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The Department offers postgraduate programmeswhich lead to the following degrees:

• Master in Pure Mathematics• Ph.D in Mathematics - Pure Mathematics• Master in Applied Mathematics• Ph.D in Mathematics - Applied Mathematics• Master in Applied Statistics• Ph.D in Statistics• Mathematics Education

FACULTY OF PURE AND APPLIED SCIENCES

Department of Mathematics and Statistics

Postgraduate Studies ProgrammeThe programmes are supervised by the PostgraduateProgrammes Coordinator who can be either thechairperson of the Department or a faculty memberappointed by the Departmental Board. TheCoordinator is the chairperson of the PostgraduateStudies Committee. The other members are alsoappointed by the Departmental Board. Aninterdepartmental committee coordinates theinterdepartmental postgraduate programme.

Admission to PostgraduateProgrammesThe number of postgraduate students to be admittedis announced separately for each specific programmeat the Master or Doctorate level.The criteria for evaluation and ranking of thecandidates are the following:• Prior university training in an appropriate field ofstudy and a transcript of the degree. Appropriatefields of study are Mathematics, Statistics or otherrelated subjects such as Computer Science, Physics,Engineering, etc.

• Recommendation letters (at least two) fromuniversity professors

• Personal interview (if necessary)• Other qualifications, such as exams, awards,distinctions, etc.

• Sufficient knowledge of the English language(recommended)

• Candidates with insufficient knowledge ofmathematics will be required to attend a numberof undergraduate courses, in addition to thoserequired by the regulations of the Department.

For more information on the Admission andAttendance Regulations – Application Requirementsconsult the Office of Postgraduate Studies, AcademicAffairs and Student Welfare Services (tel.22894021/61) or the Department’s Secretariat.

MASTER IN APPLIED MATHEMATICSRegulationsTo obtain a Master degree in Applied Mathematicssuccessful completion of a minimum of 90 ECTS isrequired.Each course corresponds to 10 ECTS, the Masterthesis to 15 ECTS and Seminars to 5 ECTS.A postgraduate student may attend at most twoSeminars.

Indicative Programme of StudiesOptions ECTS/Course Total ECTS6 Compulsory Courses 10 603 Elective Courses 10 30TOTAL 90or6 Compulsory Courses 10 602 Elective Courses 10 202 Seminars 5 10TOTAL 90or6 Compulsory Courses 10 601 Elective Course 10 101 Seminar 5 5Master Thesis 15 15TOTAL 90

List of CoursesCompulsory CoursesCategory ITwo of the following:MAS 601 Measure and IntegrationMAS 604 Functional AnalysisMAS 606 Function Theory of One ComplexVariableCategory IITwo of the following:MAS 603 Partial Differential EquationsMAS 621 Numerical Linear AlgebraMAS 671 Numerical Solution of Ordinary

Differential EquationsMAS 673 Finite Element MethodsCategory IIITwo of the following:MAS 613 Ordinary Differential EquationsMAS 672 Numerical Solution of Partial Differential

EquationsMAS 677 Topics in Numerical Analysis IMAS 678 Topics in Numerical Analysis IIMAS 679 Topics in Numerical Analysis IIIMAS 680 Seminar in Applied Mathematics IMAS 681 Seminar in Applied Mathematics IIMAS 682 Classical MechanicsMAS 683 Fluid DynamicsMAS 684 Topics in Applied Mathematics IMAS 685 Topics in Applied Mathematics IIMAS 686 Topics in Applied Mathematics IIIMAS 687 Topics in Differential EquationsMAS 688 Topics in Differential EquationsMAS 689 Topics in Differential EquationsNote: Category III also includes all the courses

offered under the Categories I and II.

Department of Mathematics and Statistics74


Elective CoursesMAS 602 Fourier AnalysisMAS 605 Elliptic Partial Differential Equations of

Second OrderMAS 608 Evolution Differential Equations with

Partial Derivatives of Second OrderMAS 611 Harmonic AnalysisMAS 617 Topics in Mathematical Analysis IMAS 618 Topics in Mathematical Analysis IIMAS 619 Topics in Mathematical Analysis IIIMAS 620 Approximation TheoryMAS 633 General Relativity

PH.D. IN MATHEMATICS –Applied MathematicsFor the fulfillment of a Doctor of Philosophy Degreethe requirements are:

1. Successful completion of 120 ECTSat the postgraduate level in accordance with theprovisions of the programme of studies of theDepartment. Students with a Master degree arepartially or fully exempted from this requirement.

2. Comprehensive Examination (CE)Candidates elect two areas from among the fourareas included in the Syllabus Content for theWritten Comprehensive Examination, on whichthey will be examined. Students should successfullycomplete the CE at the latest by the fifth semester oftheir studies.

3. Oral ExaminationThe candidate for the Ph.D. must present a researchproposal for a Doctoral Thesis before a three-member committee. The presentation takes placeat the latest one year after the admission of thestudent as a doctoral candidate.

4. Doctoral ThesisA doctoral thesis can be submitted only after thecompletion of at least four semesters in thepostgraduate programme at Ph.D. level, and afterthe student has successfully completed thecomprehensive examination and obtained therequired ECTS.

5. Defence of the ThesisDefence of the thesis takes place before a five-member committee.

For more information on the comprehensiveexamination, the oral examination, the DoctoralThesis and the Defence of the Thesis, see theAdmission and Attendance Regulations – ApplicationRequirements consult the Office of PostgraduateStudies, Academic Affairs and Student WelfareServices (tel. 22894021/61) or the Department’sSecretariat.

The Syllabus for the WrittenComprehensive Examination (WCE)APPLIED MATHEMATICSNewton equations, central forces, rotating axissystems, particle systems, motion of solids, Eulerequations. Generalized coordinate systems,holonomic systems, Lagrange equations. Hamiltonequations, equations of normal transformations,symmetries and conservation laws, Hamilton-Jacobitheory. Introduction to special relativity. Theorems ofStokes and Gauss, Calculus of variations, specialfunctions, integral equations, asymptotic analysis.

MATHEMATICAL ANALYSIS (APPLIED)Basic theory of metric space. σ-algebras, measures,outer measure. Borel measure on the real line.Measurable functions, integration.

Convergence. Product measure and n-dimensionalLebesque measure. Polar coordinates and signedmeasures, the Radon-Nikodym theorem. Basic theoryof Lp spaces.

PARTIAL DIFFERENTIAL EQUATIONSFirst order quasi – linear equations: the method ofcharacteristics. Existence (Cauchy – Kovalenski) anduniqueness (Holmgren) theorems. Theory ofdistributions: the dual of a differential operator, weaksolutions, fundamental solutions. Construction offundamental solutions. Second order equations:classification and canonical forms, solution of initialand boundary value problems for elliptic, parabolicand hyperbolic equations. Separation of variables,Fourier series.

NUMERICAL ANALYSISNumerical solution of non-linear equations. Matrixand vector norms. Direct and iterative methods forlinear systems.Computation of eigenvalues and eigenvectors.Polynomial interpolation (Lagrange and Hermite).Numerical quadrature (Newton and Gauss rules).Linear multistep methods for initial value problemsin ODEs. Initial value and finite difference methodsfor boundary value problems in ODEs. Numericalsolution for first and second order hyperbolic PDEs(method of characteristics, finite differencestechniques). Numerical solution of parabolic PDEs.Numerical solution of the one and two-dimensionalheat equation. Finite differences methods forLaplacian and Poisson problems.Metric spaces, normed linear spaces and innerproduct spaces. Banach fixed point theorem andapplications. Best approximation in normed linearspaces and inner product spaces. Finite differenceelement methods for elliptic problems. Variatonal,Ritz and Galerkin method. Error analysis.

MASTER IN PURE MATHEMATICS

RegulationsTo obtain a Master degree in Pure Mathematicssuccessful completion of a minimum of 90 ECTS isrequired.Each course corresponds to 10 ECTS, the Masterthesis to 15 ECTS and seminars to 5 ECTS.A postgraduate student may attend at most twoseminars.Regular meetings of the teaching staff will take placefor the programme of Pure Mathematics(Undergraduate and Postgraduate) where it will bedecided which courses will be offered and by whomthey will be taught.

Indicative Programme of StudiesOptions ECTS/Course Total ECTS4 Compulsory Courses 10 405 Elective Courses 10 50TOTAL 90or4 Compulsory Courses 10 404 Elective Courses 10 402 Seminars 5 10TOTAL 90or4 Compulsory Courses 10 403 Elective Courses 10 301 Seminar 5 5Master Thesis 15 15TOTAL 90

List of CoursesCompulsory CoursesMAS 601 Measure and IntegrationMAS 606 Function Theory of one ComplexVariableMAS 632 Riemannian GeometryMAS 627 Group Representation Theory I or MAS

624 Introduction to CommutativeAlgebra

Elective CoursesMAS 602 Fourier AnalysisMAS 604 Functional AnalysisMAS 605 Elliptic Partial Differential Equations

with Partial Derivatives of Second OrderMAS 607 Function Theory of Several Complex

VariablesMAS 608 Evolution Differential Equations with

Partial Derivatives of Second OrderMAS 611 Harmonic AnalysisMAS 612 Measure and ProbabilityMAS 617 Topics in Mathematical Analysis IMAS 618 Topics in Mathematical Analysis IIMAS 619 Topics in Mathematical Analysis IIIMAS 620 Approximation TheoryMAS 622 Algebraic Coding TheoryMAS 623 Number TheoryMAS 624 Introduction to Commutative AlgebraMAS 625 Theory of GroupsMAS 626 Field and Galois TheoryMAS 627 Group Representation Theory IMAS 628 Group Representation Theory IIMAS 629 Topics in Algebra IMAS 630 Topics in Algebra IIMAS 631 Differential TopologyMAS 633 General RelativityMAS 634 Algebraic Topology I

MAS 635 Lie groups and Lie AlgebrasMAS 636 Algebraic Topology IIMAS 637 Spectral GeometryMAS 638 Spin GeometryMAS 639 Algebraic GeometryMAS 640 Topics in Geometry IMAS 641 Topics in Geometry IIMAS 642 Topics in Geometry IIIMAS 643 Seminar in Pure Mathematics – AnalysisIMAS 644 Seminar in Pure Mathematics – AnalysisIIMAS 645 Seminar in Pure Mathematics – Algebra IMAS 646 Seminar in Pure Mathematics – AlgebraIIMAS 647 Seminar in Pure Mathematics –Geometry IMAS 648 Seminar in Pure Mathematics –Geometry IIMAS 682 Classical Mechanics

Ph.D. IN MATHEMATICS –Pure MathematicsFor the fulfillment of a Doctor of Philosophy Degreethe requirements are:

1. Successful completion of 120 ECTSat the postgraduate level in accordance with theprovisions of the programme of studies of theDepartment. Students with a Master degree arepartially or fully exempted from this requirement.

2. Comprehensive Examination (CE)The Comprehensive Examination comprises twosections:A. Written Comprehensive Examination (WCE):the written examination consists of an examessay divided in four parts.



B. Oral Comprehensive Examination (OCE): ontwo subjects designated by the Research Advisorof the student. The student is examined orallybefore a three-member committee appointed bythe Postgraduate Studies Committee of theDepartment after recommendation by theResearch Advisor.

3. Oral ExamThe requirements are the same as for the Ph.D. inApplied Mathematics (see relevant paragraph).

4. Doctoral ThesisThe requirements are the same as for the Ph.D. inApplied Mathematics (see relevant paragraph).

5. Defence of the ThesisThe requirements are the same as for the Ph.D. inApplied Mathematics (see relevant paragraph).

The Syllabus for the WrittenComprehensive Examination (WCE)PART ONEThe system of real numbers. Continuity anddifferentiation of univariate real – valued functions.Riemann integration.Metric spaces, compactness andconnectedness. Theorems of Bolzano – Wierstrass,Heine – Borel and Baire category. Convergence ofsequences of functions, uniform continuity. σ-algebras outer measures, Borel and Lebesquemeasures. Measurable functions. Fatou’s Lemma,Monotone Convergence Theorem and DominatedConvergence Theorem. Signed measures and Radon– Nikodym Theorem. Product measures and Fubini,Tonelli Theorems. Basic Theory of Lp spaces andRadon measures. Applications to Probability Theory(Random variables, Law of Large Numbers,conditional expectations, Central Limit Theory).

PART TWOComplex plane and stereographic projection. MobiusTransformations.Cauchy-Riemann equations, harmonic functions.Elementary analytic functions. Complex Integrationand Cauchy Integral Representation Theorem.Essential Theorems (Morera, Liouville, FundamentalTheorem of Algebra). Taylor and Laurent series.Residues.Maximum Principle, Schwarz Lemma, ArgumentPrinciple, Rouche Theorem. Conformal mappingsand Riemann Mapping Theorem. Infinite series andinfinite products. Theorems of Weierstrass andMittag-Leffler for entire analytic functions.

PART THREEGroups and homomorphisms. Free groups, generatorsand relations. Finitely generated abelian groups.Group actions. Sylow theorems and p-groups. Simplegroups. Normal series. Extensions. Rings andhomomorphisms. Ideals. Factorization incommutative rings. Modules and exact sequences.Free modules. Tensor product of modules. Modulesover principal ideal domains. Jordan canonical form.Representations. Semisimple rings. Fields, fieldextensions. Separable and normal extensions. Thefundamental theorem of Galois theory. Solvability byradicals.

PART FOURTopological and differentiable manifolds, basicexamples and properties. Fundamental group.Tangent spaces. Partition of unity. Regular (non-singular) Values. Sard's Theorem. Vector fields, flows.Frobenius's Theorem. Differential forms. Stoke'sTheorem. Riemannian manifolds. Connections andgeodesics. Exponential map and regular coordinates.Gauss Lemma andHopf-Rinow Theorem. Curvature.

Gauss-Bonnet Theorem and the Theorem ofHadamard-Cartan.

MASTER IN APPLIED STATISTICSTo obtain a Master degree in Applied Statisticssuccessful completion of a minimum of 93 ECTS isrequired.

Indicative Programme of StudiesECTS

1st SemesterMAS 650 Mathematical Statistics 10MAS 655 Survey Sampling 10MAS 658 Simulation and Data Analysis 10MAS 850 Seminar in Applied Statistics I ** 12nd SemesterMAS 653 General Linear Models* 10MAS 659 Multivariate Analysis* 10MAS Elective Course I+ 10MAS 851 Seminar in Applied Statistics II ** 13rd SemesterMAS 657 Analysis of Discrete Data* 10MAS Elective Course II+ 10MAS Elective Course III+ 10MAS 852 Seminar in Applied Statistics III ** 1TOTAL 93OptionsMAS 654 Nonparametric Statistics* 10MAS 656 Time Series Analysis* 10MAS 660 Probability Theory 10MAS 661 Topics in Statistics I 10MAS 662 Topics in Statistics II 10MAS 663 Topics in Statistics III 10MAS 664 Bayesian Statistics* 10MAS 665 Computational Statistics* 10

MAS 666 Biostatistics* 10MAS 670 Theory of Statistics 10

Notes:* In these courses, the use of statistical software is anintegral part.

** Amandatory course. Students will attend colloqiumlectures. A pass/fail course. Students must enroll inthe course every semester.

+ (a) Two classes from Options I, II and III can bereplaced by a Master thesis. The subject of thethesis should be related to Statistical Science.The thesis is carried out under the supervision ofa faculty member of the Department.

(b) If a student does not choose the thesis option,then option III can be replaced by eitherIndependent Study (MAS 667)or by practical training in the private or publicsector (MAS 668).

Ph.D. IN STATISTICSFor the fulfilment of a Doctoral degree in statistics,the following are required :

1. Successful completion of 60 ECTSat postgraduate level, in accordance with theprovisions of the programme of studies of theDepartment. Students with a Master degree arepartially or fully exempted from this requirement.The 60 ECTS should be completed as follows:• At least 10 ECTS in Probability Theory (MAS 660)• At least 10 ECTS in Statistical Theory(MAS 670)• At least 10 ECTS in Simulation and Data Analysis(MAS 658)

The remaining 30 ECTS may be completed withany postgraduate courses offered by theDepartment, including reading courses.

2. Comprehensive Examination (CE)Successful completion of the following CEs with agrade of 7.5 or better:• CE in Probability Theory (MAS 760) – 0 ECTS• CE in Statistical Theory (MAS 770) – 0 ECTS• CE in Simulation and Data Analysis (MAS 758) –0 ECTS

The CE in Probability Theory (MAS 760) andStatistical Theory (MAS 770) correspond to thefinal exams for MAS 660 and MAS 670. The CE inSimulation and Data Analysis (MAS 758) iscomprised of an open lecture on a project involvingdata analysis and computations.

3. SeminarAll doctoral students must enrol in the Seminar ofApplied Statistics for at least 6 semesters.

4. Doctoral ThesisThe requirements are the same as for the Ph.D. inApplied Mathematics (see relevant paragraph).

5. Defence of the ThesisThe requirements are the same as for the Ph.D. inApplied Mathematics (see relevant paragraph).

The Syllabus Content for theComprehensive ExaminationPROBABILITY THEORYAxiomatic FoundationMeasure theoretic probability, measure theory andintegration, σ-algebras, monotone classes, events,probability spaces, stochastic independence, 0-1 laws,the Borel-Cantelli lemmas.

Random VariablesRandom variables, distribution of a random variable,continuous and discrete random variables,distribution of a function of a random variable,random vectors.

ExpectationExpectation of a random variable, expected value andindependence, expected value as the integral withrespect to a probability measure, properties ofintegration, moments, probability inequalities,conditional expectation.

Limit TheoremsModes of convergence of a sequence of randomvariables, uniform integrability, convergence ofmoments, moment generating functions,characteristic functions, theorems of continuity andinversion, infinite divisibility laws and stable laws,central limit theorem, weak and strong laws of largenumbers.

Martingales and Random WalksProperties of randomwalk, limit theorems, definitionand properties of martingales, martingale inequalities,convergence criteria, weak and strong laws formartingales, central limit theorem for martingales.



STATISTICAL THEORYEstimation TheoryRandom sample, statistic, families of distributions,exponential families. Estimators (maximumlikelihood, least squares, moment estimators, Bayesestimators). Properties of estimators, unbiasedness,sufficiency, consistency. Unbiased estimators ofuniformly minimal variance, Fisher information,Cramer – Rao inequality. Rao – Blackwell Theoremand Theorem of Lehmann – Scheffe.

Theory of testing statistical hypothesisDecision theory, simple and composite hypothesis,test statistics, properties of tests. Neyman – Pearsonlemma, uniformly most powerful tests. Likelihoodratio tests. Hypothesis testing and confidenceintervals. Goodness-of-fit tests, tests of independence,rank tests.

For more information on the comprehensiveexamination, the oral examination, the DoctoralThesis and the Defence of the Thesis, see Admissionand Attendance Regulations – ApplicationRequirements consult the Office of PostgraduateStudies, Academic Affairs and Student WelfareServices (tel. 22894021/61) or the Department’sSecretariat.

Course DescriptionsMAS 601 Measure and IntegrationMetric spaces. σ- algebras, measures, outer measures. Borelmeasures on the real line. Measurable functions.Integration. General convergence theorems. Signedmeasures. Product measures n-dimensional Lebesqueintegral. The Radon Nikodym Theorem. Lp spaces.

MAS 602 Fourier AnalysisThe Schwarz space. Fourier transform. Plancherel’sformula. Convergence of Fourier series and integrals.Applications in partial differential equations. Distributions.Tempered distributions, compactly supporteddistributions. Sobolev spaces.

MAS 603 Partial Differential EquationsFirst order quasi-linear equations, the method ofcharacteristics. Classification and normal forms. Existencetheorem of Cauchy-Kovalevskaya and uniqueness theoremof Holmgren. Distributions and weak solutions. Hyperbolictheory, characteristics, propagation of singularities. Waveequation in one, two and three space dimensions.Conservation laws and shock waves. Elliptic theory, Laplaceand Poisson equations, fundamental solutions, harmonicfunctions. Variational formulation of elliptic boundaryvalue problems. Parabolic theory, heat equation, parabolicinitial/boundary value problems.

MAS 604 Functional AnalysisCompact operators. Spectral theory. Self adjoint operators.Closed and orthonormal operators. Spectral theorem.Semigroups.

MAS 605 Elliptic Partial Differential Equations ofSecond Order

Laplace equation, fundamental solutions, Green's function,maximum principle, Poisson kernel, Harmonic functionsand their properties, Harnack inequalities, equations withvariable coefficients, Dirichlet problem, existence andregularity of solutions.

MAS 606 Function Theory of One Complex VariableBasic facts about complex functions of one complexvariable. Differentiation. Cauchy-Riemann equations.Elementary complex functions. Complex integration andthe Cauchy Theorem. Applications of Cauchy Theorem.Meromorphic functions. Power series and Laurent series.Residues. Entire functions and Conformal mappings.

MAS 607 Function Theory of Several ComplexVariables

Basic facts about holomorphic functions of several complexvariables. Integral representations of holomorphicfunctions of several complex variables.

MAS 608 Evolution Differential Equations with PartialDerivatives of Second Order

Heat equation, fundamental solution, properties ofsolutions, weak solutions. Maximum principle, waveequations. Solutions with spherical means. Non-homogeneous problem, energy methods, weak solutions,propagation of singularities. Distributions, fundamentalsolution, L2 theory, etc.

MAS 609 Stochastic AnalysisReview of the basic notions of probability theory, stochasticintegration, Ito’s lemma, stochastic differential equations,applications (financial mathematics, formula Black-Scholes, etc.).

MAS 610 Stochastic ProcessesBasic notions of stochastic processes, Kolmogorov´stheorem, discrete and continuous time Markov processes,point processes, Brownian motion, random walk.

MAS 611 Harmonic AnalysisApproximation to the identity, weak Lp spaces,interpolation theorems. Maximal functions, harmonicfunctions, singular integrals, Littlewood-Paley theory.Function spaces.

MAS 612 Measure and Probabilityσ- algebras, measures, probability measures, measurablefunctions. Integration theory. Product measures and FubiniTheorem. Lebesque-Stieltjes measure, ordinarydistributions, characteristic functions. Sequences ofmeasurable functions and different notions of theirconvergence. Central Limit Theorem and relatedasymptotic developments. The distribution of the recursivelogarithm, Radon-Nicodym Theorem. Conditionalmathematical expectation. Martingales.

MAS 613 Ordinary EquationsExistence theorems: Picard-Lindelof and Cauchy-Peano.Uniqueness theorem when Lipschitz condition is satisfied.Smooth dependence of solutions on parameters.Extensibility of solutions. Linear systems, fundamentalsolutionmatrix, systems with periodic coefficient. Stabilityof nonlinear systems. Sturm-Liouville theory.

MAS 617 Topics in Mathematical Analysis IMAS 618 Topics in Mathematical Analysis IIMAS 619 Topics in Mathematical Analysis IIITopics in real analysis, complex analysis or differentialequations.

MAS 621 Numerical Linear AlgebraElements of matrix analysis, vector and matrix norms.Factorization and least - squares methods. Stability. Directand iterative methods for the solution of linear systems.Methods for calculating eigenvectors and eigenvalues.

MAS 622 Algebraic Coding TheoryFinite fields. Linear codes, syndrome decoding. Cycliccodes. BCH codes and Reed – Solomon codes. MDS codes.Permutation decoding.

MAS 623 Number TheoryIntroduction to algebraic number theory. Quadraticreciprocity, Gauss and Jacobi sums. Field extensions, finitefields, ideal classes. Quadratic and cyclotomic fields.Applications to Diophantine equations.

MAS 624 Introduction to Commutative AlgebraPrime andmaximal ideals. Extension. Finitely generated R– modules. Exact sequences. Tensor product of modules.Algebras. Noetherian rings and Artin rings. Dedekinddomains.

MAS 626 Field and Galois TheoryPolynomial rings. Field extensions, splitting fields.Separable extensions, normal extensions. The fundamentaltheorem of Galois theory. Roots of unity and cyclotomicpolynomials. Solution by radicals. Symmetric functionsand Abel’s theorem.

MAS 627 Group Representation Theory IRepresentations. FG-modules, FG-submodules and FG-homomorphisms. Maschke’s Theorem and Schur’s Lemma.Irreducible module. The group algebra, the centre of thegroup algebra. Characters, relation between characters andrepresentations. Character tables. Frobenius reciprocitytheorem.

MAS 628 Group Representation Theory IISemi simple rings, construction of irreducible R –modules.Splitting fields. Clifford’s theorem.Mackey DecompositionTheorem. Representations ofWeyl groups. Representationsof compact groups.

MAS 629 Topics in Algebra IMAS 630 Topics in Algebra IIMAS 631 Differential TopologyDifferentiable manifolds. Tangent space. Partition of unity.Regular points. Sard's theorem. Vector fields and flows.Frobenius Theorem. Differential forms. Stokes Theorem.De Rham's Theorem.

MAS 632 Riemannian GeometryRiemannian manifolds. Geodesics, exponential map,normal coordinates. Gauss lemma. Theorem of Hopf-Rinow. Curvature. Jacobi fields. Theorems of Bonnet-Myers, Synge-Weinstein and Hadamard - Cartan.Homogeneous and symmetric spaces.

MAS 633 General RelativityLorentz geometry. Special relativity. Newton spacetime,Minkowski spacetime. Lorentz transformation. Einsteinequations. Special solutions (Schwarzschild).

MAS 634 Algebraic Topology IHomology theory and applications. Cohomology.Universal coefficient theorem. Products. Kuennethformula. Thom isomorphism. Poincare duality.

MAS 635 Lie Groups and Lie AlgebrasDifferentiable manifolds. Tangent spaces and vector fields.Lie Groups. Exponential function. Homogeneous spaces.The Campbell-Hausdorf formula.

Ado's Theorem. Lie algebras. Ideals and homomorphisms.Solvable and nilpotent Lie algebras. Semisimple Liealgebras. Root systems. Compact Lie groups.

MAS 636 Algebraic Topology IIObstruction theory. Bundles and K- theory. Bordism.Spectral sequences. Characteristic classes.

MAS 637 Spectral GeometryLaplace operator. Minimax principle. Isoparametricinequalities. Heat kernel.

MAS 638 Spin GeometryClifford algebras. Spin groups and representations. Spinstructures. Spin connection. Spin manifolds. Diracoperator. Bochner formula. Lichnerowicz's Theorem.

MAS 639 Algebraic GeometryAlgebraic sets and the Hilbert-Nullstellensatz theorem.Affine, projective and quasi-projective varieties,morphisms, products. Local properties (smooth andsingular points), tangent space, dimension. Divisors onalgebraic curves, Riemann-Roch theorem. Bezout'stheorem and the group structure of an elliptic curve. Blowup and resolution of singularities. Lines on hypersurfaces.

MAS 640 Topics in Geometry IMAS 641 Topics in Geometry IITopics from Differential Geometry, Algebraic Geometryand Algebraic Topology.

MAS 650 Mathematical StatisticsUnivariate andmultivariate random variables, distributionfunction, joint and conditional distribution, independence,moments. Special parametric families of distributions.Estimation. Methods of finding estimators. Properties ofestimators, sufficiency, unbiasedness, consistency.Comparison of estimators. Confidence Intervals.Hypothesis testing. Simple and composite hypothesis,power function. Methods of constructing tests. Propertiesof tests, unbiasedness, consistency. Comparison of tests.Hypothesis testing and confidence intervals.



MAS 653 General Linear ModelsLinear and multiple regression, residuals and modelselection procedures, diagnostics. Analysis of variance andnon linear regression. Design of experiments, completelyrandomized designs, designs with two or more factors withinteractions. Block designs, split plot and nested designs.

MAS 654 Nonparametric StatisticsOrder statistics and their distributions. Tolerance regions.Rank and sign tests for one and two populations. Goodnessof fit tests (Kolmogorov – Smirnov, Lilliefors, Shapiro –Wilks). Siegel – Tukey and Kruskal – Wallis tests. Normaland Savage scores. Fisher exact test for 2x2 contingencytables. Mantel – Haenszel test for contingency tables.Kaplan– Meier estimator of the survival function.Jonckheere – Terpstra and page test for orderedalternatives. Nonparametric correlation coefficients(Spearman, Kendall) and measures of agreement.

MAS 655 Survey SamplingSurvey design, sampling and nonsampling errors, simplerandom sampling, stratified sampling, systematic sampling,cluster sampling, ratio estimators, regression estimators,determination of optimal sample size, bias in surveysampling, modern techniques of survey sampling.

MAS 656 Time Series AnalysisStochastic processes, weak and strong stationarity. Trendand seasonal behavior of time series. Sampleautocorrelation function and sample partial autocorrelationfunction. Prediction. Parametric families of stochasticprocesses. ARMA, ARIMA and SARIMA models.Properties, estimation and examples. ARCH and GARCHprocesses, properties of estimators and examples.

MAS 657 Statistical Analysis of Discrete DataTypes of discrete data. Contingency tables and inference(testing independence and homogeneity). Measures ofassociation. Loglinear models for contingency tables. Logitmodels. Distribution and Inference for categorical data.Asymptotic theory of goodness-of-fit X2 tests. Logisticregression.

MAS 658 Simulation and Data AnalysisIntroduction to syntax, commands, input/output files.Descriptive statistics, explanatory data analysis, regressionanalysis and analysis of variance, statistical inference(testing hypotheses, goodness of fit tests). Resampling,Simulation. Importance sampling.

MAS 659 Multivariate AnalysisRandom vectors, measures of center and variation inmultivariate moments. Multivariate normal distribution.Tests for normality. Estimation of the mean vector and thevariance analysis, independence, multivariate – covariancematrix. Wishart and Hotelling distributions. Statisticalinference. Union – Intersection Test. Confidence regions.Multivariate analysis of variance and multivariateregression analysis. Least squares method and Wilksdistribution. Analysis of covariance. Principal components,Factor analysis, Discriminant analysis, Cluster analysis.

MAS 660 Probability TheoryMeasure spaces and σ-algebras, independence, measurablefunctions and random variables, distribution functions,Lebesgue integral and expectation, convergence concepts,law of large numbers characteristic functions, central limittheorem, conditional probability, conditional expectation,martingales, central limit theorem for martingales.

MAS 661 Topics in Statistics IMAS 662 Topics in Statistics IIMAS 663 Topics in StatisticsIIITopics from probability theory, statistical theory and theirapplications, such as categorical time-series, non-parametric and semi-parametric statistics, U-statistics,Bootstrap methods, survival analysis, wavelets and theirapplications in statistics and time-series analysis, analysis ofspatial data, analysis of functional data.

MAS 664 Bayesian StatisticsSubjective probability, Bayes rule, prior and posteriordistributions, conjugate and non-informative priors,pointwise estimation and credible intervals, hypothesestesting, introduction to Bayesian decision analysis,

introduction to empirical Bayes analysis, introduction toMarkov chain Monte Carlo techniques.

MAS 665 Computational StatisticsNumerical linear algebra: Multiple regression, Choleskydecomposition, diagnostics and colinearity, principalcomponents and eigenvalue problems.Nonlinear statistical methods: Maximum likelihoodestimation, Newton-Raphson and related methods,multivariate data and the Newton Raphson method,optimization techniques (unconditional and underconstraints) EM algorithm.Numerical Integration and Approximation: Newton-Coatesmethod, spline interpolation, Monte Carlo integration,general approximation methods.Probability Density Estimation: Histogram, linear and nonlinear smoothing, splines.Bootstrap.

MAS 666 BiostatisticsDefinition of epidemiology and types of epidemiologicalstudies. Descriptive statistics: graphical and numericalmethods for medical data. Measures of association andcorrelation. Measures of risk and rate. Inference for mean,proportions indicators and coefficients of correlation.Nonparametric tests (Fisher’s exact test, McNemar test,etc.). Diagnostic methods, sensitivity and specificity.Numerical methods in clinical epidemiology, ROC curves.Meta - analysis. Censored data. Survival and hazardfunctions. Nonparametric estimation (Kaplan –Meier andNelson – Aalén estimators). Methods of comparison of twosurvival functions (Log – rank, Breslow Peto – Peto tests).Semiparametric estimation (Cox proportional hazardsmodel, partial likelihood). Parametric estimation(exponential, Weibull, log – logistic and lognormal models,proportional odds model). Frailty models.

MAS 667 Statistical ProjectThis course requires the completion of a project on aspecific statistical problem. The course gives students theopportunity to engage in applications of statisticalmethodology, to develop and cultivate their research ability,

to broaden their knowledge of statistical methodology andto become familiar with various scientific areas where thestatistical methodology is applied. This aim is achievedeither through the research projects of the faculty membersor through projects undertaken by the department forcollection and analysis of data. Moreover, the students andparticularly those wishing to enter the doctoral program,have the opportunity to familiarize themselves with theresearch interests of their academic advisor and possiblypublish original results.

MAS 668 Practical TrainingStudents are placed in organisations in the private or publicsector in order to acquire experience in topics that areclosely related to their graduate programme of studies. Atthe end of the training period, the performance of studentsis evaluated based on a written report by the managementof the host organisation.

MAS 670 Statistical Theory

Stochastic convergence, estimation, asymptotic propertiesof estimators, efficiency, testing hypotheses, asymptotic pro-perties and efficiency of testing procedures, convergencein metric spaces, stochastic processes.

MAS 671 Numerical Solution of Ordinary DifferentialEquations

One-step andmultistepmethods for initial value problems.Runge – Kutta methods. Numerical solution of two-pointboundary value problems.

MAS 672 Numerical Solution of Partial DifferentialEquations

Parabolic equations, the heat equation. Stability. The Crank– Nicolson method, ADI methods. Hyperbolic equations,the Courant – Friedrichs – Lewy condition. Ellipticequations, the Poisson equation. Iterative methods for thesolution of linear systems.

MAS 673 Finite Element Methods

Sobolov spaces. Ritz-Galrkin approximation. Variationalformulation of elliptic boundary value problems. Finite

element spaces. Polynomial approximation in Sobolevspaces. N-dimensional variational problems.

MAS 677 Topics in Numerical Analysis IMAS 678 Topics in Numerical Analysis IIMAS 679 Topics in Numerical Analysis IIITopics in Computational Mathematics and ApproximationTheory.

MAS 682 Classical MechanicsLie Groups and Lie Algebras. Equations of motion(Newton, Lagrange). Poisson structures, Integrable systems,Lax pairs, bi – Hamiltonian systems, Todu lattices.Symmetries of Differential Equations, Noether Theorem.

MAS 683 Fluid DynamicsEquations of motion. Viscous flows. Stokes flows. Non-Newtonian and viscoelastic flows.

MAS 684 Topics in Applied Mathematics IMAS 685 Topics in Applied Mathematics IIMAS 686 Topics in Applied Mathematics IIITopics from different areas of Applied Mathematics.

MAS 687 Topics in Differential Equations IMAS 688 Topics in Differential Equations IIMAS 689 Topics in Differential Equations IIITopics from Ordinary Differential Equations and PartialDifferential Equations.

Research Interests of AcademicStaff• Tasos ChristofidesProfessorU-Statistics, Probability Inequalities, Sampling, StochasticOrders.• Cleopatra ChristoforouLecturerPartial Differential Equations, Applied Analysis,Continuum Physics and Hyperbolic Systems ofConservation and Balanced Laws. Zero Viscosity Methodand Shock Waves.

• Pantelis DamianouProfessorLie Groups, Hamiltonian Systems, Differential Geometry,and Number Theory.• Konstantinos FokianosAssociate ProfessorCategorical Time Series, Semiparametric Statistics,Analysis of Spatial Data, Analysis of Large Data Seb,Bioinformatics.• Georgios GeorgiouProfessorNumerical Analysis, Numerical Solution of partialdifferential equations, Numerical simulation of Newtonianand viscoelastic flow, Hydrodynamic stability,Computational Oceanography.• Andreas KarageorghisProfessorNumerical Analysis, Computational Mathematics,Boundary and Spectral Methods for the NumericalSolution of Differential Equations.• Alexandros KaragrigoriouAssociate ProfessorStatistical Modelling, Model Selection Criteria, TimeSeries, Bio-Statistics.• Stamatis KoumandosProfessorHarmonic analysis, Orthogonal polynomials, Specialfunctions, Approximation Theory.• George KyriazisAssociate ProfessorApproximation Theory, Harmonic Analysis.• Emmanouel MilakisLecturerPartial Differential Equations, Free Boundary Problems,Geometric Measure Theory.• Christos PallikarosAssociate ProfessorGroup Representation Theory, Representations of HeckeAlgebras.



• Efstathios PaparoditisProfessorTime Series Analysis, Bootstrap Methods, MultivariateAnalysis, Non-parametric Statistics.• Evangelia SamiouAssociate ProfessorRiemannian Geometry.• Theofanis SapatinasAssociate ProfessorNon-parametric Volarity Estimation, Continuous TimeForecasting, Estimation and Inference in FunctionalMixed– Effects Models, Theory and Practice of Wavelets inStatistics and Time Series, Non-parametric Regression andInverse Problems.• Yiorgos-Socratis SmyrlisAssociate ProfessorPartial Differential Equations, Numerical Analysis, FluidDynamics.• Christodoulos SophocleousAssociate ProfessorMathematical Physics, Non-Linear Optics andNon-LinearPartial Differential Equations.• Nikos StylianopoulosAssociate ProfessorNumerical Analysis (Numerical Linear Algebra,Numerical Solution of P.D. E’s) and ComputationalComplex Analysis (Conformal Mapping, Approximationin the Complex Plane, Orthogonal Polynomials).• Alekos VidrasProfessorComplex Analysis (Multidimensional Residues, MeanPeriodicity), Carleman Formulas, Bohr phenomena.• Christos XenophontosAssociate ProfessorNumerical Analysis, Computational Mathematics,Numerical Solution of partial differential equations, FiniteElement Methods.

ContactDepartment SecretaryTel.: 22892600Fax: 22892601

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