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# Department of Mathematics

**Secretary: **Agapiadis Anestis

**Secretary's Office:
Tel.: ** (+30) 26510-07190, 07428, 07492

**Fax:**(+30) 26510-07005

**E-mail:**

**Website:**http://www.math.uoi.gr/

## General - Aim of the Department

Mathematics, which in its initial development stage mainly constituted a series of empirical rules for mathematical calculations, has today become a necessity in our lives, decisively and rapidly penetrating every contemporary field of scientific activity. The Mathematical Science is mainly characterised by the method of the proof and the search for and description of mathematical concepts and laws, which are necessary in the description of contemporary reality. The two main directions in mathematics are theoretical and applied mathematics. Theoretical mathematicians aim at the best, strictest and most efficient foundation and advancement of mathematical theories. Applied mathematicians attempt to create and apply advanced mathematical methods in order to study the various phenomena that interest them.

## Structure of the Department - Sections

The Department of Mathematics covers subject areas of mathematical science and is subdivided into four Sections: the Mathematical Analysis Section, the Algebra and Geometry Section, the Probability,

Statistics and Operations Research Section and the Applied Mathematics and Engineering Research Section.

**1. Mathematical Analysis Section**

Mathematical Analysis is the subject of the Mathematical Analysis Section and is one of the most extensive and profound branches of Mathematics. Although it is harder today to demarcate this branch than in the past, one could say that Mathematical Analysis begins with the introduction of the concept of the "limit" and the subsequent infinitesimal analytic method, and further expands radially and inexhaustibly in all directions. The mission of the Mathematical Analysis Section is the initiation of all students in the concepts and methods of Mathematical Analysis and at the same time the cultivation and growth of knowledge in the field through the quest of new ideas and methods.

An invaluable contribution of Mathematical Analysis is the supply of creative and effective tools to other fields of Mathematics, from purely theoretical to completely applied fields. Some of the basic and interdependent directions of Mathematical Analysis are the Theory of Real Functions, the Theory of Complex Functions, Topology, Differential Equations, the Theory of Measure and Integration, Functional Analysis, etc.

The exact study of a physical or mechanical and generally of a dynamical system, which describes the development of a phenomenon or the control of a certain population situation, can take place through continuous or Discrete Differential Equations.Such equations can provide information that refers to the general behaviour of solutions, as for example is the description and ascertainment of stability, approximation, periodicity, etc.

As is natural, the closer the theoretical model is to the natural phenomenon, the closer we come to its exact study through the model. For example, we will have a better approach to reality if we take into consideration the phenomenon's history. Thus, we come to the so-called Delay Differential Equations, which constitute an extensive and rather complex class of Functional Differential Equations. In this general case, the study is carried out by examining the convergence of the paths of abstract systems that are observed in general topological spaces. The study of such spaces, which facilitates comprehension of natural problems, is the subject of Functional Analysis, Topology and the Measure Theory.

**2. Algebra and Geometry Section**

The Algebra and Geometry Section includes the following fields of Mathematics: Abstract Algebra, Differential Geometry, Number Theory, Mathematical Logic, Differential and Algebraic Topology, Algebraic Geometry, etc.

Algebra developed mainly in the 19th and 20th centuries and its aim was the solution of specific problems in Geometry, Number Theory and the Theory of Algebraic Equations. It also contributed to a better understanding of the existing solutions to such problems. Today, Algebra's contribution to other sciences, such as that of Computer Science, is invaluable.

Differential Geometry constitutes one of the main branches of mathematics and deals with the study of metric concepts on manifolds, such as metrics and curvature. The classic period of Differential Geometry was the 19th century, during which the local theory of curves and surfaces - now known as elementary Differential Geometry - developed as an application of Infinitesimal Calculus. In the 20th century the field developed rapidly, based on the recent achievements of the theory of Partial Differential Equations, Algebraic Topology and Algebraic Geometry. The dynamics and fruitfulness of Differential Geometry is also a result of its interaction with other sciences, such as Physics (Theory of Relativity), etc.

**3. Probability, Statistics and Operations Research Section**

Probability, Statistics and Operations Research constitute the area of research of the third Section of the Department of Mathematics. Probability and Statistics is the branch of Mathematics which is concerned with the concept of uncertainty (probability), the design of experiments and sampling methods, the collection and analysis of measurements (numerical data) and the extraction of inferences. It also deals with the study of random phenomena, the development of stochastic models for the purpose of describing various natural, social, biological and other phenomena, and generally with the theory and applications of stochastic processes. Subjects such as opinion polls, demographic surveys, quality control, sampling surveys, clinical trials, retrospective and prospective medical studies, etc. belong to the field of Probability and Statistics.

Operations Research is the field of Mathematics that deals with the optimisation of multivariate functions under various types of constraints and the study of stochastic systems like queuing systems, inventory control, human resource systems, population models, etc. Operations Research has its roots in theoretical mathematics and finds applications in all areas of human activity where problems of modelling and optimisation occur. The faculty of the Section are also interested in the study and understanding of the applications of their science to the problems of medicine, chemistry, agriculture, psychology and education.

**4. Applied Mathematics and Engineering Research Section**

The research interests of the fuculty of the fourth Section are directed towards the scopes of Mechanics, Computational Mathematics and Informatics.

**Mechanics **is the oldest branch of Applied Mathematics, since it developed at the same time and following intense interaction with Classical Analysis and very often by the same researchers. For many years it constituted the preferential - perhaps even exclusive - field of application of new mathematical ideas. Today mechanics continues to constitute a branch of Applied Mathematics. The research development of mechanics in our time is mainly taking place in the field of Continuum Mechanics. Most of the problems posed by contemporary technology in Mathematics are formulated in the "language" of Continuum Mechanics.Mechanics has an enormous scope, as it extends from the mathematical description of a problem (modelling) and its "proper placement" to its solution (analytical - approximative). This determines the interaction potential of Mechanics with almost all branches of pure and applied mathematics.

**Computational Mathematics **is a branch of Applied Mathematics which basically aims at producing, analysing and using effective numerical (computational) methods (algorithms) for solving mathematical problems, and consequently, real practical problems encountered in various sciences. With the numerical methods, which constitute completely determined finite processes, and by means of a computer, one searches for the most accurate possible numerical (approximative) solutions to mathematical problems with the least possible computational cost.

The areas of **Computer Science **include: Symbolic Computations (or symbolic and algebraic processing), Artificial Intelligence (automatic programming, natural language processing), Computational Linguistics (contextual languages), Parallel Algorithms.

## Seminar Rooms and Laboratories

The following Seminar Rooms and Laboratories have been allocated to the Department of Mathematics:

**Seminar Rooms**

Algebra, Geometry, Mathematical Analysis

**Laboratories**

Numerical Analysis, Mathematics, Mechanics, Microcomputers, Probability and Statistics

## Subject Areas

The subject areas that are coordinated by the Sections of the Department of Mathematics in the School of Sciences are defined as follows:

**1. Mathematical Analysis Section:**

Real analysis, Theory of measure and integration, Complex analysis, Harmonic analysis, Topology, Mathematical logic, Functional analysis, Differential equations, Applied analysis, Mathematical analysis applications in other disciplines.

**2. Algebra and Geometry Section:**

Number theory, Field theory and polynomials, Commutative rings and algebras, Algebraic geometry, Linear and multilinear algebra, Associative rings and algebras, Nonassociative rings and algebras, Category theory and Homological algebra, K-theory groups and generalisations, Topological groups and Lie groups, Geometry, Convex and Discrete geometry, Differential geometry, Algebraic topology, Manifolds and cell complexes, Integral and manifold analysis, Geometric analysis, Mathematical logic and foundations, Algebraic theory of automata and languages, Algebra and geometry applications.

**3. Probability, Statistics and Operations Research Section:**

Probability and applications, Mathematical statistics, Applied statistics, Market research, Biostatistics, Behavioural sciences statistics, Stochastic processes, O.R. stochastic models, Mathematical programming, Operations research, Insurance mathematics, Financial mathematics, Econometrics.

**4. Applied Mathematics and Engineering Research Section:**

**(i) Numerical Analysis: **Error analysis, Numerical simulation, Numerical approach, Numerical linear algebra, Numerical solution of nonlinear equations and systems, Mathematical programming - optimisation and modification techniques, Numerical solution of ordinary differential equations and partial differential equations, Difference and functional equations, Integral equations, Numerical methods in Fourier analysis.

**(ii) Mechanics: **Material point mechanics and material point systems, Continuum mechanics, Elasticity, Fluid mechanics, Waves in continuous media, Heat transfer, Biomechanics.

**(iii) Computer science: **Theoretical computer science, Algorithm theory, Symbolic mathematical computations, Parallel computations, Databases, Programming languages, Artificial intelligence, Expert systems, Computational linguistics, Automatic natural language processing, Digital logic circuit design, Technical simulations.

## Members of Academic Staff

### Mathematical Analysis Section

**George Karakostas, **Professor, Differential quations (with continuous or discrete variables), Control Theory, Volterra Integral Equations, Population Dynamics, Dynamical Systems

**Athanasios Katsaras, **Professor, Functional Analysis

**Sotirios Ntouyas, **Professor, Differential Equations

**Christos Philos, **Professor, Differential Equations, Integral Equations, Difference Equations, Continuous and Discrete Models

**Ioannis Sficas, **Professor, Differential Equations

**Ioannis Stavroulakis, **Professor, Differential Equations, Difference Equations, Functional Equations, Partial Differential Equations

**Panagiotis Tsamatos, **Professor, Differential Equations

** Gerasimos Barbatis, **Assistant Professor, Differential Equations

**Chrysostomos Petalas, **Assistant Professor, Functional Analysis

**Ioannis Pournaras, **Assistant Professor, Differential Equations

**Theodoros Vidalis, **Assistant Professor, Topology, Functional Analysis, Theory of Measure

### Algebra and Geometry Section

**Christos Baikoussis, **Professor, Riemann Geometry

**Theodoros Bolis, **Professor, Group Theory - Analysis

**Thomas Hasanis, **Professor, Differential Geometry (Riemann Geometry, Submanifold Theory, Minimal Submanifolds)

**Themistocles Koufogiorgos, **Professor, Riemann Geometry - Contact Manifolds

**Nikolaos Marmaridis, **Professor, Algebra (Representation Theory - Homological Algebra)

**Apostolos Beligiannis, **Associate Professor, Algebra (Representation Theory - Homological Algebra)

**Apostolos Thoma, **Associate Professor, Algebraic Geometry, Anticommutative Algebra

**Theodoros Vlachos, **Associate Professor, Differential Geometry (Riemann Geometry, Submanifold Theory, Minimal Submanifolds)

**Anestis Fyraridis, **Assistant Professor, Algebraic Theory of Automata

**Epaminondas Kehagias, **Assistant Professor, Algebraic Topology - Invariant Theory

**Konstantinos Mexis, **Lecturer, Algebraic Theory of Automata

### Probability, Statistics and Operations Research Section

**Kosmas Ferentinos, **Professor, Statistical Information Theory, Statistical Inference, Biostatistics

**Sotirios Loukas, **Professor, Statistical Distributions, Statistical Inference, Simulation, Survival Analysis, Non-Parametric Statistics, Data Analysis

**Konstantinos Karakostas, **Associate Professor, Sampling Theory, Linear Models, Statistical Analysis of Numerical Data, Estimation

**Christos Lagaris, **Associate Professor, Stochastic Processes, Operations Research Stochastic Models, Service Systems

**Konstantinos Zografos, **Associate Professor, Statistical Information Theory, Parametric Statistical Inference, Measures of Dependence and Contingency

**Konstantina Skouri, **Lecturer, Operations Research

### Applied Mathematics and Engineering Research Section

**Sophocles Galanis, **Associate Professor, Numerical Linear Algebra (Iterative Methods for Solving Linear Systems)

**Dimitrios Noutsos, **Associate Professor, Numerical Linear Algebra (Iterative Methods for Solving Linear Systems)

**Christos Perdikis, **Associate Professor, Elasticity and Fluid Mechanics

**Andreas Raptis, **Associate Professor, Fluid Mechanics

**Apostolos Yeyios, **Associate Professor, Computational Mathematics - Numerical Linear Algebra (Iterative Methods)

**Nikolaos Glynos, **Assistant Professor, Symbolic Mathematical Computations, Artificial Intelligence, Databases

**Anna Psimarni, **Assistant Professor, Numerical Linear Algebra (Iterative Methods for Solving Linear Systems)

**Ioannis Stamatiou, **Assistant Professor, Algorithm Analysis, Databases

**Socrates Baltzis, **Lecturer, Automatic Natural Language Processing (NLP)

**Andreas Leonditsis, **Lecturer, Databases

**Ioannis Tsomokos, **Lecturer, Theory of Automata

The Department's teaching duties are supplemented by temporary teaching staff.

## Career prospects - New fields of specialisation

Graduates of the Department of Mathematics can find employment:

- As teachers in secondary education.
- As scientific and research personnel at centres and services in the public and private sectors.
- At public sector corporations and organisations such as the Hellenic Telecommunications Organisation, the Public Power Corporation, Local Government, etc.
- At insurance companies and market research and marketing companies.
- In the industry and at public and private sector banks.
- At computer centres.
- In sectors that engage in secure data transfer systems, especially cryptology, cryptography and code theory.
- At companies engaged in computer graphics.

## Postgraduate studies

As of 1995 the Department has been running a Postgraduate Study Programme (PSP) which focuses on mathematical sciences as they develop and evolve in the modern age through various branches and individual fields of specialisation.

The aim of the PSP is to promote knowledge and develop research and applications in all branches of mathematical sciences by training scientists, who are needed in order to cover the educational, research and developmental needs of the country.

Through the PSP the Department awards:

a) MSc degrees in the following fields of specialisation:

- Mathematics (Analysis - Algebra - Geometry)
- Applied Mathematics and Mechanics
- Statistics and Operations Research
- Computational Mathematics and Computer Science

b) Doctoral Degrees