The previous status (September, 2008),

a lot of changes since then!

** **

**Graduate School of Mathematics and **

**Computer Science**

** **

**Head: Prof. Gábor
Czédli DSc**

Faculty of Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary**
**Phone: +36-62-544-093, fax: +36-62-544-548,

E-mail: czedli@math.u-szeged.hu

WWW: http://www.math.u-szeged.hu/~czedli

** **

**Vice head:
Prof. Zoltán Fülöp DSc
**Faculty
of Science, Institute of Informatics

6701 Szeged, Hungary, P. O. Box 652.

Phone: +36-62-544-302, fax: +36-62-546-397,

E-mail: fulop@inf.u-szeged.hu

WWW: http://www.inf.u-szeged.hu/~fulop

In the Bolyai (Mathematical) Institute of the University of Szeged mathematical education and research started in 1921 when the two world famous masters of mathematical analysis, Frigyes Riesz and Alfréd Haar reorganized the Institute after the relocation of the University of Kolozsvár (Cluj-Napoca), which was a consequence of the territorial loss of Hungary following World Word I. During this early period they developed a mathematical school in Szeged. Very soon the Institute became an internationally recognized centre of science. Riesz and Haar also founded “Acta Scientiarum Mathematicarum”, the first mathematical journal in foreign language in Hungary. This journal earned a world-wide reputation very soon. The “Acta” played and continues to play a crucial role in the circumtance that the Library of Bolyai Institute possesses one of the richest collections of mathematical literature in Central Europe. Nowadays, more than 250 current mathematical journals, almost 20 thousand books and more than 20 thousand volumes of journals are available for the professors, researchers, and students interested in mathematics.

Today the Bolyai Institute consists of six chairs: Algebra and Number Theory, Analysis, Applied and Numerical Mathematics, Geometry, Set Theory and Mathematical Logic, Stochastics. The Institute has 50 academic staff members within a wide range of research interests. Bolyai Institute have regularly organized international algebraic conferences (in Universal Algebra, Lattice Theory, and Semigroup Theory) since 1971, and have organized the I.-VIII. International Colloquium on the Qualitative Theory of Differential Equations (1978-2007).

Systematic education in computer science was launched within the Mathematical Institute by László Kalmár in 1957. The first major computer science programme started in 1963. An important date in the history of computer science at the University of Szeged is 1967, when the Department of Computer Science was established. The first head of the department was László Kalmár. The Institute of Informatics was established as an independent unit of the Faculty of Sciences in 1990. Currently it consists of five departments: Department of Applied Informatics, Department of Computer Algorithms and Artificial Intelligence, Department of Foundations of Computer Science, Department of Image Processing and Computer Graphics, and Department of Software Engineering. The Institute of Informatics is in close partnership with the Research Group of Artificial Intelligence of the Hungarian Academy of Sciences both in research and teaching. Since 1993 the Institute has been running a doctoral program in computer science.

The computer science students have access to 10 high capacity servers via more than 250 workstations. The baud rate of the internet connection is 1 Gb. The education programmes, the research projects, and the mobility of students and professors of the Institute are supported by cooperations with over 30 higher education institutes from Europe, Japan, and the United States. The Institute of Informatics has a library which holds about 5000 volumes and subscribes over 200 scientific journals, both mainly in English language. The journal Acta Cybernetica has been edited and published since 1969 by the Institute. Since 1998 a biannual conference has been organized for doctoral students in computer science.

** **

**Educational Programmes**

**1. Mathematics**

*Programme
director:*** Prof. Gábor Czédli DSc**

Faculty of Science, Bolyai
Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary**
**Phone: +36-62-544-079, fax: +36-62-544-548,

E-mail: czedli@math.u-szeged.hu

WWW: http://www.math.u-szeged.hu/~czedli

The PhD programmes
in Mathematics take minimum 3 years, the average duration is 4-5 years. The students
have to choose at least 8 courses such that at least 3 of them are general or
basic courses. The Bolyai Institute offers 5 general courses (*Algebra*,*
Theory of Measure and Integral*, *Topology,* *Discrete Mathematics,
Probability Theory*), 23 basic courses and several dozens of specialized
courses. The lectures are taught either by the members of the Bolyai Institute
or by invited experts. If the enrolment is low, a course may be held as a
reading course with regular consultations, possibly in English.

**1.1. Algebra**

*Coordinator:*** Assoc. Prof.
László Zádori PhD**

*Research topics *

1. Semigroup theory, structure of regular semigroups and their generalizations, classes of regular semigroups, the structure of simple semigroups.

2. Lattice theory, congruence lattices and Mal’cev conditions, related lattices, lattices and their generalizations.

3. Universal algebra, clones and relations, structure theory of finite algebras, varieties, commutator theory, finitely and nonfinitely based algebras and varieties, algebraic methods in algorithmic problems.

4. The antique and medieval roots of the secondary (high) school curriculum in mathematics.

__Supervisors:__ B. Csákány DSc,
G. Czédli DSc, L. Klukovits CSc, M. Maróti PhD, L. Megyesi CSc, L. Szabó CSc,
Á. Szendrei DSc, M. B. Szendrei DSc, L. Zádori PhD.

*Three
representative courses:*

Lattice theory (G. Czédli)

*Rudiments of lattice theory. Algebraic and
subalgebra lattices. Distributive lattices: Stone’s and Birkhoff’s
representation theorems, the structure of finite distributive lattices.
Birkhoff’s and Dedekind’s characterizations. Three-generated free distributive
and modular lattices. Congruences. Intervals, sublattices and decompositions in
modular lattices. Geometric and complemented modular lattices. Projective
geometries versus geomodular lattices. Varieties of lattices. *

Semigroup theory (M. B. Szendrei)

*Transformation semigroups and free semigroups.
Ideals and Rees congruences. Green's equivalences, the structure of D-classes.
Simple and 0-simple semigroups, principal factors, the Rees theorem for
completely 0-simple semigroups. Completely regular semigroups, semilattices of
groups. Inverse semigroups, the Wagner-Preston representation theorem, the
natural order. *

Universal algebra (Á. Szendrei)

*Algebras, term functions, polynomial functions. Subalgebras,
isomorphisms, homomorphisms, and the general isomorphism theorems. Direct
product, other concepts of product. Subdirect representation and Birkhoff's
theorem. Closure operators and closure systems. Congruence lattices. Free
algebras and varieties. Birkhoff's HSP theorem and Birkhoff's completeness
theorem. Equivalence of varieties. Equational properties of varieties,
Mal'tsev's and Pixley's theorems. Magari's theorem. Minimal varieties.
Ultraproducts and congruence distributive varieties. Varieties generated by a
primal algebra. Quasiprimal algebras and discriminator varieties. Finite basis
theorems**.*

**1.2.
Analysis**

*Coordinator:*** Prof. Ferenc
Móricz DSc**

*Research topics*

1. Approximation theory and orthogonal polynomials.

2. Fourier series, integrals and orthogonal series.

3. Functional analysis.

4. Operator theory.

5. Summability and Tauberian theorems.

6. Some interesting problems of the analysis and their treatment during teaching.

__Supervisors:__ F. Móricz DSc, V. Totik MHAS, L. Kérchy DSc, J. Kosztolányi PhD,
L. Leindler MHAS, J. Németh CSc, Z. Németh PhD, L. Stachó CSc, J. Szalay CSc.

*Three
representative courses:*

Contractions of Hilbert spaces II (L. Kérchy)

*Operator-valued analytic functions. Inner and
outer functions, factorization theorems. Scalar multiple. The characteristic
function of a contraction. Unitary equivalent function models of contraction.
The connection between spectra and characteristic functions. Characterizing
invariant subspaces of contractions with regular factoriazations of
characteristic functions. Invariant subspaces of C _{11}-contractions.
Weak contractions. *

Complex harmonic analysis (F. Móricz)

*H ^{p} and h^{p} spaces on the
unit disk of the complex plane, their characterization by Poisson integral,
analytic completion of a harmonic function, Helly’s selection theorem, Fatou’s
theorem on the existence of the radial limit. .The Jensen and Poisson-Jennsen
formulas, distribution of the zeros of an analytic function, Blaschke products,
F. Riesz’ and Nevalinna’s factorization theorems, convergence to the limit
function in L^{p}-norm. The Riesz brothers’theorem and its equivalent
reforumulations.*

Approximation by polynomials (V. Totik)

* Trigonometric polynomials, Nikolskii’s
theorems, Dzjadik’s inverse theorems, characterization of the best
approximation by algebraic polynomials in terms of φ- modulus, discrete
operators, potential theory and approximation by polynomials; approximation by
changing weights, orthogonal polynomials and weighted approximation by
polynomials; Müntz’ theorem and its generalizations.*

**1.3. Dynamical Systems**

*Coordinator:*** Prof. László
Hatvani MHAS**

*Research topics*

1. Qualitative theory of ordinary, functional, and partial differential equations: nonlinear oscillations; existence and stability of periodic solutions; almost periodic solutions.

2. Lyapunov stability and its applications; stability in population dynamics; dependence of the solutions from parameters for partial differential equations; radially symmetric solutions of elliptic partial differential equations.

3. Existence and smoothness of invariant manifolds for functional differential equations; stable subharmonic solutions of periodic functional differential equations; existence of heteroclinic orbits; infinite dimensional dissipative dynamical systems; Hopf bifurcation and centre manifolds; inertial manifolds; exponential attractors.

4. Numerical solutions; computer simulation of dynamical systems.

5. Modern didactical methods in the teaching of dynamical systems.

__Supervisors:__ L. Hatvani MHAS,
J. Hegedűs CSc, J. Karsai PhD, T. Krisztin DSc, G. Röst PhD, Terjéki CSc.

*Two representative
courses:*

Ordinary differential eqations I, II (L. Hatvani)

*Differential equations on manifolds. Existence
and uniqueness theorems. Differential equations in spaces of infinite
dimension. Linear systems. Infinitesimal generator. Integral manifolds.
Linearization, Hartman_Grobman’s theorem. Perturbation theory. Non-autonomous
systems. Periodic and almost periodic equations. The method of averaging.
Boundary value problems. Sturm-Liouvilee theory. Second order equations,
oscillation. Limit sets and limit cycles. Poincaré-Bendixson theorem.
Stability. Lyapunov’s methods. Invariance principles. First order partial
differential equations. Hamilton-Jacobi’s theorem. *

*Functional
differential equations I, II (T. Krisztin)*

*The abstract theory of phase spaces,
trajectories, solutions. Existence and uniqueness theorems. Continuous
dependence on the initial data .New phenomena in comparison with the ordinary
differential equations. Continuability and compactness of solutions. Linear
functional differential equations. Oscillation for first and second order
equations. Stability. Integro-differential equations. Neutral equations.
Geometric theory of autonomous equations. The existence of periodic solutions.
Applications from biology, mechanics, and other sciences. .*

**1.4. Geometry and Combinatorics**

*Coordinator:*** Assoc. Prof.
Péter Hajnal CSc**

*Research topics**: *

1. Differential geometry, Lie groups and Lie algebras; analysis on manifolds; web geometries.

2. Discrete geometry and combinatorics; finite geometry; combinatorial and convex geometry; latin squares and 3-webs.

3. Graph theory; combinatorial complexity theory.

4. Elementary geometry and combinatorics.

__Supervisors:__ P. Hajnal CSc,
J. Kincses CSc, Á. Kurusa CSc, F. Fodor PhD, G. Gévay PhD, Gy. Kiss PhD, G. P.
Nagy PhD, T. Ódor CSc, L. Szilassi PhD, T. Szőnyi DSc.

*Three representative
courses:*

Graph theory (P. Hajnal)

*Connectivity: the connectivity of directed
graphs, nowhere zero flows. Matching: Gallai-Edmonds’ structure theorem, Edmonds’ polytop. random methods for determining ν(G), the number of matchings in a
graph, permanent, the proof of Van der Waerden’s conjecture. Coloring of
graphs: Hajós’ theorem, Kneser’s graph and its chromatic number, the chromatic
number of ***R**^{d}*. Independent subsets in graphs: τ-critical graphs, vertex
packing polytope, perfect graphs, the Shannon capacity of graphs. Eigenvalues
of graphs, random walks on graphs, expanding parameter of graphs.*

Combinatorial methods in geometry (Gy. Kiss)

*Block designs. Parameters of block designs and
divisibility conditions. Steiner systems. Hadamard matrices. Solvable block
designs. Baranyai’s theorem. Finite projective geometries. Latin squares. Combinatorial
properties of finite projective geometries. Coordinatization of finite planes,
the theorems of Desargues and Pappus. Finite affine planes. Finite reflection groups.
Coxeter groups and complexes. Buildings. *

Convex geometry (J. Kincses)

*Combinatorial properties of convex sets.
Charatheodory, Radon and Helly’s theorems; their generalizations and
applications. Separability of convex sets and duality. Approximation of convex
sets, Blaschke’s choice theorem. Operations with convex sets, mixed volume. The
isoperimetric theorem. Convex bodies with constant width. Evaluation of convex
sets. Zonoids. *

* *

**1.5. Stochastics**

*Coordinator:*** Prof. András
Krámli DSc**

*Research topics *

1. Asymptotic distributions in probability theory: domains of attraction and partial attraction; St. Petersburg games and merging approximations; weak and strong laws; almost sure asymptotic distributions.

2. Statistical estimation and testing: statistical extreme value theory; weighted correlation tests; empirical distributions and transforms; parametric and nonparametric bootstrap; estimation under random censorship and long-range dependence.

3. Statistical physics: hyperbolic dynamical systems (hard ball systems); infinite particle systems; mathematical theory of phase transitions.

4. Famous problems of
probability and statistics from the 17^{th}-19^{th} century and
their didactical relations.

__Supervisors:__ A. Krámli DSc, E.
Huhn CSc, K. Stékliné Boda PhD, L. Viharos PhD.

*Two representative
courses:*

Foundations of stochastic processes (A. Krámli)

*Discrete time and discrete space stochastic
processes. Selected topics from the theories of Markov chains, birth-and-death
processes, renewal processes, branching processes and discrete time martingals.
*

The statistics of extreme-value distributions (L. Viharos)

*Estimation of parameters of extreme-value
distributions from domain of attractions. Tail index estimators. Model
validation. Appications in hydrology and in the field of insurance and
financial mathematics.*

Lattices and
invariants,* E. K. Horváth,* G. Czédli and B. Csákány, 2005.

Asymptotic
Bernstein-type inequalities, *B. Nagy,* V. Totik, 2006.

Bifurcation theory
of periodic functional-differential equations, *G. Röst*, T. Krisztin,
2006.

Smoothness of
Green’s function and density of sets, *F. Toókos*, L. Hatvani and V.
Totik, 2006.

The equilibrium
measure and the Saff conjecture, *D. Benkő*, V. Totik, 2006.

Tauber type
theorems for ordinary and statistical convergence and statistical limits, *Á.
Fekete*, F. Móricz, 2007.

Absolutely
convergent Fourier series and classes of functions spaces, *V. Fülöp*, F.
Móricz, 2007.

Decidability
problems in algebra, *M. Maróti*, Á. Szendrei, 2007.

Minimal clones, *T.
Waldhauser*, B. Csákány and Á. Szendrei, 2007.

Orders, conjugacy
classes, and coverings of permutation groups, *A. Maróti*, L. Pyber and Á.
Szendrei, 2008.

Orthodox
semigroups and semidirect products, *M. Hartmann*, B. M. Szendrei, 2008.

**2. Informatics**

*Programme director:***
Prof. Zoltán Fülöp DSc**

Faculty of Science, Institute of Informatics

6701 Szeged, Hungary, POB 652.

Phone: +36-62-544-302, fax: +36-62-546-397,

E-mail: fulop@inf.u-szeged.hu

WWW: http://www.inf.u-szeged.hu/~fulop

** **

The program, of which the duration is 3 years, prescribes the accomplishment of 180 credits, active participation in the Institute's seminars, and the conduction of research under the supervision of a thesis adviser appointed by the Council of the Program. The completion of 8 courses is included, the courses embrace a number of fields in computer science. A course may be offered as a reading course if enrollment is low. In such cases consultation is provided. The language of education in the three-year program is mainly Hungarian, but invited professors teach in English.

**2.1.
Theoretical Computer Science**

*Coordinator:*** Prof. Zoltán Ésik
DSc**

** Research topics:** From the theory
of automata and formal languages, which is now considered a classical field, to
the most up-to date topics such as advances in computational complexity, term
rewriting, tree automata and tree transformations, mathematical semantics of
programming languages, process algebras, theory of fixed points, temporal
logic, semirings.

__Supervisors:__ E Csuhaj-Varjú,
Z. Ésik DSc, Z. Fülöp DSc, F. Gécseg MHAS and S. Vágvölgyi PhD.

*Two representative courses:*

Automata and formal logic (Z. Ésik)

*First-
and second-order logic on words.*
*Büchi's theorem for second-order logic and regular languages. Fagin's
theorem: the class NP and the existential second-order logic. The second-order
logic and the polynomial hierarchy. The Ehrenfeucht-Fraisse game. Mc Naughton's
theorem for aperiodic automata and first-order definable languages. Hierarchies
of first-order definable languages. The equivalence of temporal logic and
first-order logic. Kampf's theorem. First order logic and counting. Modular
quantification. Büchi's theorem for regular omega-languages. First- and
second-order logic on trees. The theorems of Thatcher, Wright and Rabin.
Regular omega-languages and their syntactic monoids. Regular omega-languages
and logic. The decidability of the monadic S1S. The definability of the
star-free and the omega-languages in first-order logics.*

Tree Automata (F. Gécseg)

*Recognizable
tree languages. Variants of tree automata and the comparison of their
recognizing capacity. Equivalent specifications of recognizable tree languages:
regular tree grammar, congruence of finite index, finitely generated
congruence, regular expression. Connection between recognizable tree languages
and derivation trees of context-free grammars. Top-down, bottom-up, attributed
an macro tree transducers; comparison of their transformational capacity.
Composition and decomposition of tree transformation classes. Infinite
hierarchies formed by powers of tree transformation classes. Description of
monoids generated by fundamental tree transformation classes. Weighted tree
automata and tree transducers. Tree automata on infinite trees. Rabin's theorem.
Applications to logic. *

**2.2. Operation Research and Combinatorial
Optimization**

*Coordinator:*** Prof. Tibor Csendes DSc**

** Research topics:** Theory of economic decisions, multiciriteria
decision making, group decisions, fuzzy theory, learning algorithms, global
optimization, reliable numerical procedures, interval inclusion functions,
process network synthesis, bin packing algorithms, online data compression
procedures, scheduling problems and set partitioning.

__Supervisors:__
J. Dombi CSc, T. Csendes DSc, J. Csirik DSc, G. Galambos CSc, Cs. Imreh PhD and
A. Pluhár PhD.

*Two representative courses:*

Global optimization (T. Csendes)

*Forms of global optimization problems, their
complexity, classes of global optimization methods, grid search, random search,
simulated annealing, evolutional approaches, stochastic and multistart
algorithms, Lipschitz constant based procedures, DC-optimization, interval
B&B methods, applications.*

Game theory (A. Pluhár)

*Basic
concepts, von Neumann theorem for finite games. Matrix games and thier
connection to Linear Programming. Minimax ttheorem. Reduction methods (dominance,
saddle point). Non-zero sum games and thier applications. Nash equilibrium.
Basic concepts of n-player games, imputations, core and stable sets. Simple
games. The LP characterization of the core. Shapley theorem and computation of Shapley
values. Stable matchings and the kernel of directed graphs. Group decison
making. Arrow’s theorem. The basis of the Conway theory. Combinatorial games,
examples. The Erdős-Selfridge theorem and its generalizations**.** *

** **

**2.3. Applications of Computer Science**

*Coordinator:*** Prof. János Csirik DSc**

** Research topics** range from software engineering, to artificial
intelligence, robotics and picture processing. Software engineering topics:
advanced programming paradigms, theory of compilers, compilation of embedded
systems, legacy system analysis, software maintenance and reengineering,
object-oriented design and development, web programming, databases and data
mining, network protocols, formal specification and testing of protocols,
distributed programming. Artificial intelligence: frame and rule based
knowledge representation, machine learning algorithms (decision trees,
inductive logic programming, genetic algorithms, neuron networks), complexity
of the machine learning algorithms, natural language processing, speech
recognition. Image processing and medical applications: image processing in
nuclear medicine, image reconstruction from projections, discrete tomography,
picture archiving and communication systems, segmentation of medical images,
image registration, skeletonization, thinning and its applications, geographic
information systems, intelligent mobile robots, fuzzy control in robotics,
remote control, navigation of autonomous systems.

__Supervisors:__ Z. Alexin PhD,
Á. Beszédes PhD, J. Csirik, DSc, J. Dombi CSc, R. Ferenc PhD, T. Gyimóthy PhD, Z.
Hantos DSc, M. Jelasity PhD, Z. Kató PhD, E. Katona PhD, E. Máté CSc, I.
Matijevics PhD, Gy. Mester PhD, K. Palágyi PhD, Gy. Turán PhD.

*Three
representative courses: *

Embedded systems (T. Gyimóthy)

*Introduction to
embedded systems (SW/HW architectures). Embedded operating systems (Linux,
VxWorks). Embedded software development (On-chip debugging support* *JTAG, ICD, ICE). Hardware interfaces (UART, ICC,
USB, Network interface). Particular embedded topics: real time concepts,
multitasking, communication of processes, memory management, error finding
methods.*

Skeletonization in image processing (K. Palágyi)

*Introduction, the
skeleton and its properties. Skeletonizations techniques (distance
transformation, Voronoi diagram, thinning). Thinning algorithms in 2 and 3
dimensions. Applications.*

Peer to peer and self organizing algorithms (M. Jelasity)

*Properties of large size peer to peer networks.
Structured overlay networks. Superpeer networks. Content distribution networks.
Biology inspired algorithms. Amorphous computing. Problems in the development
and evaluation of P2P algorithms.*

**Representative
dissertations from Informatics **

**(title,
author, supervisor, year):**

Soliton automata: a computational model on the principle of graph
matchings, *M. Krész,* M. Bartha, 2005.

Modelling and Reverse Engineering C++ Source Code, *F. Rudolf*,
T. Gyimóthy 2005.

Source code analysis and slicing for supporting program
understanding, *Á. Beszédes*, T. Gyimóthy 2005.

Packing of Equal Circles in a Square - bounds, repeated patterns
and minimal polynomials,* P. G. Szabó*, T. Csendes, 2006.

Posterior-Based Speech Models and their Application to Hungarian
Speech Recognition, *L. Tóth*, J. Csirik, 2006

Discrete tomographic and PACS image processing systems* A. Nagy*,
A. Kuba, 2006.

Shape preserving Tree Transducers, *Zs. Gazdag*, Z. Fülöp,
2006

Development, comparison and application of global optimization
methods and investigation of atomic cluster problems, *T. Vinkó*, T.
Csendes, 2006

Binary Tomography Using Geometrical Priors: Uniqueness and
Reconstruction Results* P. Balázs*, J. Csirik, A. Kuba, 2007

Registration methods and its medical applications, *A. Tanács*,
A. Kuba, 2007

The "Gas of Circles" Model and its Application to Tree
Crown Extraction,* P. Horváth*, Z. Kató, I. Jermyn, 2007

Interval methods for competitive location problems, *B. Tóth*,
T. Csendes, 2007

Verified computer assisted methods for the investigation of the
stability and chatoic behaviour of dynamic systems, *B. Bánhelyi. *T.
Csendes, 2008

Learnability and Characterization Results for Classes of Boolean
Functions, *B. Szörényi*, Gy. Turán, 2008

Higher Dimensional Automata, *Z. Németh*, Z. Ésik,
2008

Logic and tree automata, *Sz. Iván*, Z. Ésik, 2008