April 30, 2010

**Graduate (Doctoral) School **

**in **

**Mathematics and Computer Science**

**http://www.math.u-szeged.hu/phd/**

**Director:
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<at>math.u-szeged.hu

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

**Associate
Directors: **

Faculty of Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary

Phone: +36-62-544-088, fax: +36-62-544-548,

E-mail: hajnal<at>math.u-szeged.hu

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

**Assoc.
Prof. ****László Zádori**** DSc**

Faculty of
Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary**
**Phone: +36-62-543-881, fax: +36-62-544-548,

E-mail: zadori<at>math.u-szeged.hu

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

Mathematical education and research at the Bolyai (Mathematical) Institute of the University goes back to Frigyes Riesz and Alfréd Haar, 1921. These two world famous masters of mathematical analysis founded “Acta Scientiarum Mathematicarum (Szeged)” in 1922, the first mathematical journal in foreign language in Hungary. The “Acta” played and continues to play a crucial role in the circumstance that the Library of Bolyai Institute offers more than 250 current mathematical journals, almost 20 thousand books and more than 20 thousand volumes of journals to professors, researchers and students. 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 with a wide range of research interests.

The doctoral education in the Bolyai Institute started in 1993. Between 2002 and 2008, the School consisted of two educational programmes: mathematics and informatics. The school reached its present form in 2009.

** **

**Educational Programmes**

Each of the five PhD
educational programmes, listed below, takes 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*), 27 basic courses and several
dozens of specialized courses. Two basic courses and 11 specialized courses are
devoted to the Didactics of Mathematics. The lectures are taught mostly by the
members of the Bolyai Institute or, sometimes, by invited experts. If the
enrolment is low, a course may be held as a reading course with regular
consultations, possibly in English.

**1. Algebra**

*Programme
director:*** Assoc. Prof. ****László Zádor****i
DSc**

Faculty of
Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary**
**Phone: +36-62-543-881, fax: +36-62-544-548,

E-mail: zadori<at>math.u-szeged.hu

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

*Research topics *

1. Semigroup theory: regular semigroups and their classes and generalizations, inverse 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 DSc.

*Basic courses:*

MDPT211. Semigroup theory (Mária B. Szendrei, DSc)

*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. Fundamental inverse
semigroups, Munn’s theorem. Commutative semigroups*

Literature: Grillet, P. A.: Semigroups: An Introduction to the Structure Theory, Marcel Dekker, 1995.

Howie, J. M.: Fundamentals of Semigroup Theory, Clarendon Press, 1995.

MDPT212. Lattice theory (Gábor Czédli, DSc)

*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. *

Literature: G. Czédli G.: Lattice Theory (Hungarian)

G. Grätzer:
General Lattice Theory* *

MDPT213. Universal algebra (László Zádori, DSc)

*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. *

Literature: Burris–Sankappanavar: Introduction to Universal Algebra

McKenzie–McNulty–Taylor: Algebras, Lattices, Varieties.

MDPT214. Group theory (Ágnes Szendrei, DSc)

*The multiplicative
group of fields. Permutation groups: primitive and multiply transitive
permutation groups, wreath product, Frobenius groups. Free groups: rank,
defining relations, Reidemeister-Schreier theory. Solvable groups. p-groups.
Nilpotent groups. The transfer. The Burnside problem. Matrix groups. Finite
simple groups. Subgroup lattices. *

Literature: Aschbacher: Finite Group Theory.

Hall, M. Jr.:The Theory of Groups.

Huppert: Endliche Gruppen.

Lyndon-Schupp: Combinatorial Group Theory.

**2. Analysis**

*Programme
director:*** Prof. ****Ferenc
Móricz**** DSc**

Faculty of
Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary**
**Phone: +36-62-544-082, fax: +36-62-544-548,

E-mail: moricz<at>math.u-szeged.hu

*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, L. Kérchy DSc, L. Leindler MHAS, J. Németh CSc, L.
Stachó CSc, V. Totik MHAS.

*Basic courses:*

MDPT 221. Chapters from the Complex Function Theory (László Leindler, MHAS)

*Mittag-Leffler’s theorem on the decomposition of meromorphic
functions into partial fractions, decomposition of *cot
*π**z. Weierstrass factorization theorem of entire functions,
factorization of *sin *π**z.
The gamma function. Approximation by rational functions, Runge’s theorem. Hardy
H ^{p} spaces of analytic functions on the open unit disk. Nontangential
limit on the unit circle, Fatou’s theorem. Riesz brothers’ theorem, Szegő’s
theorem. Blaschke products, inner and outer functions, canonical factorization.
Banach algebra of the functions that are analytic on the open unit disk and
continuous on the closed unit disk. Invariant subspaces of the shift operator
in the Hilbert space H^{2}.*

Compulsary literature: B. Szőkefalvi-Nagy: Introduction to real functions and orthogonal expansions.

K. Hoffmann: Banach spaces of analytic functions

Recommended
literature: P.
L. Duren: Theory of *H ^{p}* spaces

J. Garnett: Bounded analytic functions

P. Koosis:
Introduction to *H ^{p}* spaces

W. Rudin: Real and complex analysis

MDPT 223. Introduction to the Approximation Theory (Vilmos Totik, MHAS)

*Approximation by positive operators, Korovkin’s theorem, the
Weierstrass and Weierstrass-Stone theorems. Moduli of continuity and
smoothness, Jackson’s theorem, direct theorems. Estimation of the derivatives,
Bernstein’s theorem and the inverse theorems of approximation theory.
Characterization of the best approximating polynomials, extreme signatures. L ^{p}-approximation.
Saturation of Bernstein polynomials, parabola method.*

Compulsary literature: G. G. Lorentz: Approximation of functions

G. G. Lorentz–R. DeVore: Approximation theory

Recommended literature: N. I. Akhiezer: Lectures on the theory of approximation

P. Borwein–T. Erdelyi: Polynomials and Polynomial Inequalities

G. G. Lorentz–M. Von Golitschek-A. Makovez: Approximation Theory II.

I.P. Natanson: Constructive function theory

M. Timan: The approximation of real functions

MDPT 224. Fourier Series I. (Zoltán Németh, PhD)

*Completeness of the trigonometric system. Bessel’s inequality,
Parseval’s formula. Convergence of Fourier series: Riemann-Lebesgue lemma, the
Dini test, the principle of localization, the Dirichlet-Jordan test, the
Lebesgue constants. Summability of Fourier series: theorem of Fejér and its
consequences, theorem of Lebesgue, the Lebesgue points of an integrable
function. Divergence of Fourier series: examples of Fejér and Lebesgue. Special
trigonometric series with coefficients tending monotonically to zero.** *

Compulsary literature: A. Zygmund: Trigonometric series

Recommended literature: N. K. Bary: A treatise on trigonometric series

R. E. Edwards: Fourier series

B. Szőkefalvi-Nagy: Introduction to real functions and orthogonal expansions.

MDPT 225. Functional Analysis (László Kérchy, DSc)

*Orthonormal systems in Hilbert spaces, dimension of a Hilbert space.
Convergence of Fourier series, Cesàro and Abel summability. The
Hahn-Banach theorem and its applications: the Banach limit, integral and
measure. The Banach-Steinhaus theorem, the open mapping theorem and the closed
graph theorem: their applications to Fourier series. Duals of the L ^{p}
spaces, reflexivity. The dual of the space of continuous functions, the Riesz
representation theorem. The Weierstrass-Stone approximation theorem.*

Compulsary
literature: L.* *Kérchy:
Real and Functional Analysis, Polygon, Szeged, 2008 (Hungarian)

Recommended literature: F. Riesz–B. Szőkefalvi-Nagy: Functional analysis

W. Rudin: Real and complex analysis

B. Szőkefalvi-Nagy: Introduction to real functions and orthogonal expansions.

*Three
representatives from the specialized courses:*

MDPT3203. Contractions of Hilbert spaces II (László Kérchy, DSc)

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

Compulsary literature: B. Szőkefalvi-Nagy–C. Foias: Harmonic analysis of operators on Hilbert spaces

Recommended
literature: H. Bercovici: Operator
theory and arithmetic in *H*^{infinity}.

C. Foias–A.F. Frazho: The commutant lifting approach to interpolation problems

K. Hoffmann: Banach spaces of analytic functions

MDPT3214. Complex harmonic analysis I (Ferenc Móricz, DSc)

*Representation of
holomorphic functions on the open unit disk by Poisson integrals. Holomorphic
completion of harmonic functions, Herglotz integral. H ^{p} and h^{p}
spaces on the unit disk of the complex plane. Characterization of the h*

Compulsary
literature: P.
L. Duren: Theory of *H ^{p}* spaces

Recommended literature: Garnett: Bounded analytic functions

K. Hoffmann: Banach spaces of analytic functions

P. Koosis:
Introduction to *H ^{p}* spaces

A. Torchinsky: Real variable methods in harmonic analysis

A. Zygmund: Trigonometric series

MDPT3225. Approximation by polynomials (Vilmos Totik, MHAS)

*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. *

Compulsary literature: Z. Ditzian–V. Totik: Moduli of smoothness

G. G. Lorentz–R. DeVore: Approximation theory

M. Timan: The approximation of real functions

Recommended literature: N. I. Akhiezer: Lectures on the theory of approximation

G. G. Lorentz: Approximation of functions

I.P. Natanson: Constructive function theory

**3. Dynamical Systems**

*Programme
director:*** Prof. ****László
Hatvani**** MHAS**

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: hatvani<at>math.u-szeged.hu

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

*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, T. Csendes DSc, J. Karsai PhD, T. Krisztin DSc,
B.G. Pusztai PhD, G. Röst PhD.

*Basic courses:*

MDPT231-232. Ordinary differential eqations I-II (László Hatvani, MHAS)

*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-Liouville
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
theory. *

Compulsary literature: H. Amann, Ordinary Differential Equations, DeGruyter, 1990.

V. I. Arnold, Ordinary Differential Equations, Springer-Verlag, 1992.

J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, 1969.

Recommended literature: D. V. Anosov, V. I. Arnold, Dynamical Systems I, Ordinary Differential Equations and Smooth Dynamical Systems, Springer-Verlag, 1991.

C. Chicone, Ordinary Differential Equations with Applications, Springer, 1999.

Ph. Hartman, Ordinary Differential Equations, Birkhäuser, 1982.

M. Hirsch, S. Smale, Differential Equations, Dynamic Systems and Linear Algebra, Academic Press, 1974.

M. A. Naimark, Linear Differential operators, Nauka, 1969 (in Russian).

V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Dover Publications, 1954.

J. Palis, W. DeMelo, Geometric Theory of Dynamical Systems, Springer-Verlag, 1982.

V. A. Pliss, Integral Manifolds of Periodic Systems of Differential Equations, Nauka, 1977 (in Russian).

MDPT233. Partial differential equations I (Tibor Krisztin, DSc)

*Distributions (generalized functions). Sobolev spaces. Fourier
transform of distributions. Fundamental solutions of PDE’s. Partial
differential operators. Classical and generalized solutions. Hypo-elliptic
differential operators. Well-posed problems in half space for linear systems of
PDE’s. Existence, uniqueness and stability of solutions of boundary value
problems for elliptic, hyperbolic, and parabolic PDE’s in Sobolev spaces.** *

Compulsary literature: V. Sz. Vlad’imirov, B Introduction to the theory of partial differential equations.

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 20, AMS, Providence, Rhode Island, 1998.

Recommended literature: O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer- Verlag, 1985.

I. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1986.

MDPT234-235. Dynamical systems I-II (László Hatvani, MHAS)

*Existence and smoothness of invariant manifolds. Behaviour of
solutions near fix points and periodic orbits. Linearization. Orbital
stability. Poincaré maps. Averaging. Limit sets. Asymptotically smooth maps and
semi-groups. **α**-contractive semi-groups. Stability of invariant sets. Dissipative
systems. Global attractors. Fix point theorems. Morse-Smale maps. Dimension of
a global attractor. Periodic flows. Gradient systems. Examples: retarded FDE’s,
neutral FDE’s, parabolic and hyperbolic PDE’s.** *

Compulsary literature: M. Hirsch and S. Smale, Differential Equations, Dynamic Systems and Linear Algebra, Academic Press, 1974.

J. Palis, W. DeMelo, Geometric Theory of Dynamical Systems: an Introduction, Springer- Verlag, 1982.

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, 1990.

Recommended literature: J. Guckenheimer and P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, 1983.

J. Hale, L. Magalhaes, W. Oliva, An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory, Springer-Verlag, 1984.

J. Hale, Asymptotic Behavior of Dissipative Systems, AMS, 1986.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981.

M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds, Springer-Verlag, 1977.

V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Dover Publications, 1954.

H. L. Smith, Monotone Dynamical Systems, AMS, 1995.

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997.

*A representative
from the specialized courses: *

Functional differential equations I, II (Tibor Krisztin, DSc)

*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. *

Compulsary literature: O. Dickmann, S. A. Van Gils, S. M. Verduyn Lunel, H.-O. Walter, Delay Equations, Springer, 1995.

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, 1977.

Recommended literature: T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, 1985.

G. Gripenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, 1990.

I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, 1991.

Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Infinite Delay, Springer-Verlag, 1991.

V. B. Kolmanovskii, V.R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986.

T. Krisztin, H.-O. Walter, J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, AMS, 1999.

S. H. Saperstone, Semidynamical Systems in Infinite Dimensional Spaces, Springer-Verlag, 1981.

**4. Geometry, Combinatorics and Theoretical Computer
Science**

*Programme
director:*** Assoc. Prof. ****Péter Hajnal****
CSc**

Faculty of Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary

Phone: +36-62-544-088, fax: +36-62-544-548,

E-mail: hajnal<at>math.u-szeged.hu

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

*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.

5. Theoretical computer science

__Supervisors:__ P. Hajnal CSc, J. Barát PhD, T. Csendes DSc, J. Csirik DSc, J.
Dombi CSc, Z. Ésik DSc, F. Fodor PhD, Z. Fülöp DSc, G. Gévay PhD, Cs. Imreh
PhD, J. Kincses CSc, Gy. Kiss PhD, G. P. Nagy PhD, A. Pluhár PhD, Gy. Turán
CSc.

*Basic courses:*

MDPT241.Combinatorial methods in geometry (György Kiss, PhD)

*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. *

Recommended literature: M. Jr. Hall, Combinatorial theory, Waltham, Mass. 1967.

Gy. Kiss and T. Szőnyi: Finite Geometries, Polygon, Szeged, 2001 (Hungarian)

P. Hajnal: Systems of sets, Polygon, Szeged, 2002 (Hungarian)

Brown, Buildings, Springer-Verlag, London, 1989.

MDPT242. Riemannian geometry (Árpád Kurusa, CSc)

*Riemannian geometry: metric, Levi-Civita connection, curvature.
Geodesics: n**ormal **neighborhood*, *convex
neighborhood*, *normal
coordinates*, *geodesic
variation. **Jacobi fields, the Jacobi theorem on
the characterization of minimizing geodesics via conjugate points. Hopf-Rinow
theorem. Hadamard theorem.** Morse index theorem. curvature tensor*. *Ricci curvature*, *scalar curvature*, *sectional curvature. Spaces of
constant **curvature.** *

Compulsary literature: M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992.

J. Milnor, Morse Theory, Princeton University Press, 1963.

Recommended literature: W. Klingenberg, D. Gromoll, W. Meyer: Riemannsche Geometrie im Grossen, Springer, 1968.

J. Cheeger, D. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland, 1975.

MDPT243. Convex bodies and classical integral geometry (János Kincses, CSc)

*Basic properties of convex sets. Charatheodory,
Radon and Helly’s theorem. Separability, Euler’s relation, duality.
Approximation of convex sets, Blaschke’s choice theorem. Mixed volume,
Brünn-Minkowski theorem, **Fenchel-Alexandrov inequality.** Density for points and lines, cinematic
density, planar integral formulas, Steiner’s formula, quermass integrals.
Blaschke and Poincaré’s fundamental formulas. Curve integrals and their
applications.** *

Compulsary literature: L.A. Santalo, Integral Geometry and Geometric Probability, Encyclopedia of Math., Addison–Wesley, London, 1976.

T. Bonnesen, W.Fenchel, Theorie der konvexen Körper, Springer, Berlin, 1934.

W. Blaschke, Vorlesungen über Integralgeomtrie, Berlin, 1955.

H. Busemann, Convex surfaces, Interscience, London, 1958.

MDPT244. Computational geometry (Péter Hajnal, CSc, and Ferenc Fodor, PhD)

*Special data structures for geometric problems. Geometric
search. Coding polytopes, arrangements. Allowable sequences. Range search,
incremental algorithms. Complexity of cell structures. Convex hull algorithms
in two and three dimensions. Average case analysis of algorithms. Geometry of
linear programming. Point location in arrangements. Maximal inscribed sphere,
minimal volume enclosing simplex. Maximization of vector sums. Similarity
search. Algorithms for determining Voronoi diagram, Delaunay triangulation
Triangulations. Closest neighbor problem Determining the minimal length
spanning tree. Shape of point sets. Separation and intersection problems.Design
of algorithms. *

Recommended literature: M. de Berg, M. van Kreveld, M. Overmas, O. Schwarzkopf: Computational Geometry, 2nd. revised edition, Springer 2000.

H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer, New York, 1987.

MDPT245. Geometric algebra (Gábor P. Nagy, PhD)

*Affine and projective spaces. Desargue theorem and
coordinatizing with fields. Pappos theorem and commutativity of the
coordinatizing field.The characteristic of the coordinatizing field and the
Fano configuration. **Collinearity*-*preserving
functions, semilinear maps. Sympletic and othogonal geometry. The structure of
the symplectic and orthogonal groups. Clifford algebra.*

Compulsary literature: E. Artin, Geometric Algebra, Princeton University, 1957.

R. Baer, Linear Algebra and Projective Geometry, Academic Press, 1952.

Recommended literature: D. R. Hughes, F. C. Piper: Projective Planes, Springer, 1970.

J. Dieudonné, La Géométrie des Groupes Classiques, Springer, 1955.

Gy. Kiss and T. Szőnyi: Finite Geometries, Polygon, Szeged, 2001. (Hungarian)

MDPT246. Algebraic topology (János Kincses, CSc)

*Simplicial complexes and homotopy. Baricentric division and
simplicial approximation theorem. The fundamental group and computational
methods for it. 2-dimensional triangulable manifolds and their
characterization. Singular homology groups and algorithmic methods for them:
simplicial homology, exact sequences. Homology with arbitrary coefficients.
Lefschetz fixed point theorem. Cohomology groups and their computation.
Alexander-Poincare duality. Homotopy theory of CW-complexes Whitehead theorem
and cellular approximation. Homology and cohomology of CW-complexes.Hurewitz
theorem. Products in cohomology.*

Recommended literature: S. Eilenberg, N. Steenrod, Foundations of Algebraic Topology, Princeton, 1952.

E. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966.

C. R. F.Maunder, Algebraic Topology, Van Nostrand Reinold, London, 1970.

W. S. Massey, Singular Homology Theory, Springer, 1980.

*A representative from
the specialized courses: *

MDPT3402.**
**Graph
theory (Péter Hajnal, CSc)

*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.*

Recommended literature: L. Lovász and M.D. Plummer, Matching theory, Akadémiai Kiadó, Budapest, 1986.

B. Bollobás, Modern graph theory, Graduate Texts in Mathematics 184., Springer- Verlag, New York, 1998.

Reinhard Diestel, Graph theory, Second edition, Graduate Texts in Mathematics 173., Springer-Verlag, 2000.

**5. Stochastics**

*Programme
director:*** ****Prof.**** András
Krámli ****DSc**

Faculty of
Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary**
**Phone: +36-62-544-097, fax: +36-62-544-548,

E-mail: kramli<at>informatika.ilab.sztaki.hu

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

*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, P. Major MHAS, Gy. Pap DSc, K. Stéhlikné Boda
PhD, L. Viharos PhD.

* *

*Basic courses:*

MDPT251-252. Probability Theory I-II (Gyula Pap, DSc)

*Bernoulli's strong law of large numbers
and the DeMoivre-Laplace theorem. Kolmogorov's foundation of probability
theory. Random variables, random vectors and their distribution, distribution
function. Stochastic processes: Kolmogorov's existence theorem. Independence and product spaces. Discrete, absolutely continuous and singular
distributions; Lebesgue's decomposition. Convolutions. Expectation, moments,
variance and correlation coefficient. Important special distributions. Types of
convergence. Laws of large numbers, 0-1 theorems, three series theorem. Weak
convergence, convergence in distribution. Helly's theorem, tightness.
Characteristic functions. Central limit theorems. Multivariate normal
distributions and multivariate central limit theorems. Local central limit
theorems and asymptotic expansions. Conditional probability, conditional
expectation and conditional distributions. Random walks. Martingales, Markov
chains and stationary sequences. Brownian motion, Gaussian processes: existence
and continuity. Nondifferentiablity of the Wiener process. Laws of iterated
logarithms, fluctuations. Renewal processes, Poisson process. Combinatorial
methods for the random walk, the arc sine law.*

Literature: Billingsley: Probability and Measure, New York, 1986

S. Csörgő: Fifty-three Lectures on Probability, Ann Arbor, 1991 (lecture notes)

Feller: Introduction to Probability Theory and its Application I, II, New York, 1968, 1971

Kallenberg: Foundations of Modern Probability, New York, 1997

Petrov: Sums of Independent Random Variables, Berlin, 1975

Spitzer: Principles of Random Walks, New York, 1964

MDPT25-2543. Mathematical statistics I-II (András Krámli, DSc)

*I.
Empirical distributions, the Glivenko-Cantelli theorem. Exponential family of
distributions. The Fisher information. Theory of point estimates: sufficient
statistics, Fisher-Neyman factorization theorem, minimal sufficient
statistics, unbiased estimates, consistency, admissibility. The
Blackwell-Kolmogorov-Rao theorem, completeness. The Cramér-Rao inequality,
efficiency. Estimation methods: the momentum method, the method of minimal
divergence, the maximum-likelihood method. The asymptotic properties of the maximum-likelihood
method: consistency, asymptotic normality and efficiency. Bayes-estimates:
admissibility, unbiasedness, the minimax property. Confidence intervals: exact
and asymptotic methods.*

*II. Analysis of
contingence tables: the loglinear model. Bias reduction, a short summary of
,,jackknife'' and ,,bootstrap'' methods. The basic notions of statistical
hypotheses testing: tests, significance, power of a test, the Neyman-Pearson
lemma, unformly most powerful unbiased tests. Monotone likelihood ratios,
locally best, invariant, and similar tests. χ ^{2} tests, the
classical tests for the parameters of the gaussian distribution, goodness of
fit tests. The estimation of the parameters of the multidimensional gaussian
distribution. Regression, linear regression, method of lest squares, analysis
of variances.*

Literature: Bickel, Doksum: Mathematical Statistics, Oakland 1977

Borovkov: Mathematical Statistics, Amsterdam 1998

Cramér: Mathematical Methods of Statistics, Princeton, 1946

Efron: Bootstrap methods: Another look at the Jackknife, Ann. Statist. 7 (1979) 1-26

Kullback: Information Theory and Statistics, New York 1959

Lehmann: Theory of Point Estimation, New York 1983

Lehmann: Testing Statistical Hypotheses, New York 1983

MDPT255. Introduction to the ergodic theory (András Krámli, DSc)

*Mean and pointwise
ergodic theorems. The theory of maps with discrete spectrum. Examples: the
rotation of the unit circle, Bernoulli shift, baker's automorphism, Arnold's CAT. The problems related to the continued fractions.** *

Literature: Halmos: Lectures on ergodic Theory, Tokyo, 1956

Kornfeld, Fomin, Sinai: Ergodic Theory, New York, 1982

Khinchin: Kettenbrüche, Lepzig, 1956

MDPT256. Introduction to the Kolmogorov-Arnold-Moser theory (András Krámli, DSc)

*The n-body problem
of classical mechanics, perturbation of Hamiltonian systems, the problem of
small divisors. The proof of the twist-lemma and its application to the
restricted 3-body problem.** *

Literature:
* *Moser: Lectures on Hamiltonian Systems, New York, 1968

Siegel, Moser: Lectures on Celestial Mechanics, New York, 1965

*Two representatives
from the specialized courses:*

MDPT3508. Foundations of stochastic processes (András Krámli, DSc)

*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. *

Literature: Doob: Stochastic Processes, London, 1953

Gikhman, Skorokhod: Introduction to the Theory of Random Processes, Mineola, NY, 1996

Takács: Stochastic Processes London, 1966

Harris: Theory of Branching processes, New York, 1999

MDPT3506. Extreme-value distributions (László Viharos, PhD)

*The maximum of
independent and identically distributed random variables: Gnedenko’s theorem on
possible limiting distributions. Domain of attractions of extreme-value
distributions. Extension for dependent variables; correlated Gaussian
sequences. *

Literature: de Haan: Regular Variation and Sample Extremes, Amsterdam, 1975

Galambos: The Asymptotic Distribution of Extreme Order Statistics, Malabar, 1987

**Didactics of Mathematics**

*Coordinator:*** Assoc.
Prof. ****József Kosztolányi****, CSc**

Faculty of
Science, Bolyai Institute

H-6720 Szeged, Aradi vértanúk tere 1. Hungary**
**Phone: +36-62-544-095, fax: +36-62-544-548,

E-mail: kosztola<at>math.u-szeged.hu

This is not a separate educational programme. Each of the five educational programmes above includes topics and courses in the Didactics of Mathematics.

__Supervisors:
__J. Kosztolányi PhD, J. Karsai
CSc, L. Klukovits CSc, L. Pintér CSc, I. Szalay CSc, L. Szilassi PhD.

*Major courses** (outside the
previous five programmes) :*

MDPT261. Selected chapters from the methodology of teaching algebra, number theory, geometry, and combinatorics in high schools and universities (József Kosztolányi, PhD)

MDPT262. Selected chapters from the methodology of teaching analysis, probability theory and statistics in high schools and universities (József Kosztolányi, PhD)

*One representative from the
specialized courses:*

MDPT3115. Teaching algebra at universities in the twentieth century (Lajos Klukovits)

*The position of
algebra at the beginning of the twentieth century. Prerunners and the
breakthrough of modern algebra. The comparison of Van der Waerden, Bourbaki,
Jacobson, Rédei and Birkhoff’s books. Great inventions of the twentieth
century: lattices, categories, universal algebra, homological algebra, etc.,
and their role in teaching. Algebra in Hungarian education and the
corresponding books from G. König till nowadays. *

Literature: Birkhoff–MacLane: Algebra

Birkhoff–Bartee: Modern Algebra in Computer Science

Corry: Modern Algebra and the Rise of Mathematical Structures

Van der Waerden: Moderne Algebra

Rédei: Algebra

Jacobson: Lectures on Abstract Algebra; General Algebra

Selected chapters from Bourbaki’s books

Selected chapters from Mathematical Intelligencer

Selected chapters from publications on the hystory of mathematics

**Representative dissertations**

(Title, *author*,
supervisor, year. The dissertations are available at

*http://www.math.u-szeged.hu/phd/* )

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.

On
stability conditions of operator semigroups,* Z. Léka*, L. Kérchy, 2009.

Linear
combinations of iid random variables from the domain of geometric partial attraction
of a semstable law, *P. Kevei*, S. Csörgő, 2009.

Pointwise convergence of Fourier and conjugate
series of periodic functions in two variables, *Á. Jenei*, F. Móricz,
2009.

Online algoritms for combinatorial problems, *J.
Nagy-György*, P. Hajnal, 2009.

Combinatorical completely 0-simple semigorups
and free spectra, *K. Kátai-Urbán*, L. Megyesi and Cs. Szabó, 2009.

Monoidal intervals, *M. Dormán*, Á.
Szendrei, 2010.

Approximation
of convex bodies by polytopes,* V. Vígh*, F. Fodor, 2010.