and logarithmic functions; Complex powers; Contour integration; Cauchy-Goursat theorem; Cauchy’s intergral formula; Morera’s theorem; Liouville’s theorem; Maximum modulus principle; Taylor and Laurent series; Zeros and singularities; Residue theorem; Evaluation of trigonometric and improper integrals; Argument principle; Rouche’s theorem; Open mapping theorem; Conformal mappings.

Assessment: End of semester examination.

Dependencies: None

Syllabus:Fundamental Concepts: Predicates and quantifiers; Basic set theory; The generalized notion of a function, surjections, injections and bijections, algebra of functions; Generalized union and intersection of sets; Generalized product of sets, the axiom of choice, choice functions and the generalized product of sets; The image and the inverse image of a set under a function.Relations: Inverse of a relation, composition of relations; Equivalence relations and partitions.Ordering relations: Partial ordering, total ordering, well ordering; Maximal and minimal elements, upper bound and lower bound, least upper bound and greatest lower bound; Zorn’s lemma, Hausdorff maximal principle, the well-ordering principle and their equivalence to the axiom of choice.Finite and infinite sets: Finite sets, denumerable sets, countable and uncountable sets; Theorems on finite sets; Schroder- Bernstein theorem; Theorems on infinite sets; Cardinality.Metric spaces: Definition and examples, distance between sets, diameter; Basic definitions (interior points, open set, limit point, closed set etc.); Subspaces of a metric space; Neighbourhoods, bases, first countability, second countability; Continuity, uniform continuity, homeomorphisms; Equivalent metrics, uniformly equivalent metrics.

Assessment: End of semester examination.

**PM 3055: Topology II (45L, 3C)**

Dependencies:PM 3504

Syllabus :Metric spaces: Convergence: Convergent sequences, Cauchy sequences, complete spaces, Cantor’s intersection theorem, dense sets and separable spaces, nowhere dense sets, Baire’s category theorem, sequences and continuity, extension theorems, isometries and completion of spaces.Compactness: Compactness of a space and the compact subsets of a space, theorems on compact subsets of a space, theorems on compact subsets of the space Rn, compactness of the Cantor set; Equivalence of compactness, sequential compactness, and the Bolzano Weirstrass property; Compactness and the finite intersection property, continuity and compactness.Connectedness: Separated sets, disconnected and connected sets, connected subsets of R, continuous functions and connected sets.Topological spaces: Definition and examples, subspace topology, comparison of topologies; Definitions in topological spaces (boundary point, dense set, etc.); Bases and sub-bases; The product topology X x Y, order topology, lower limit topology and upper limit topology; Compactness and connectedness (definitions only); Continuity, homeomorphism, topological property; First countable and second countable spaces; Convergence in first countable spaces.

Assessment: End of semester examination.

**PM 3056: Real Analysis I (45L, 2C)**

Dependencies: None

Syllabus:Sequences: Subsequences; Cluster point of a sequence; Limsup and liminf; Proof of Cauchy condition; Bolzano-weierstrass theorem. Series: d’Alembert’s ratio test and Cauchy’s root test in terms of limsup and liminf; Existence of radius of convergence of a power series.Sequences and series of functions: Pointwise convergence and uniform convergence of a sequence of functions; Weierstrass M’ test and Dirichlet’s test for uniform convergence of a sequence of functions.Functions: Boundedness of a function (continuous functions being bounded on a closed interval, continuous functions on a closed interval taking maximum and minimum values at points in the interval); Uniform continuity (functions being continuous on a closed interval); Rolle’s theorem, mean value theorem, Taylor’s theorem and Maclaurin’s theorem (all these theorems with proof), l’Hospital’s rule; Functions defined by series (exponential, logarithmic, trigonometric and hypobolic functions).Functions of several variables: Limits, repeated limits, continuity, partial derivatives and differentiation.

Assessment: End of semester examination.

**AM 3002: Computer applications in discrete mathematics (30L/30P, 3C)**

Dependencies**:** A certain number of students will be selected by the total mark of the 1st and 2nd year Applied Mathematics core courses. In addition to this, any student doing Mathematics Special or the joint special ‘Mathematics, Statistics with Computer Science’ could also follow the course.

Syllabus**:**Most of the following languages and packages will be used in applying discrete mathematics – JAVA (only introductory material), 3D modeling packages, Linux and Window packages; Connecting to databases using relevant packages. Any language or package can be replaced by another equivalent language or package.Advanced Topics: Complexity of algorithm; Introduction to graph theory; Huffmann Codes; Encoding matrices; Decoding tables; Breadth first search and depth first search for spanning trees; Prim’s algorithm and Kruskal’s algorithm for minimum spanning trees; Shortest path algorithm; Euclidean algorithm; Sorting algorithms and sorting trees; Search algorithms; Complexity of algorithms; Analysis of algorithms; Infix, prefix and postfix forms; Binary strings; Traveling salesman problem; Algorithms with applications in Java, C++. Automata, grammars and languages; Finite-state machines; Finite state automata; Languages and grammars; Contest-free grammars and context-sensitive grammars; String fractals.

Assessment**:** End of semester examination.

**AM 3004: Mathematical modeling in Economics and Business (45L, 3C)**

Syllabus**:**Introduction: Introduction to economics and business; Role of mathematics in economics and business; General study of demand, supply and market equilibrium.Economic models: Static and comparative-static analysis of market models, inventory models, input-output models and selected macro economic models; Effect of taxation on static market models; Dynamic analysis (in continuous and discrete time) of market models, inventory models, input-output models, financial models and some macro economic models; Effect of taxation on dynamic market models.Elasticity and other economic concepts: Elasticity of demand and supply; Point and cross elasticities; Analysis of single product and joint products cost, revenue, average cost, price, profit functions etc.; Marginal analysis, consumer’s surplus and producer’s surplus; Optimisation of revenue, cost and profit functions of single product and joint products.Consumer demand theory: Derivation of utility functions; Maximizing utility functions with and without budget constraints; Derivation of demand functions; Indifference curves; Marginal rate of substitution and contract curves (Edgeworth box).Introduction to game theory: Zero-sum matrix games; Single and mixed strategy games; Optimal strategies; Dominance; Simple applications.

Assessment**:** End of semester examination.

**AM 3007: Computer Applications in Combinatorics (30L,30P, 3C)**

Dependencies**:** None

Syllabus**:**Problems related to counting, Problems related to cyclic order, Pigeonhole principle, Ordered and unordered selections, Counting problems related to partition functions, Algorithms in generating combinations and permutations, Generalized permutations and combinations, Introduction to VC++, Working with Timers, Menus, Toolbars, Status bars and Dialog boxes. Using Visual C++ (swing), creating a user interface to solve counting problems. Examples of permutations and Combinations in applied probability, Applications of permutations and combinations using Visual C++ and other programming languages, Generalized multiplications, Binormial expansion, Binormial coefficients, Binormial expansion and its relation with combinations, Multinomial coefficients and Multinomial expansions, Combinatorial identities (for some special cases), Counting problems. Proofs to combinatorial identities using permutations and combinations, Problems related to combinatorial identities. VC++ may be replaced by another Object Oriented Language.

Note**:** Any special students having to take AM 3002 as a compulsary course may opt to take AM 3007 instead. A student is not allowed to take this course if he or she is taking AM 3002

Assessment**:** End of semester examination.

**AM 3050: Mathematical Methods (45L, 3C)**

Dependencies:None

Syllabus:Gamma and Beta functions; Elliptic integrals; Fourier analysis; PDE’s; Solutions of linear PDE’s with homogeneous and non-homogeneous boundary conditions; Variable separable methods; Fourier transforms, Laplace transforms; Fourier sine and cosine transforms; Calculus of variation; Hankel transforms, Melin transforms; Chebyshev polynomials; Hermite polynomials.

Assessment: End of semester examination.

**AM 3051: Numerical Analysis II (45L, 3C)**

Dependencies: Strongly recommended – AM 2005

Syllabus:Systems of linear equations: (a) Direct methods: Easy to solve systems; Forward and backward substitutions; Gaussian elimination; Pivoting strategies; Matrix factorization methods; Survey of software. (b) Iterative methods: Norms of vectors and matrices; Basic concepts of iterative methods; Simple iterative methods; Jacobi method; Gauss-Scheidle method; Conjugate gradient method; SOR-methods; Convergence and error estimates; Survey of software. Cubic spline interpolation.Initial value problems for ordinary differential equations: Existence and uniqueness results; Linear multi-step methods; Consistency; Zero stability; Absolute stability and convergence; Survey of software.Boundary value problems for ordinary differential equations: The linear shooting method; Linear shooting method for non-linear problems; Finite difference methods for linear problems; Survey of software.

Assessment: End of semester examination.

**FM 3002: Actuarial Mathematics II (45L, 3C)**

Dependencies**:** None

Syllabu**s:**The economics of insurance, Individual risk models for a short term, Collective risk models for a single period, Collective risk models over an extended period,Application of risk theory, Insurance models including expenses, Non-forfeiturebenefits and dividends, Special annuities and insurance, Advance multiple theory.

Assessment**:** End of semester examination

**MS 3001: Introduction to Game Theory (45L, 3C)**

Dependencies**:** None

Syllabus**:**Introduction to: Static and Dynamic games with complete information, Static games with incomplete information, Payoff matrix, applications.Static games with complete information : Standard games, zero-sum games, Prisoner’s dilemma, battle of sexes, coordinate games, Chicken or Hark versus Dove.Basic Theory and Applications : Normal form games, Nash equilibrium, Iterated elimination of strictly dominated strategies. Cournot Model of Duopoly, Bertrand Model of Duopoly.Mixed strategies**:** Game theory applications in industry, politics, etc..

Assessmen**t:** End of semester examination

## Level 4

**PM 4001: Commutative Algebra I and Category Theory (60L, 4C)**

Dependencies: None

Syllabus:Ring Theory: Definition of a ring, examples and basic properties; Subrings; Ideals; Quotient rings; Homomorphisms; Isomorphism Theorems; Field of quotient of an integral domain; Prime, maximal and primary ideals; Divisibility theory; Euclidean rings; Polynomial rings.

Assessment: End of semester examination.

**PM 4002: Fields and Galois’ Theory (60L, 4C)**

Dependencies: None

Syllabus:Fields and Galois’ Theory: Algebraic and transcendental extensions; Finitely generated and finite dimensional towers; Algebraic numbers; Gaussian integers; Quadratic integers; Applications; Rule and compass constructions; Galois groups of polynomials; Galois correspondence and applications; Finite fields; Insolvability of quintic equations; Fundamental theorem of algebra.

Assessment: End of semester examination.

**PM 4003: Measure Theory (60L, 4C)**

Dependencies: None

Syllabus: σ – algebras of sets; Additive set functions and measures; Lebesgue outer measure; Measurable sets and Lebesgue measure; Borel sets; Non-measurable sets; Measurable functions; Structure of measurable functions; Lebesgue integration; Fatou’s lemma; Lebesgue monotone convergence theorem; Lebesgue dominated convergence theorem; Modes of convergence; Connection between Riemann and Lebesgue integrals. Lp spaces.

Assessment: End of semester examination.

**PM 4004: Real Analysis (60L, 4C)**

Dependencies: None

Syllaus: Normed vector spaces: Definition; equivalent Norms; Norms that arise from inner products; Norms defined on Rn.Sequence and functions spaces: Norm convergence of these spaces; completeness; Limits in functions spaces; Continuous functions on compact sets; Equicontinuous families of functions; Completion of a Normed space.Series: Non absolute convergence; Absolute convergence in Normed vector spaces.Series of functions: Absolute and uniform convergence; Interchangability of limits; Differentiability and integrability of series of real functions.Integration of vector valued functions: The extension theorem for linear maps; The integral of step maps and the extension of the integral to regulated maps; Properties of the integral; The derivative and relations between integration and differentiation; Interchanging derivatives and integrals (also involving improper integrals)

Assessment: End of semester examination.

**PM 4005: Topological Spaces (60L, 4C)**

Dependencies:None

Syllabus:Topological spaces: Continuous functions; Product topology; Metric topology; Connectedness; Components; Totally disconnected spaces, Locally connected space; Areawise connected spaces; Compactness; Limit point compactness; Local compactness; Tychonoff theorem; Countability axioms; Separation axioms; Urysohn lemma; Tietze extension theorem; Urysohn metrization theorem.

Assessment: 40% inclass assignments and 60% end of semester examination.

**PM 4006: Functional Analysis (60L, 4C)**

Dependencies:None

Syllabus:Complete metric spaces: Contraction mapping theorem and Baire’s category theorem.Normed linear spaces: Finite and infinite dimensional spaces; convergence; completeness and compactness; linear operators and bounded linear operators; Uniform boundedness theorem; Hahn Banach theorem (without proof); Compact linear operators; Linear functional and bounded linear functional; Generalized functions; dual spaces; weak convergence; Space of bounded linear operators and bounded linear functionals; Convergence.Inner product spaces: Inner products and properties; Orthogonal complements; direct sums; orthogonal sets and sequences.Hilbert spaces: Properties; closest point theorem and applications; Bounded linear operators and bounded linear functionals on Hilbert spaces; Riesz representation theorem and Lax- Milligram theorem (without proofs); Adjoint, self adjoint, unitary and normal operators.Applications: Differential equations, optimization, approximation theory, etc.

Assessment: End of semester examination.

**PM 4007: Research Project (240P, 8C)**

Dependencies: None

Syllabus:There will be two assignments in Analysis and two assignments in algebra. An assignment could be either a problem assignment or a reading assignment. Problems will be assigned from the undergraduate material of the subject.

**PM 4050: Complex Analysis (60L, 4C)**

Dependencies:None

Syllabus: The origin of complex numbers; The algebra, geometry and topology of complex numbers. Functions of a complex variable: Limits, continuity, analyticity and Cauchy-Riemann equations; Harmonic functions; Julia and Mandelbrot sets.Elementary functions: Exponential, trigonometric, hyperbolic and logarithmic functions; Complex powers; Contour integration; Cauchy-Goursat theorem; Cauchy’s intergral formula; Morera’s theorem; Liouville’s theorem; Maximum modulus principle; Taylor and Laurent series; Zeros and singularities; Residue theorem; Evaluation of trigonometric and improper integrals; Argument principle; Rouche’s theorem; Open mapping theorem; Conformal mappings.

Assessment: End of semester examination.

**PM 4051: Topology II (45L, 3C)**

Dependencies: PM3054

Syllabus: Metric spaces: Convergence: Convergent sequences, Cauchy sequences, complete spaces, Cantor’s intersection theorem, dense sets and separable spaces, nowhere dense sets, Baire’s category theorem, sequences and continuity, extension theorems, isometries and completion of spaces. Compactness: Compactness of a space and the compact subsets of a space, theorems on compact subsets of a space, theorems on compact subsets of the space Rn, compactness of the Cantor set; Equivalence of compactness, sequential compactness, and the Bolzano Weirstrass property; Compactness and the finite intersection property, continuity and compactness. Connectedness: Separated sets, disconnected and connected sets, connected subsets of R, continuous functions and connected sets. Topological spaces: Definition and examples, subspace topology, comparison of topologies; Definitions in topological spaces (boundary point, dense set, etc.); Bases and sub-bases; The product topology X x Y, order topology, lower limit topology and upper limit topology; Compactness and connectedness (definitions only); Continuity, homeomorphism, topological property; First countable and second countable spaces; Convergence in first countable spaces.

Assessment: End of semester examination.

**AM 4001: Discrete Optimization with Computer Application (60L, 4C)**

Dependencies:AM1005,AM3002

Syllabus:Integer and linear programming; Case studies with applications using the above packages. A variety of real life problems will be discussed. Assignment problems, transshipment problems, fixed- charged problems, machine/ work scheduling problems, inventory models, production process models, capital budgeting problems, financial planning problems, multi period financial problems, network models, branch and bound method, cutting plane algorithm, implicit enumeration and maximum flow problems. Dynamic programming and genetic algorithms with applications. Applications in object oriented programming languages like C#.Languages: Most current computer languages will be introduced.Packages: Simulation packages to solve linear and non linear optimization problems (e.g. Lindo, Lingo, SQL, Spreadsheets)

Assessment: End of semester examination.

**AM 4002: Quantitative Methods (60L, 4C)**

Dependencies: AM 3004

Syllabus:Economics Models: Static and Dynamic analysis (in continuous and discrete time) of single and multi- market models; Equilibrium analysis- existence, uniqueness and stability of equilibrium of models stated above. None- negative matrices and input- output models: Properties and some results on non- negative and M- matrice; Static and dynamic input- output models (open and closed ones); Input- output models as a linear- programming model.Decision theory and utility theory: Decision making under uncertainty; decision trees; Expected monetory value(EMV); Expected value of perfect information (EVPI); problem with EMV; Utility theory- utility axioms, concequences of sensible preferences (utility functions, theorems, implications and derivation of utilities), risk attitudes, Bayes decision under a given utility. Introduction to consumer demand theory and theory of production (A mathematical approach): Consumer theory, costs and perfectly competitive firms, monopoly and imperfect competition. Game Theory: Introduction to static and dynamic games with complete and incomplete information; Some economic applications such as bargaining, auctions and bidding, job market signaling etc. Stochastic matrices and Markov chains: Introduction and applications such as involving market shares, replacement policies, etc. Other Applications: Some economic and business applications using linear and non- linear programming; Problems such as cutting stocks, resource allocation, equipement replacement, job assignments, portfolio analysis, etc.

Assessment: End of semester examination.

**AM 4003: Actuarial Mathematics (60L,4C)**

Dependencies:None

Syllabus: The students will be taught and be made thoroughly familiar with the notations used in actuarial science. Survival distributions and life tables: The theory of survival models: Life expectancy, life insurance, descrete and continuous life insurance

Assessment: End of semester examination.

**AM 4004: Non-linear programming (60L, 4C)**

**Dependencies:**None

**Syllabus:** Unconstrained optimization via calculus: The Hessian of a function; Positive and negative, semidefinite and definite matrices; Coercive functions and global minimizers; Iterative methods. Convex sets and convex functions: Convex sets in economics- linear production models; Convexity and the arithmetic- geometric mean inequality- an introduction to geometric programming; Unconstrained geometric programming. Least square optimization: Least square fit; Subspaces and projections; Minimum norm solutions of undetermined linear systems; Generalized inner products and norms- the portfolio problem. The Karush- Kuhn- Tucker conditions: Separation and support theorems for convex sets; Convex programming; The Karush- Kuhn- Tucker theorem; Dual convex programs; Penalty functions and the penalty method; Lagrange multipliers and the Karush- Kuhn- Tucker theorem for mixed constraints; Quadratic programming.

**Assesments:** End of semester examination.

**AM 4005: Theory of Interest and Corporate Finance (60L, 4C)**

**Dependencies:**None

**Syllabus:** The purpose of this course is to introduce students to the mathematical applications in the financial industry (both actuarial and corporate finance). Part I: Theory of interest: Accumulation/ amount function, discounting/accumulating with compound interest, effective annual rate of discount with compound interest, equivalent rates of interest/ discount, nominal annual rates of interest/discount, force of interest, cash flow terminology, cash flow valuation, net present value, equivalent cash flows, basic

** AM 4006: Partial Differential Equations and their Applications in Financial Derivatives(60L,4C)**

Dependencies**:** AM 1001, AM 2001, AM 2005, AM 3051

Syllabus**:** Partial differential equations: Conservation laws, classifications, elementary analytical methods, initial/ boundary value problems. Diffusion equation: Fundamental solution, similarity solution, qualitative behavior of diffusion initial value problems, Cauchy problem with infinite domain, Initial boundary value problems in the semi- infinite domain, Green’s function, homogeneous boundary value problem with inhomogeneous boundary condition. Hyperbolic equations: Characteristic methods, initial value problems with non- continuous initial data, introduction to weak solutions. Basic option theory: Call option, put option, Asian option, Black – Sholes model and its derivatives. Numerical methods: Discretization of derivatives, boundary conditions, grids, finite difference methods for initial/ boundary value problems, consistency, stability, convergence, applications of finite difference methods in financial derivatives.

**Assesments:** End of semester examination.

**AM 4007: Research Project (180P, 6C)**

**Dependencies:** None

**Syllabus:** The students will do a research project typically on some mathematical and computational** **aspects of a real life problem arising from an industrial, financial or biological background and will write a dissertation on it. A part of the project could be carried out in collaboration with external bodies.

**Assessment:** Lecturer’s evaluation – 20%, Presentation – 30%, Report – 50%

**AM 4008: Advanced Optimization (60L, 4C)**

**Dependencies:** None

**Syllabus: **Inventory Models: Introduction, reasons for holding inventories, cost of inventories, deterministic inventory models in continuous and discrete times, shortages, buffer stock, instantaneous demand, continuous demand, price breaks. Stochastic inventory models. Queuing Models: Introduction, cost of queues, arrival and service time models, birth – death processes, service in random order models. Replacement models: Introduction, replacement of items that deteriorate, increase in maintenance and repair costs, change in value of money, item. Deterministic replacement models in continuous and discrete time, and stochastic replacement models.

**Assesments:** End of semester examination.

** AM 4011: Research Project (180P, 6C)**

Dependencies:None

**Syllabus:** The students will do a research project typically on some mathematical and computational** **aspects of a real life problem arising from an industrial, financial or biological background and will write a dissertation on it. A part of the project could be carried out in collaboration with external bodies.

**Assessment:** Lecturer’s evaluation – 20%, Presentation – 30%, Report – 50%

**AM 4012: Industrial Training (120P, 4C)**

Virtually all mathematics special students become skilled at logical, analytical thinking and at formulating and modeling and solving problems. But, communication skills, teamwork and other soft skills are not always as highly developed in typical math programs. Often the best way to enhance these skills is to spend some time actually working in an industrial environment.

Objectives:The purpose of industrial training program is to offer special degree students in Mathematical Finance, Finance, Business and Computational Mathematics and Mathematics and Statistics with Computer Science an opportunity to

• Gain valuable real world work experience before their graduation.

• Experience concrete practical applications of principles learned in mathematics courses, and

• Do their research projects in industrial working environment.

The AM 4012 also provides an opportunity for employers to evaluate students as potential employees. The training and orientation invested in the students enhance their employability potential. Additionally, the program can foster closer interaction between the employers and the university.

Evaluation: Each intern is required to submit a written report at the end of his/her training. Evaluation of AM 4012 based on the written report (40%), presentation (50%) and the progress report (10%) by the industry supervisor.

**AM 4013: Case Study in Mathematical Modeling (90P, 3C)**

Dependencies: None

Syllabus:Individual or group of students will be assigned Case Studies in Mathematical Modeling in various fields, e.g.: Biology, Finance, Economics etc. of six month duration. A report submitted on the case study will be examined at a seminar presentation.

Assessment: Assessment based on the written report (40%), presentation (40%) and continuous progress (20%).

**FM 4001: Applied Functional Analysis (60L, 4C)**

Dependencies: None

Syllabus:Introduction to Normed Linear Spaces, Linear Transformations, Hilbert Spaces.Analysis of abstract equations: ODE, Stochastic differential equations, PDE.Spectral Theory and Applications.Applications: Stability Theory, Linear Systems Theory, Optimization problems, Numerical Methods.

Assessment: End of semester examination.

**FM 4002: Financial Mathematics Project (180P, 6C)**

Dependencies: None

Syllabus:Individual or group of students will be assigned a Mathematical Finance research project of one year duration. A dissertation submitted on the project will be examined at a seminar presentation.

Assessment: Dissertation and Seminar.

**FM 4003: Case Study in FM (90P, 3C)**

Dependencies: None

Syllabus:Individual or group of students will be assigned Case Studies in

• Mathematical Programming in Finance, or

• Actuarial Mathematics, or

• Game Theory

of six months duration. A report submitted on the case study will be examined at a seminar presentation.

Assessment: Report and Seminar.

**FM 4004: Business Accounting (30L, 3C)**

Dependencies: None

Syllabus:Introduction to Accounting Theory, Basic principles of Accounting, Book keeping and accounting, Ledger Accounting and Control Accounts, Bank Reconciliation and Financial Statements.Introduction to Management Accounting, Classification of costs, product costing, Budget and Budgetary controls, Standard costing and variences.Activity- based costing, activity drivers, cost drivers, contrast with traditional costing systems, Hierarchy of activities.Short run decisions, marginal costing, relevant costs, flexible budgets and special orders, Avoidable costs, product lines and outsourcing, Capacity costs and theory of constraints, uncertainty.

Assessment: End of semester examination.

**FM 4005: Microeconomics (30L, 2C)***

Dependencies: None

Syllabus:Introduction to Microeconomics: Introduction, Market and market mechanism. Market of goods and service: Consumer behavior and formation of demand on market of goods and services, Firm behavior and formation of supply on market of goods and services, Equilibrium on perfect competition market, Characteristics of imperfect competition, Firm behavior in imperfect competition conditions, Profit as a stimulus and alternative goals of the firm.Factor Markets: Formation of prices on factor markets, Labor market and wages, Capital market, Factor market and splitting incomes.Interactions of markets: General equilibrium, Market failure and microeconomic state policy.

Assessment: End of semester examination.

* This course will be conducted by lecturers of the Department of Economics, University of Colombo.

**FM 4006: Macroeconomics (30L, 2C)***

Dependencies: None

Syllabus:An overview of the Macro economy, The production function and aggregate supply, Foundation of aggregate demand, Open economy macroeconomics, Money and the role of asset markets, Theory of the business cycle, IS-LM: A model of the macro economy, Classical business cycle theory, A Keynesian view of the world, The Philips curve: Unemployment and inflation, Exchange rates and the model in an open economy.

Assessment: End of semester examination.

* This course will be conducted by lecturers of the Department of Economics, University of Colombo.