Code 
Title 
ECTS 
Period 
Erasmus 
School 
Level 
Learning Outcomes 
Course Content (Syllabus) 
Keywords 
General Competences 
Activities 
Class ID 
0637 
Algebraic Geometry 
10 
Spring 

Mathematics 
Postgraduate 

Affine Varieties: Algebraic Sets in Αn, Affine and quasiaffine Varieties, Hilbert's Nullstellensatz, Coordinate Rings, Noether's Spaces. Projective Varieties: Algebraic Sets in Pn, Projective Nullstellensatz, Projective Closure of an Affine Variety. Morphisms of Varieties: Regular Functions, Function Fields of One Variable, Basic properties of Morphisms, Finite Morphisms, Rational maps. Product of Varieties: Product of Affine Varieties, Product of Projective Varieties, Segree's Embedding, Image of a Projective variety. Dimension: Dimension of a Topological Space, Krull's Dimension, Dimension of the Intersection of a Variety with a hypersurface, Dimension and Morphisms. Local Properties. 
Αffine Varieties  Projective varieties  Algebraic Varieties 
Adapt to new situations, Make decisions, Work autonomously, Work in teams, Work in an international context, Generate new research ideas, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Written assigments, Exams 
600180365 
0639 
Subjects Mathematical Logic 
10 
Spring 

Mathematics 
Postgraduate 

This class will cover Model Theory, Set Theory, and Computability Theory. Every time the class is offered it will cover one of the three subjects starting with Model Theory→ Set Theory→ Computability Theory→ Model Theory→ Set Theory→ ...
For 2021 the subject to be covered is Model Theory.
Below are the descriptions of all three subjects.
Model Theory
This is a Master's level introduction to Model Theory with applications from Algebra. The topics include:\
• Definable sets
• Complete Theories, Algebraically Closed Fields
• Upwards and Downwards LowenheimSkolem Theorem
• Dense linear orders and backandforth arguments
• Quantifier Elimination
• Types, Omitting Types Theorem
• Prime and Atomic Models
• Saturated and Homogeneous Models
• Vaught's TwoCardinal Theorem
• ωStable Theories and Morley's Theorem
Bibliography: D. Marker, Model Theory: An Introduction
We will cover parts of the first 6 chapters
Prerequisites: Familiarity with Mathematical Logic and Set Theory (undergraduate level courses)
Set Theory
This class is an introduction to forcing and independence proofs. We will prove the independence of the Continuum Hypothesis (plus more). Topics include:
• The axioms of ZFC
• Boolean Algebras
• Filters, Ultrafilters, and Generic Filters.
• Booleanvalued Models
• Generic Extensions
• Forcing Theorem and Generic Extension Theorem
• Countable Chain Condition and Preserving Cardinals
• Independence of the Continuum Hypothesis and Independence of the Axiom of Choice
Bibliography: T. Jech, Set Theory
In this class we will focus on chapters 14 and 15.
Prerequisites: Set Theory, Mathematical Logic (undergraduate courses)
Computability Theory
Computability Theory is a branch of Mathematics and, in particular, Mathematical Logic. The central problem in Computability Theory is the mathematical formulation of the notion of "algorithm" and "computable function". The notions of algorithm and computable function are interesting both for Mathematics, e.g. Hilbert's 10th problem, but also for Computer Science, e.g. the Halting problem.
Topics include:
• Primitive Recursive Functions and General Recursive Functions.
• Recursion and Computation, Recursive Sets
• Church Turing Thesis
• Turing machines; Turing computable functions
• Semirecursive functions and recursive enumerable sets
• Kleene's normal form
• Recursion Theorems
• Arithmetical Hierarchy
• Nondefinability of Truth
Bibliography:
1. Michael Sipser: Introduction to the Theory of Computation
3. Harry Lewis, Christos Papadimitriou: Elements of the Theory of Computation
Prerequisites: Mathematical Logic (undergraduate course) 



600180367 
0131 
Group Theory 
5.5 
Spring 

Mathematics 
Undergraduate 

Group action on sets and groups (permutation representation, Orbits, Stabilizers, Lemma OrbitStabilizer), Transitive action, Group action by conjugation (normalizer, centralizer), semidirect product of groups (Dihedral groups), Abelian groups (Free abelian group of finite rank, torsionfree abelian group, torsion abelian group), The decomposition theorem of finitely generated abelian groups (decomposables and indecomposables), Sylow’s Theorems (counting and cyclic methods), Simple groups, Groups of small order. 
Action, Sylow Theorems, Abelian groups, Series, Soluble groups 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Work autonomously, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Tutorial, Exams 
600166546 
0132 
Set Theory I 
5.5 
Spring 

Mathematics 
Undergraduate 

Axiomatic set theory ZF  Natural numbers and transitive sets  Binary
relations  Functions  Equipollent sets  Finite sets  Ordering relations  Well ordered
sets  Operations and ordering of natural numbers  Integers  rationales  reals.


Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously 
Lectures, Reading Assigment, Exams 
600166747 
0133 
Mathematical Logic I 
5.5 
Winter 

Mathematics 
Undergraduate 

Propositional and predicate calculus  Soundness and Completeness
theorems.


Make decisions, Work autonomously 
Lectures, Reading Assigment, Exams 
600166645 
0134 
Galois Theory 
5.5 
Winter 

Mathematics 
Undergraduate 

Field extensions. Prime fields. Algebraic and Transcendental extensions. Classification of simple extensions. Constructions with ruler and compass. Algebraic closure of a field. Splitting fields. Normal and Separable extensions. Finite fields. Automorphisms of fields. Galois group and Galois extension. Fundamental Theorem of Galois Theory. Applications: solvability of
algebraic equations  The fundamental theorem of Algebra  Roots of unity.

algebraic and Galois extensions, solvability, classical problems 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work autonomously, Work in teams, Appreciate diversity and multiculturality, Respect natural environment, Demonstrate social, professional and ethical commitment and sensitivity to gender issues, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Written assigments, Exams 
600166775 
0135 
Algebraic Curves 
5.5 
Winter 

Mathematics 
Undergraduate 

Polynomials of several variables  Resultant  Plane algebraic curves  Projective algebraic curves  Intersection number  Bezout's theorem  Multiplicity and intersection number  Linear systems of curves  Cubics. 

Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work autonomously, Work in an international context, Work in an interdisciplinary team, Be critical and selfcritical, Advance free, creative and causative thinking 

600179740 
0161 
Fuzzy Sets 
5 
Spring 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work in teams, Generate new research ideas, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166753 
0165 
Algebraic Number Theory 
5.5 
Spring 

Mathematics 
Undergraduate 



Apply knowledge in practice, Make decisions, Work autonomously, Work in teams, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166750 
1061 
Introduction to Meteorology and Climatolgy 
5 
Winter 

Mathematics 
Undergraduate 

Introduction to the earth's atmosphere. Atmospheric composition. Vertical distribution of temprature and pressure. Geopotential Height and weather maps. Humidity in the atmosphere, Thermodynamics of the atmospheric air. Solar and terrestrial radiation, Energy balance. Hydrological cycle, evapotranspiration, precipitation. Distribution of the climatic elements. Climates’ classifications. The climatic classification of Koppen and Thornthwaite. 
weather, climate, atmosphere, meteorology, climatology 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Work in an interdisciplinary team 
Lectures, Reading Assigment, Tutorial, Exams 
600166641 
1062 
General and Dynamic Meteorology 
5 
Spring 

Mathematics 
Undergraduate 

Ιsobaric surfaces. Air masses. Fronts. Cyclones. Anticyclones. General circulation of the atmosphere. The geotropic wind. Gradient wind. Cyclostrophic wind. Thermal wind. Tropical cyclones, hurricanes. Continuity equation. Pressure tendency equation. Absolute, relative and potential vorticity. 

Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work in an interdisciplinary team, Advance free, creative and causative thinking 
Lectures, Laboratory Work, Exams 
600166746 
1063 
Seismology 
5 
Winter 

Mathematics 
Undergraduate 

ELASTIC THEORY,STRESS AND STRAIN,RECORDING INSTRUMENTS OF EARTHQUAKES, SEISMIC WAVES AND THEIR PROPAGATION, SEISMOMETRY, MAGNITUDE AND ENERGY OF EARTHQUAKES, DISTRIBUTION OF THE SEISMIC ACTIVITY IN THE EARTH, HOW EARTHQUAKES GENERATED, GLOBAL TECTONICS, EARTHQUAKES PREDICTION, MACROSEISMIC RESULTS OF THE EARTHQUAKES 
seismicity, seismic waves propagation, seismograph, seismometer, earthquake magnitude 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work in teams, Design and manage projects, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Seminars, Laboratory Work, Exams 
600166597 
1064 
Theoretical Mechanics 
5 
Winter 

Mathematics 
Undergraduate 

Veloity and accelaration of a point mass. Newton's laws. Inertial systems. Special law of Galileo;w relativity. Universal law of gravitation. Axiom of conservation of the momentum and the angular momentum. Work and kinetic energy. Forces that are produced by a potential. The theorem of conservation of energy. Integrals of motion.
One degree of freedom systems. Limits of the motion. Harmonic oscillation. Harmonic oscillation with resistance. Forced oscillationsresonance.
Cental forces. Integrals of momentum and angular momentum. Equivalent onedimensional motion (effective potential). Differential equations of motion. Differential equations of second order with respect to time. Differental equation of second order with respect to the angle, differential equation of first order. Limits of the motion. Circular orbits. Stability of circular orbits. Attracting forces inverse proportional to the square of the distance. Equations of the obits. Escape velocity. Laws of Kepler for the motion of the planets. 
Mechanics 
Apply knowledge in practice, Work autonomously, Advance free, creative and causative thinking 
Lectures, Tutorial, Exams 
600166644 
1066 
Continuum Mechanics 
5 
Spring 

Mathematics 
Undergraduate 

Curvilinear coordinate systems. natural and inverse natural basis. Covariant and contravariant vectrors and tensors. Metric tensor. Continuous media. Euler and Lagange variables. Accelaration of a particle. Deformation tensor. Geometric aspect of the deformation tensor. Evaluation of the deformation tensor from the translation vector. Main directions of the deformation tensor. Flow lines and trajectories of particles. Circulation of the velocity. Typical velocity fields and flows.
Continuity equation. The stress vector. The stress tensor. The Euler equations of motion. Balance of the momentum. Balance of the angular momentum. Main elements of the stess tensor. Maximum shear stress. Equilibrium condititions.

Continuous media 
Apply knowledge in practice, Make decisions, Work autonomously, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166670 
0532 
Matrix Theory 
5.5 
Spring 

Mathematics 
Undergraduate 

1. Introduction 2. Canonical Forms (Invariant polynomials, elementary divisors, Smith canonical form, first and second canonical form, Jordan canonical form, applications) 3. Matrix Functions (Interpolatory polynomials, matrix components, matrix sequences and series, relations between matrix functions, applications) 4. Matrix Norms 5. Generalized Inverses (Hermite canonical form, MoorePenrose generalized inverse, solving linear systems using generalized inverses, best approximate solution, least square generalized inverse, applications) 
Canonical Forms, Matrix Functions, Matrix Norms, Generalized Inverses 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Work autonomously, Work in teams, Work in an interdisciplinary team, Generate new research ideas, Advance free, creative and causative thinking 
Lectures, Laboratory Work, Reading Assigment, Exams 
600166743 
0533 
Determinisstic Methods of Optimization 
5.5 
Spring 

Mathematics 
Undergraduate 

Introduction, Rate of convergence of an optimization method, Optimization methods for problems with univariate objective functions (methods: Cauchy, Newton, improved Newton, golden sections, inrerpolation of second degree), Optimization methods for problems with multidimensional objective functions (methods: Newton, improved Newton, conjugate directions, FletcherReeves) 
nonlinear optimization 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously 
Lectures, Reading Assigment, Exams 
600166741 
1161 
Special Topics 
5 
Spring 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Generate new research ideas, Be critical and selfcritical, Advance free, creative and causative thinking 
Reading Assigment, Written assigments, Exams 
600166667 
1162 
Special Topics 
5 
Winter 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Generate new research ideas, Be critical and selfcritical, Advance free, creative and causative thinking 
Reading Assigment, Written assigments, Exams 
600166594 
0332 
Classical Differential Geometry II 
5.5 
Spring 

Mathematics 
Undergraduate 
Upon successful completion of the course, students will be able to:
1) Use the notion of covariant derivative between vector fields and understand its functionality on a surface.
2) Compute and identify geodesic curves of a surface. They will also be able to ompute the geodesic curvature.
3) Apply the Gauss  Bonnet Theorem.
4) Distinguish between the spaces of constant curvature and their individual features.
5) Demonstrate these features on the basic models of such spaces.

Reminder of basic constructions of Differential Geometry (principal and mean curvature, Gauss curvature, normal and geodesic curvature), covariant derivative and geodesic curves, length functional, Clairaut's Theorem, local and global GaussBonnet Theorem, surfaces of constant curvature, topological structure of surfaces, Euler characteristic. 
Curvature, GaussBonnet Theorem, Euler characteristic. 
Apply knowledge in practice, Make decisions, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Generate new research ideas, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Tutorial, Exams 
600166745 
0433 
Classical Control Theory 
5.5 
Winter 

Mathematics 
Undergraduate 

Introduction to the concepts of systems signals and Automatic Control, (brief historical review, basic structure of feedback control, examples)  Mathematical concepts and tools for the study of continuous and discrete time signals and systems (Laplace transform, ztransform, applications, block diagrams and signal flow graphs) 
Classification of signals and systems. Continuous and discrete time signals and systems  Time invariance, linearity  Classical analysis of systems and control in the time and frequency domains  Linear time invariant singleinput, singleoutput systems described by ordinary, linear diferential equations  Input output relation and the transfer function description of a linear time invariant system  Free forced and total response of systems in the time domain  Stability of linear time invariant systems and algebraic stability criteria  Routh test for stability  Frequency response of linear time invariant systems  Closed loop systems  Root locus  Nyquist stability Criterion  Stabilizability and Stabilization of systems via precompensation and output feedback  Synthesis of controlers and parametrisation of stabilising controlers. 
classical control theory, feedback control systems 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Work autonomously, Work in teams, Advance free, creative and causative thinking 
Lectures, Laboratory Work, Tutorial 
600166629 
0402 
Numerical Analysis 
5.5 
Winter 

Mathematics 
Undergraduate 
After having successfully completed the course, the students will be able to:
• calculate the error in representing numbers in computer memory and in computer arithmetic
• use numerical methods to calculate the values of polynomials and the solution of equations
• perform numerical differentiation and integration
• approximate functions and estimate the approximation error 
Errors  Number systems and number representation  Floating point arithmetic – Evaluation of polynomials  Interpolation and approximation with difference methods  Interpolation by Lagrange, Newton and Hermite polynomials  Error analysis – Numerical differentiation  Numerical integration by rectangle, midpoint, trapezoid, corrected trapezoid, Simpson, Richardson and Romberg methods  Numerical solution of nonlinear equations by methods of bisection, Regulafalsi, NewtonRaphson and secant. The fixed point iteration method. Convergence criteria. 
Errors, Machine representation, Polynomials, Interpolation, Numerical Differentiation, Numerical integration, Numerical solution of equations 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Design and manage projects, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Laboratory Work, Reading Assigment, Exams 
600166628 
0431 
Computational mathematics 
5.5 
Spring 

Mathematics 
Undergraduate 
After having successfully completed the course, the students will be able to:
• approximate functions with piecewise polynomials
• solve linear systems numerically
• apply numerical methods for calculating eigenvalues and eigenvectors of matrices
• solve numerically odes and systems of odes 
Interpolation and approximation with piecewise polynomials and splines , Numerical linear algebra: Gauss elimination for linear systems pivoting, LU factorization and an introduction to the stability of systems and algorithms, norms of vectors and matrices, condition number, Iterative methods, Introduction to the numerical solution of eigenvalueeigenvector problem, Numerical solution of ODEs (existence and uniqueness of initial value problem). Euler method, Taylor method, RungeKutta methods and multistep methods. 
Piecewise polynomials, Splines, Hermite, Numerical solution of linear systems, Numerical calculation of eigenvalues and eigenvectors, Numerical solution of odes, Numerical solution of systems of odes 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Design and manage projects, Advance free, creative and causative thinking 
Lectures, Laboratory Work, Reading Assigment, Exams 
600166769 
0432 
Theoritical Informatics II 
5.5 
Spring 

Mathematics 
Undergraduate 

Complete minimization of finite automata. Contextfree grammars. Syntactic trees. Contextfree languages. Properties of contextfree languages. Relation among recognizable and contextfree languages. Pushdown automata. 
Grammars, Contextfree languages 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Generate new research ideas, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166770 
0464 
File Structures 
5 
Winter 

Mathematics 
Undergraduate 

Introduction to algorithms  Problem Complexity and its measures  Search algorithms  Selection algorithms  Sorting algorithms  Hashing  Data Structures and Operations  Arrays  Lists  Stacks  Queues  Trees  Heaps  Graphs 
Algorithms, Complexity, Data Structures 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166638 
0207 
Introduction to Real Analysis 
5.5 
Winter 

Mathematics 
Undergraduate 

Real numbers  Countable and uncountable sets  Sequences and series 
permutations of series  representations of real numbers  The Cantor set and Cantor’s
function  Special classes of functions (monotone, bounded variation, absolutely
continuous, convex)  Sequences and series of functions  uniform convergence and
applications  nowhere differentiable continuous functions  space filling curves 
equicartinuity  Azzela’sAscoli’s theorem  Weierstrass approximation theorem 
Lebesque’s measure.

real numbers, countable sets, uncountable sets, Cantor set, sequences and series functions, . 

Lectures, Reading Assigment, Exams 
600166622 
0231 
Measure Theory 
5.5 
Spring 

Mathematics 
Undergraduate 

Lebesgue measure on the real line  Measurable functions  Lebesgue
integral  Monotone and dominated convergence theorems  Comparison of integrals of
Riemann and Lebesgue  The fundamental theorem of Calculus for Lebesgue integral 
Abstract measure theory  Signed and complex measures  Product measures  Fubini’s
theorem.


Apply knowledge in practice, Make decisions, Work autonomously, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166761 
0232 
Functional Analysis 
5.5 
Spring 

Mathematics 
Undergraduate 

Metric spaces, the theorem of Baire. Normed linear spaces, Banach spaces  Finite dimensional spaces, compactness  Inner product spaces,
Hilbert spaces  Linear operators and linear functionals  Duality  The HahnBanach theorem  The principle of uniform boundedness 
The open mapping and closed graph Theorems. 


Lectures, Reading Assigment, Exams 
600166758 
0234 
Fourier Analysis 
5.5 
Spring 

Mathematics 
Undergraduate 

Trigonometric series  Fourier coefficients  Fourier series  Convergence of
Fourier series  Theorems of Dini and Dirichlet  Summability of Fourier Series  The
space of square integrable functions and Fourier series Parseval identity  Applications.
Trigonometric series  Fourier coefficients  Fourier series  Convergence of
Fourier series  Theorems of Dini and Dirichlet  Summability of Fourier Series  The
space of square integrable functions and Fourier series Parseval identity  Applications.



Lectures, Reading Assigment, Exams 
600166759 
0235 
Partial Differential Equations 
5.5 
Winter 

Mathematics 
Undergraduate 

First order Partial Differential Equations  Classification of Equations and
Characteristic  Initial Value problems  Systems of PDEs  Classification and Initial
Value Problems.


Apply knowledge in practice, Make decisions, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166633 
0506 
Stochastic Strategies 
5.5 
Winter 

Mathematics 
Undergraduate 

stochastic problems  Stochastic networks  Stochastic problems of tools
replacement and repairing  Renewal theory  Inventory.


Apply knowledge in practice, Make decisions, Work autonomously, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166624 
0534 
Mathematical Statistics 
5.5 
Winter 

Mathematics 
Undergraduate 

Distributions of functions , of random variables  The exponential family  Sufficiency of a statistic
for a parameter or for functions of parameters. The RaoBlackwel theorem 
Completeness and uniqueness  Unbiased estimators with minimum variance  The
CramerRao inequality  Efficient statistics  Consistent statistics  Maximum likelihood
and moment estimators and their properties  Prior and posterior distributions and Bayes
estimators  The minimax principle  Interval estimation. General methods for
construction of confidence intervals  Approximate confidence intervals  Confidence
regions. 
Point estimation, Interval estimation, Maximum likelihood, Unbiased estimators with minimum variance, Bayes 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work autonomously, Work in teams, Work in an interdisciplinary team, Generate new research ideas, Advance free, creative and causative thinking 
Lectures, Tutorial, Exams 
600166653 
0535 
Stochastic Operational Research 
5.5 
Winter 

Mathematics 
Undergraduate 

Renewal theory. Stochastic processes for population models. Markov Decision processes. Semi Markov processes. 
Renewal theory, Markov processes, stochastic models 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Design and manage projects, Advance free, creative and causative thinking 
Lectures, Tutorial, Exams 
600166655 
0562 
Stochastic Methods in Finance 
5 
Winter 

Mathematics 
Undergraduate 

Introduction to probability theory  rates, time value of money – Options
and derivatives  Options evaluation  Conditional mean value  Martingales  Selffinanced
processes  Brownian motion  The BlackSchool model  Stochastic differential
equations  Stochastic integration  Evaluation of the European option.

Options, derivatives, Options evaluation, Conditional mean value, Martingales, Selffinanced processes 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Design and manage projects, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166640 
0564 
Time Series 
5 
Spring 

Mathematics 
Undergraduate 

Time series characteristics, στατιοναριτυ, autocorrelation function, linear stochastic models: AR (p), MA (q), ARMA (p, q), finding the order of a linear model, nonstationary ARIMA models (p, d, q), methodology of Box & Jenkins, methods of predicting time series. 

Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Appreciate diversity and multiculturality, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Fieldwork, Reading Assigment, Exams 
600166733 
0566 
Sampling 
5 
Spring 

Mathematics 
Undergraduate 
1) They can manipulate theoretical concepts of estimators and sampling designs
1) They can design and participate in conducting sampling.
2) They may budget the size and structure of the sample.
3) Ηandle Tables of Random Numbers (TRN)

Population and Sample. Definition of Sampling and usefulness of Probability Sampling. Eστιματορσ: Basic Properties and τειρ role in the Planning of sampling. Main types of Sampling techniques: A) Simple Random Sampling (SRS), B) Stratified Sampling (StS) (various versions of), C) Systematic Sampling (SyS) (Introductory, Cyclic Law, SyS in 2dimensional populations, Best Selection of samples) D) Cluster Sampling (CluS) (equal size clusters and introduction to techniques with non equal size). Comparison of sampling methods. Applications of Sampling in Economy, Ecology and Politics. Indicators, Indices. Classic application examples from the literature and daily practice. Management of Irresolute voters electronic or not. 
Sampling. 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Respect natural environment, Demonstrate social, professional and ethical commitment and sensitivity to gender issues 
Lectures, Reading Assigment, Exams 
600166739 
0666 
Line Geometry 
10 
Winter 

Mathematics 
Postgraduate 

A. Introduction: CayleyKlein geometies and the ErlangenProgram of Felix Klein. The ndimensional affine space. and the ndimensional projectiv space.
B. Ruled surfaces: Conical, cylindrical and tangential ruled surfaces. The right helicoid, ruled surfaces of Ch. E. Catalan. Conoidal ruled surfaces. Developability condition and developable ruled suρfaces. The parameter of distribution and the striction line. The moving frame of E. Kruppa. Derivative equations of G. Sannia. Complete system of invariants. Envelope of an 1parameter family of planes. Accompanying developable ruled surfaces. Ruled surfaces of constant slope. Ruled surfaces of constant parameter of distribution. Closed ruled surfaces. Linear span.
C. Plücker's line coordinates of a straight line in P^3. The hypersurface of the second order of Plücker and the PlückerKlein mapping. Straight lines and 2dimensional generators of the Plücker hypersurface. Linear complexes of straight lines. Polarity systems. Linear congruences of straight lines.

Ruled surfaces, line geometry, Plücker's line coordinates 
Apply knowledge in practice, Make decisions, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Generate new research ideas 
Lectures, Reading Assigment, Tutorial, Written assigments, Exams 
600177925 
0136 
Number Theory 
5.5 
Spring 

Mathematics 
Undergraduate 

Arithmetic Functions, Primary Decomposition, Polynomial Congruences, Primitive Roots, Quadratic Residues, Continuous Fractions, Diophantine Equations. 
Arithmetic Functions, Primary Decomposition, Polynomial Congruences, Primitive Roots, Quadratic Residues, Continuous Fractions, Diophantine Equations. 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Generate new research ideas, Respect natural environment, Demonstrate social, professional and ethical commitment and sensitivity to gender issues, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment 
600166715 
0465 
Error Correcting Codes 
5.5 
Winter 

Mathematics 
Undergraduate 



Apply knowledge in practice, Make decisions, Work autonomously, Work in teams 
Lectures, Reading Assigment, Exams 
600166632 
0434 
Cryptography 
5.5 
Spring 

Mathematics 
Undergraduate 

Basic notions  Historical examples  The cryptosystem RC4 The cryptosystem DES  Basic computational number theory  The cryptosystem RSA  The cryptosystem of Rabin  The protocol of DiffieHellman  The cryptosystem of ElGamal  The cryptosystem of MasseyOmura  Hash functions  The digital signature RSA  ElGamal  DSA. 
Symmetric cryptography, Public key cryptography  Digital signature  Hash functions  Computational Number Theory 
Apply knowledge in practice, Adapt to new situations, Make decisions, Work autonomously, Work in teams 
Lectures, Reading Assigment, Exams 
600166763 
0570 
Theory of information and chaos 
5 
Spring 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Work in teams, Work in an international context, Generate new research ideas, Appreciate diversity and multiculturality, Respect natural environment, Demonstrate social, professional and ethical commitment and sensitivity to gender issues, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166727 
0137 
Advanced Topics in Algebra 
5.5 
Winter 

Mathematics 
Undergraduate 


Modules 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Work autonomously, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Tutorial, Exams 
600166648 
0471 
Applied Geometry 
5 
Spring 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work in teams, Generate new research ideas, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166703 
0869 
EIDIIKA THEMATA I (AIII) 
10 
Spring 

Mathematics 
Postgraduate 



Apply knowledge in practice, Make decisions, Work autonomously, Generate new research ideas, Advance free, creative and causative thinking 
Lectures, Seminars, Reading Assigment, Tutorial, Exams 
600180380 
0571 
Data Analysis 
5 
Spring 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Work in teams, Work in an interdisciplinary team, Generate new research ideas, Design and manage projects, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Laboratory Work, Reading Assigment, Exams 
600166730 
0572 
Conbinatorics and Graph Theory 
5 
Spring 

Mathematics 
Undergraduate 
1) Knowledge and use of special methods on Counting, Distributing, Partitioning and Divisioning
2) Knowledge and use of Graphs and Random Graphs on the representation of complex systems 
1. COUNTING TECHNIQUES: Product Principle, The Sum Principle, Permutations  Lists  Combinations, with and without repetition, Binomial Coefficients, Pigeonhole or Dirichlet Principle, InclusionExclusion Principle, derangements, Reflection Principle, Routs in Grids, The lexicographic permutation order.
2. SPECIAL TOPICS IN COUNTING: The Pascal’s Triangle and the Fibonacci Numbers, Diophantine equations and Partitions, Distribution Problems (Beads in Cells, Stirling's, Bell, Catalan Numbers), Generator Functions.
3. GRAPHS: Basic Concepts (order, size, connectivity, direction, neighbors, walk, path, trail, circle, complement, bipartite graphs, operations, degree, geodesics, distance, diameter, radius), Properties, Matrices, Isomorphism, line graph), Subgraphs, Paths, Trees, Factors, Bridges, Theorems of Kirchhoff, Dirac, Menger. Special Graphs (plane graphs, Euler, Hamilton, ncubes, Gray Codes, Ramsey Numbers ), Colors (basic theorems, color polynomials, algorithms).
4. INTRODUCTION TO Random Graphs: ErdösRényi Networks (Degree Distribution, Average degree, Giant Component, Average Distance, Clustering Coefficient, Transitivity), Introduction to Small World and Scale Free Networks, Introduction to Real Networks, , Centrality: degree, eigenvalue, closeness, betweenness), Examples Using the R Language (Collaboration Networks, Social, Financial, Online, etc.) 
combinatorics, graphs, random graphs 
Apply knowledge in practice, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team 
Lectures, Reading Assigment, Exams 
600166729 
0531 
Statistical Learning and knowledge processing 
5.5 
Winter 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work autonomously, Work in teams, Work in an interdisciplinary team 
Lectures, Laboratory Work, Reading Assigment, Exams 
600166652 
0307 
INTRODUCTION TO GEOMETRY II 
5.5 
Spring 

Mathematics 
Undergraduate 





600158854 
0308 
GEOMETRY AND GROUPS 
5.5 
Winter 

Mathematics 
Undergraduate 
Students taking this course will not only aquire further mathematical knowledge, but they will deepen and better understand the interrelations between the Algebra and Geometry courses they took in the first two years of the undergraduate program. In addition, they will familiarize with the interdisciplinary approach to mathematical problems and enforce their mathematical perception. The techniques used in the course are fundamental for many mathematical disciplines but are also used extensively for many applications of any kind of motion simulation and graphic design. 
(Note: Special attention will be given to the cases n=2,3.)
The group Aff(n). Short reminder on the isometries of the plane and space. The group ISO(n). Subgroups of Isometries (discrete, finite, fixed point). Circle and the group SO(2). Spherical geometry (spherical coordinates, triangles, great circles). Isometries of the sphere, the groups O(3), SO(3). Stereographic projection, real projective line, Mobius tranformations. SL(2,R) and RP(1), the group PSL(2,R). Complex projective line, SL(2,C) and action on CP(1), Riemann sphere, the group PSL(2,C). Hyperbolic plane, real projective plane and SL(3,R). 

Work autonomously, Work in teams 
Lectures, Reading Assigment, Exams 
600166776 
0508 
STOCHASTIC PROCESSES 
6 
Spring 

Mathematics 
Undergraduate 



Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Make decisions, Work autonomously, Design and manage projects, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams, Other / Others 
600158852 
0102Α 
Introduction to Algebra 
6 
Winter 

Mathematics 
Undergraduate 
Upon successful completion of the course, students should have
 Understand the basic concepts of Mathematical Structures
 Understand the basic concepts of Number Theory
 solves Algebra's computational and theoretical problems
 solves computational and theoretical problems of Number Theory

Sets, Functions. Equivalence relations and order relations. Operations in a set.
The set of natural numbers. Mathematical Induction. Principle of good order.
Countable sets. Newton's identities.
Groups, Ring Bodies: Definitions and Examples. The ring of integers.
Divisibility. Prime numbers. The Euclidean Algorithm. GCD, LCM. Fundamental
theorem of number theory. The ring of modn congruences. The
Zp field. Linear congruences. Multiplicative functions. 
Sets, functions, relations, natural numbers, mathematical induction, divisibility, linear congruences, multiplicative functions. 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work autonomously, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Tutorial, Exams 
600166613 
0430Α 
Introduction to Computer Programming 
6 
Winter 

Mathematics 
Undergraduate 
• Learning basic syntax and semantic rules in a high level programming language like Fortran 90/95/2003 or C + +.
• Empasis is given to the development of algorithms for the solution of basic mathematical problems. 
The objective of this course is to teach basic programming principles in one of the programming languages Fortran 90/95/2003 or C++.
Computer hardware  Computer software  Programming languages  An introduction to problem solving with Fortran 90/95 or C++  The structure of a program  Simple input and output  Control structures  Iterations  Array processing (one dimensional and multidimensional matrices)  Functions  Subroutines  Modules  IMSL libraries  File organization (sequential files, direct access files) Applications to mathematical problems. 
computer programming 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Work autonomously, Work in teams, Work in an interdisciplinary team, Design and manage projects, Be critical and selfcritical 
Lectures, Laboratory Work, Reading Assigment, Exams 
600166605 
0106Α 
Algebraic Structures I 
6 
Winter 

Mathematics 
Undergraduate 

Groups, Subgroups, Group generated by a set, Homomorphisms of groups, Permutation groups, Lagrange Theorem, Order of a group and order of an element, Euler's Theorem, Fermat's Theorem, Wilson's Theorem and their applications in arithmetics, Normal subgroups, Quotient group, Isomorphism theorems, Cyclic groups, classification of cyclic groups and their applications (Primitive roots (mod n)), product of subgroups, Direct product of groups. 
Groups, Quotient group, Isomorphism theorems, Cyclic groups, Classification 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Work autonomously, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Tutorial, Exams 
600166646 
0203Α 
Calculus III 
6 
Winter 

Mathematics 
Undergraduate 

Functions of several variables  Limits and continuity  Partial derivatives 
Differentiation of scalar and vector functions  The chain rule  Higher order partial
derivatives  Directional derivatives  Taylor’s formula  Extremes of real valued
functions  Lagrange multipliers  The implicit function theorem and the inverse function
theorem.

functions of several variables, partial derivatives, Taylor's formula, Lagrange multipliers. 
Apply knowledge in practice, Make decisions, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166630 
0204Α 
Topology of Metric Spaces 
6 
Winter 

Mathematics 
Undergraduate 

Basic notions of the Set Theory  Metric spaces  Topology of metric spaces
 Convergence of sequences  Continuous functions  Compactness and Connectedness of
metric spaces.
Description: Texts of Algebra, Analysis, Geometry, Set Theory and Computer Science.
Instructors: N. Anastasiou, F. Kapetaniou, M. Papageorgiou.
Algebraic Structures 
metric space, topology of metric space. 
Apply knowledge in practice, Retrieve, analyse and synthesise data and information, with the use of necessary technologies 
Lectures, Reading Assigment, Exams 
600166599 
0502Α 
Probability Theory I 
6 
Winter 

Mathematics 
Undergraduate 
1. Acquaintance with the ProbabilisticStochastic Thought.
2. know how to use the combinational analysis methods in solving problems of probabilities.
3. use conditional probabilities, total probability, Bayes rule, Poincare theorem, product law and apply them to probability problems.
4. know the notion of distribution function, probability function and probability density function, how to calculate them for discrete and continuous random variables and how to manipulate functions of random variables,
5. can calculate parameters of distributions (mean, variance and other moments), calculate and manipulate probability generator function and moment generator function,
6. know and use basic univariate discrete distributions: uniform, Bernoulli, binomial, Poisson, geometric, hypergeometric, and continuous distributions: uniform, exponential, normal, gamma, betta and trinomial bivariate distribution

Historical problems. Randomnes, the sample distribution space, events, Venn diagrams. Classical definition of mathematical probability, statistical regularity, axiomatic foundation of probability  Finite sample distribution spaces, combinatorics, geometric probabilities  Conditional probability, independence  Univariate random variables, distribution functions, function of a random variable, moments, momentgenerating function, probability generating function, discrete bivariate distributions  Useful univariate distributions: Discrete (Bernouli, Binomial, Hypergeometric, Geometric, Negative Binomial, Poisson), Continuous (Uniform, Normal, Exponential, Gamma)  Applications. 
probability, random variables, distribution functions, momentgenerating andprobability generating functions 
Apply knowledge in practice 
Lectures, Reading Assigment, Exams 
600166637 
0205Α 
Calculus IV 
6 
Spring 

Mathematics 
Undergraduate 

Multiple integrals  Line integrals  Surface integrals  The integral
theorems of Vector Analysis.


Apply knowledge in practice, Adapt to new situations, Make decisions, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166754 
0505Α 
Probability Theory II 
6 
Spring 

Mathematics 
Undergraduate 
Upon successful completion of the course students will be able to:
1. calculate the marginal random variables of a multidimensional random variable;
2. calculate the conditional random variables of a multidimensional random variable;
3. calculate moments of multidimensional random variables;
4. calculate and use the moment generator function of a multivariate random variable;
5. use/apply the Central Limit Theorem. 
The algebra of events  Probability Space  The axioms of Probability 
Random variables  The notion of stochastic distribution  Multidimensional random
variables  Multidimensional distribution functions  Marginal distributions 
Denumerable multidimensional random variables  Continuous multidimensional
distributions  Multidimensional normal distribution  Stochastic independence 
Conditional Probability  Conditional density  Conditional distributions  Mean values
for multidimensional random variables  Conditional mean values  Regression line 
Mean square error  Random variable transforms  Compound distributions  Inequalities
 Multiple Correlation coefficient  Ordered random variables  Characteristic functions 
The sum of independent random variables  Characteristic functions of multidimensional
random variables  Moment generating functions  Probability generating functions 
Limit theory of random variables  Convergences  Relations between convergences 
Central Limit Theorem  Laws of large numbers  The log log law. 
Multidimensional random variables, Conditional Probability, Random variable transform, Convergences, Central Limit Theorem 
Apply knowledge in practice, Adapt to new situations, Make decisions, Work autonomously, Work in an international context, Generate new research ideas, Appreciate diversity and multiculturality, Respect natural environment, Demonstrate social, professional and ethical commitment and sensitivity to gender issues, Be critical and selfcritical, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Tutorial, Exams 
600166740 
0303Α 
Classical Differential Geometry I 
6.5 
Winter 

Mathematics 
Undergraduate 
Upon successful completion of the course, students will:
1) Be able to recognize parametrizations of standard curves and surfaces.
2) Be able to check whether a curve is given with its natural parameter and if not, how to reparametrize it so.
3) Be able to calculate the basic geometric data, like the curvature, of a plane curve.
4) Be able to calculate the basic geometric data, like the curvature and torsion of a space curve.
5) Be able to classify space curves and manipulate the Frenet frame of a space curve.
6) Be able to check whether a parametrization corresponds to a smooth surface and compute its tangent plane.
7) Be able to compute distance between two points on a surface, length of a curve, and angle of two surface curves.
8) Be able to manipulate the basic geometric data of a surface like the normal vector and orientation, the Gauss map and shape operator, the first and second fundamental form, Gauss curvature, and mean curvature.
9) Be able to check which analytical data correspond to smooth surfaces (and in which way). 
Theory of Curves: The concept of the curve in the differential geometry. The moving frame. The Frenet formulae. The fundamental theorem (existence and uniqueness). Osculating cycle. Plane curves.
Theory of surfaces: The concept of surface in differential geometry. Curves on a surface. The first and the second fundamental form. Gauss, mean curvature and principal curvatures. Christoffel symbols. The Gauss map and equations of Gauss and Weingarten. Theorema Egregium of Gauss. The fundamental theorem (existence and uniqueness). 
Theory of curves, theory of surfaces 
Apply knowledge in practice, Make decisions, Work autonomously, Work in teams, Work in an international context, Work in an interdisciplinary team, Generate new research ideas, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Tutorial, Exams 
600166650 
0208Α 
Complex Analysis 
6 
Spring 

Mathematics 
Undergraduate 

Complex numbers, the complex plane, topology of the plane, elementary
complex functions  Holomorphic functions, CauchyRiemann equations  The complex
integral, Cauchy's theorem and integral formula  The maximum principle, theorems of
Morera and Liouville, the Schwarz lemma  Power series, the identity theorem.  Laurent
series, singularities, residues.


Retrieve, analyse and synthesise data and information, with the use of necessary technologies, Adapt to new situations, Make decisions, Work in teams, Advance free, creative and causative thinking 
Lectures, Reading Assigment, Exams 
600166683 
0569Α 
Statistical Inference 
5.5 
Spring 

Mathematics 
Undergraduate 
Upon successful completion of the course students will:
1. be familiar with the theoretical background of case tests;
2. be able to apply NeymannPearson's fundamental lemma to construct hypothesis testing;
3. be able for constructing Uniformly Strong Hypothesis Tests;
4. be familiar with the monotone likelihood ratio property and the generalized likelihood ratio;
5. become familiar with the relationship between confidence intervals and hypothesis tests and with the theoretical background of the hypothesis tests in the general linear model and the analysis of variance. 
Introduction to testing hypothesis  Selecting the test procedure  Testing
simple hypothesis  NeymanPearson’s fundamental lemma  Uniformly most powerful
tests  Tests for the parameters of one or two normal populations  Likelihood ratio tests. Tests for the parameters of the general linear model. 
hypothesis, tests, NeymanPearson’s lemma, Likelihood ratio tests 
Apply knowledge in practice, Make decisions, Work autonomously, Work in teams, Generate new research ideas, Advance free, creative and causative thinking 
Lectures, Tutorial, Exams 
600166726 