A. GENERAL INFORMATION *

I. THE INSTITUTION *

II. PRACTICAL INFORMATION *

B. Study Guide *

I. UNDERGRADUATE MATHEMATICS PROGRAM *

II. THE POSTGRADUATE PROGRAM *

1. THE INSTITUTION

The exact dates are announced at the end of the previous academic year:

End of September Beginning of winter semester

End of December End of winter semester

January Mid- year examinations

Beginning of February Beginning of spring semester

Middle of May End of spring semester

May - June Examinations

September Re-sits

1. Student application form and learning agreement (provided by

the Home University).

2. Two passport photographs.
3. Verified transcript of academic record.

Ms Eva Michelidaki: Institutional coordinator,

Tel: +30 81-393177, Fax +30 81-210073

E-mail: michel@admin.uch.gr

Ms Anastasia Diakatou: Student exchanges,

Tel: +30 81-393180, Fax +30 81-210073

E-mail: dioni@admin.uch.gr

Ms Sophia Toufexi: Student grants, administration

Tel: +30 81-393176, Fax: +30 81-210073,

E-mail: toufex@admin.uch.gr

The permit is issued:

1. UNDERGRADUATE MATHEMATICS PROGRAM

1. PROGRAM OF STUDIES

The department of Mathematics offers courses, which are divided into two groups:

• Group 1 includes the obligatory courses. The students are expected to attend these courses during the first two years of their study.
• Group 2 comprises the optional courses.

Students may follow courses offered by other departments as well. A number of those may count towards the units necessary for the acquisition of the degree. During the last semesters, students are able to attend postgraduate courses.

1. THE COURSES

In the table below the corresponding columns represent the following:

1. The course number
2. The title of a course
3. Academic Credit Units (A.C.U)
4. Hours of lectures
5. Lab or tutorial hours
6. ECTS Units
7. Course prerequisites
8. Courses strongly recommended to have been previously attended

Each hour of course attendance per week corresponds to 1 Academic Credit Unit, while one to three lab or tutorial hours correspond to 1 Academic Credit Unit.

With the name 'Topics….' are described the courses, which are not included in the study guide. Each semester the courses of this category are announced as well as their content, the hours of attendance and the course prerequisites.

1. DEGREE ACQUISITION

In order to obtain the degree in Mathematics the students should:

a) attend courses for at least 8 semesters

2. succeed in all the obligatory courses
3. succeed in at least two courses of each of the subgroups 2.1, 2.2, 2.3
4. collect at least 120 A.C.U.s of which:

1. At least 90 A.C.U.s should be collected from courses offered by the department of Mathematics or courses of mathematical content offered by other departments of the School of Sciences or the department of Economics.
2. At least 105 A.C.U.s should be collected from courses offered by the School of Sciences or by the department of Economics.

6 of the A.C.U.s mentioned in d.i may be obtained by writing a diploma thesis.

1. RECOGNITION OF COURSES OFFERED BY OTHER DEPARTMENTS

Courses taken from other departments of the University of Crete may be recognized, after authorization from the board of studies of the department.

Interested students may submit an application form to the board of studies, requesting approval two weeks (the latest) after the beginning of the semester in the department where the particular course is being taught.

 

2. DIPLOMA THESIS

Students may actively participate in research under the supervision of a suitable professor and present their results in the form of a diploma thesis. The diploma thesis is optional and counts for 6 A.C.U.s. The credits are approved after a final public presentation of the results in the form of a seminar, and grades allocated by a three-member committee, which includes the student's diploma supervisor. The diploma thesis can only be assigned at the beginning of the semester.

Students may be assigned a diploma thesis only if:

• They have succeeded in all group 1 courses
• They have succeeded in two courses at least of group 2, which are relative to the subject of the diploma thesis.

1. DESCRIPTION OF COURSES

M100 Analytical Geometry - Complex Numbers

• Vector algebra and analytical geometry in dimensions 2 and 3. Lines, planes, conic sections. Surfaces of degree 2. Cylinders, cones, surfaces of revolution.
• Familiarisation with the algebra and geometry of complex numbers. Roots of unity. Simple conformal transformations.

• Sets, paradoxes, operations with sets.
• Relations, equivalence and order relations.
• Functions. Injenctive, surjective and bijective functions. Composition. Inverse functions.
• Logical propositions. Quantifiers. Mathematical proof.
• Peano axioms. Arithmetic and ordering of natural numbers. Principle of induction. Well ordering principle.
• Cardinal numbers. Finite, countable, uncountable sets. Cantor's diagonal argument.
• Abstract algebraic structures.
• Combinatorics. Counting. Sampling with or without replacement or ordering. Allocations.

• Sequences. Limit of a sequence. Subsequences.
• Elementary functions. Limit of a function. Continuity.
• Differentiation. Mean value theorem. Applications. Derivatives of higher order. Applications.
• Power series. Taylor series of elementary functions.
• Definite integral of a continuous function. Numerical integration. Indefinite integral. Applications. Generalised integrals.

• Curves in R² and R³. Curvature, length.
• Functions of several variables. Continuity.
• Partial derivatives. Directional derivative.
• Tangent plane and normal vector to the graph.
• Mean value theorem. Chain rule.
• Higher order partial derivatives.
• Minima and maxima of a function of several variables. Lagrange multipliers.
• Implicit functions.
• Double integral. Jacobean. Change of variables.
• Polar co-ordinates.
• Triple integral. Spherical and cylindrical co-ordinates. Applications.

• Ordinary differential equations. First order equations. Methods of solution. Applications.
• Second order equations. Methods of solution. Applications.
• Systems of first order equations.
• Curvilinear integral. Theorem of Green on the plane. Applications.
• Parametric representation of surfaces. Surface integrals. Theorem of Green-Grauss, theorem of Stokes. Applications.

• Vector spaces, linear independence, Bases.
• Linear homomorphisms.
• Matrices. The matrix of a homomorphism. Change of basis.
• Systems of linear equations. Gaussian elimination.
• Determinants. Computation. Applications.
• Euclidean spaces. Orthonormal bases.
• Eigenvalues, eigenvectors. Diagonalisation.

• The real numbers as a complete ordered field.
• Sequences. Limit of a sequence. Monotonic sequences. Implications of completeness. Points of accumulation. Limsup, liminf.
• Continuity of a function. Intermediate value theorem. Existence of extrema. Uniform continuity.
• Exponential and logarithmic functions.
• Riemann integral for bounded functions.
• Differentiation of composite and inverse functions. Mean value theorem. Rules of de l’ Hospital. Convex and concave functions.

• Topology of the real line. Metric spaces. Compactness.
• Sequences of functions. Pointwise and uniform convergence.
• Continuity, integrability, differentiability.
• Dini theorem. Stone-Weierstrass Theorem.
• Series of functions. Convergence of power series.

• Commutative rings. Integral domains.
• The Integers. Order properties.
• Divisibility. Euclidean Geometry.
• Congruences. The rings Zn.
• Fields. Construction of the rationals.
• Polynomials. Zero divisors.
• Uniqueness of factorization.
• Groups. Cyclical groups. Subgroups.
• Lagrange theorem. Permutation groups.
• Homomorphisms. Conjugate elements.
• Quotient groups.

The purpose of the course is the development of the elements of probability theory, with emphasis on random variables and calculation of probabilities.

The axioms of probability theory, independence, sequences of independent trials, conditional probability, Bayes rule.

Real random variables, distribution function, probability mass function, density function, moments, moment generating function, functions of a random variable, inequalities.

Random vectors and generalisation of the above concepts.

Covariance, independence, sums of independent random variables, ordered random variables.

Conditional distributions and moments, a priori and a posteriori distributions.

Basic modes of convergence of sequences of random variables, law of large numbers, Central Limit Theorem (heuristics).

• Propositional Calculus: Tautologies, formal proofs, completeness, adequate sets of connectives.
• Predicate calculus: logical implications, formal proofs, completeness.
• First order theories.
• Elimination of quantifiers.
• Elements of model theory.

• Euclid’s axioms. Hilbert’s axioms. Compatibility.
• Absolute geometry. Euclidean geometry. Basic results. Conical sections.
• Pencils of circles. Spherical geometry. Projective geometry.
• Hyperbolic geometry. Hyperbolic distance, angle of parallelism.
• Geodesics, circles. Hyperbolic area.

• Integers and rationals. Number theoretic functions. Euler and Mobius functions.
• Linear congruences. Algebraic congruences.
• Primitive roots. Indices.
• The symbols of Legendre and Jacobi.
• Diophantine equations.

• Egyptian and Babylonian mathematics.
• Greek mathematics. Thales, Pythagoras, the famous problems of the ancient Greek mathematics.
• The elements of Euclid, after Euclid (Apollonius, Archimides…)
• Synopsis of the history of mathematics after the Hellenistic period.

• Education and mathematics.
• Curriculum. Textbooks.
• Evaluation of pupils. Examinations.
• Teaching practices.

• Introduction to Combinatorics, graphs, trees and networks.
• Elements of set theory, maps, induction, algorithms, recurrence relations.
• Basic principles of combinatorics, permutations, combinations, combinatorial identities, matching problems.
• Graphs, paths, cycles: properties and applications.
• Kinds of trees: properties and applications, models of networks.
• Boolean algebra, propositional calculus.

• The revival of the Greek mathematics in the centuries after Christ. Diofantus, Ptolemaeus, Pappus, Proklus.
• Short survey of Chinese and Indian mathematics.
• Arabian mathematics and Middle Ages.
• Mathematics of Renaissance, especially after Cardano, Tartaglia and Ferrari.
• Beginning of contemporary mathematics: Viete, Napier, Briggs, Galilei, Kepler, Cavalieri.
• The era of Fermat and Descartes.
• Several topics concerning the forerunners of Calculus, Newton, Leibnitz, Bernoulli, Euler, Lagrange, Gauss, Cauchy etc

• Books 1-6 and 11-13 from Euclid’s elements with addition of some more recent results.
• Brief survey of the attempts to prove Euclid’s postulate.

• The idea of the computable function.
• Turing’s formalization. Kleene’s formalization. Other formalizations.
• Church’s thesis. Recursive sets. Recursively enumerable sets and their characterizations.
• Coding finite sequences. Universal functions.
• The recursion theorem.
• Turing reduction, degrees of unsolvability.

• Talks are given on several topics with the goal to bring the first year student to a first contact with problems, concerning his studies, his professional work with mathematics or his research.

Functions of bounded variation. Riemann-Stieltjes integral. Measure and Lebesgue integral in R: definitions, basic properties, convergence theorems, comparison to Riemann integral.

Complex numbers. Roots. Analytic functions, Cauchy-Riemann conditions, harmonic functions. Elementary functions. Contour integrals. Theorems of Cauchy-Goursat, Morera, Liouville. Fundamental theorem of algebra. Taylor and Laurent series. Isolated singularities. Roots of analytic functions. Residues. Argument principle. The theorem of Rouche.

Local existence. Uniqueness of local and global solutions. Extension, blow up of solutions. Dependence on parameters.

Systems of equations.

Boundary value problems. Comparison theorems. Maximum principle. The theory of Sturm-Liouville. Stability of linear and nonlinear systems.

• Sturm-Liouville problems.
• Basic problems of classical PDE’s.
• Orthogonality, L² spaces, Fourier series.
• Heat equation, Laplace equation, Wave equation.
• Fourier transform.

• Curves in R³. The frame of Frenet.
• Surfaces in R³. Definitions and basic concepts. Differentiable mappings. Orientable surfaces. The first fundamental form.
• Differentiation of vector fields. Shape operator and second fundamental form. Curvature.
• Intrinsic geometry of surfaces. Isometries. Theorema Egregium.

Geometry in Rⁿ and inner product spaces. Hilbert spaces with emphasis on the geometric side of the theory and the role of completeness. Normed spaces and Banach spaces. Applications: fixed point, approximation, Hahn-Banach theorem.

Differentiability of functions of several variables. Inverse function theorem and implicit function theorem. Derivatives of higher degree. Change of variable in multiple integrals. Differential forms. The general theorem of Stokes.

• The symmetric group. Groups, subgroups.
• Lagrange’s theorem. Cyclic groups. Normal subgroups. The alternating group. Solvable groups.

• Selection from the topics: Sylow theorems, abelian groups and representation theory.

• Rings. Subrings.
• Ideals. Prime and maximal ideals. Euclidean rings. Principal Ideal Domains.
• Unique Factorization Domains. Modules. Submodules. Quotient modules.
• Morphisms and direct sums. Torsion and free modules. The Decomposition theorem.

• The notion of the group, ring and algebra.
• The algebra of polynomials. The algebra Hom (V, V).
• Cyclic subspaces of a vector space with respect to a linear map. Decomposition in cyclic subspaces of Hom (V, V).
• The Jordan forms. The Cayley-Hamilton theorem.
• Euclidean spaces. Unitary and Symplectic spaces.

• Metric spaces. Continuous maps. Topological spaces. Compact and connected spaces.

• Tychonoff’s theorem. Compactness in metric spaces.
• The separation property. Hausdorff spaces.
• Urysohn’s lemma. Homotopy. The fixed point theorem.

• The algebra of sets, relations and functions.
• Construction of the set of natural numbers. Ordered numbers and their arithmetic.
• The axiom of choice. Cardinalities and their arithmetic.

• Polyhedral and simplicial complexes. Orientation. Chains, cycles and boundaries.
• Homology groups. The Euler-Poincare theorem.
• Continuous maps. Induced maps on the homology groups.
• Brower’s fixed point theorem. Homotopic paths. The fundamental group. The relation between the homology and the fundamental group.
• Selection from topics: Covering maps, higher homotopy groups, relative homology, exact sequences.

• Finite field extensions. Algebraic numbers.
• Constructions with ruler and compass.
• The field of roots of a polynomial. The Galois group of a finite extension. The fundamental theorem of Galois theory.
• Criterion for solvability of an algebraic equation. Solving equations with radicals and the non-solvability of the generic equation.

• The fundamental theorem of linear programming.
• Duality theorem of linear programming. Simplex methods.
• Optimization with no constraints. Necessary and sufficient conditions (in Rn) for local extrema.
• Optimization of convex functionals.
• Optimization under constraints (Langrange multipliers, Kuhn-Tucker conditions, duality, sensitivity analysis)

• Introduction (floating point number systems, rounding errors)
• Numerical solution of nonlinear equations.
• Numerical integration.
• Numerical solution of linear and nonlinear systems of equations.
• Interpolation and approximation.

• Introduction to the mathematical models in Mathematical Physics.
• Examples from elasticity theory, optics, electromagnetics, continuous mechanics, heat conductivity etc.

• Various modes of stochastic convergence and their relationship. Slutsky’s theorem and variance stabilizing theorem.
• Parametric statistical models, statistical samples, statistics, sufficiency, completeness, losses risks, efficiencies.
• Estimation: Parameter space, method of moments, maximum likelihood estimates, least squares estimates, Bayes estimates and UMVU estimates. Gramer - Frechet - Ra inequality, efficiency, asymptotics, confidence intervals.
• Testing: Hypotheses, size, power and p-value, Neyman - Pearson tests, likelihood ratio tests, asymptotics, connection between tests and estimates, classical tests for normal populations, goodness of fit tests, linear regression.

• Finite difference methods for two-point problems.
• Finite difference methods for Poisson's equation.
• Finite difference methods for initial and boundary value problems, for parabolic, hyperbolic, elliptic equations.

• Numerical solution of initial value problems for ODE’s: Euler, Runge-Kutta, multistep methods.
• Consistency, stability, convergence.
• Finite difference and Galerkin methods for the two-point boundary value problem.
• Introduction to the numerical solution of PDE’s.

• Norms of vectors and matrices.
• Condition number of a matrix and its significance to the numerical solution of linear systems with Gaussian elimination.
• Iterative methods. Eigenvalue problem. Systems with sparse matrices. Linear least squares problems.

• Optimal approximations. Existence and uniqueness.
• Computation of optimal approximations in Euclidean spaces.
• Canonical equations.- Fourier expansions- Orthogonal polynomials.
• Uniform approximation: characterization of optimal uniform approximations and computation via the Remez - methods.
• Interpolation in one and two dimensions.
• Interpretation with splines. Approximation-properties of splines.
• Numerical integration: Newton - Cotes, Romberg, Gauss methods.

• Normal samples and related distributions.
• Linear Models- estimation and testing. Analysis of Variance. Use of statistical packages.
• Methods of summarizing data, graphs and plots, tests of Normality, data transforms, model estimation.
• Diagnostic statistics.
• Examples from Biology, Medicine, Econometrics etc

• Examples of simple types of stochastic processes, classification of processes, sample paths, finite dimensional distributions, stationarity, ergodicity.
• Discrete time, Markov chains: transition probabilities, classification of states, periodicity, ergodicity, absorbion.
• Continuous time, Markov chains: birth and death processes, homogeneous Poisson processes, arrival times, stopping times, compound Poisson, non-homogeneous Poisson, limit theorems.
• Martingales, inequalities, convergence.
• Renewal processes: renewal function, renewal equations, renewal theorems, limit theorems. Selected topics from diffusion processes, branching processes, ques.

1. THE POSTGRADUATE PROGRAM

The department of Mathematics offers the following postgraduate programs:

• M.Sc. in Theoretical Mathematics

• M.Sc. in Applied Mathematics

• M.Sc. in Statistics

• M.Sc. in Applied Mathematics and Statistics.

• Ph.D. in Mathematics