Doctoral School of Mathematical and Computational Sciences at the University of Debrecen Head of Doctoral School: Prof. Dr. Zsolt Páles, member of the Hungarian Academy of Sciences, University of Debrecen, Department of Analysis Postal address: H4002 Debrecen, POBox 400, Hungary Phone: +3652512900 (ext.: 22811), Fax: +3652512728 Email: pales at science.unideb.hu
Doctoral Programs Didactics – teaching of mathematical and computational sciences
Leader: Prof. Dr. Gyula Maksa, professor emeritus Email: maksa at science.unideb.hu
Description of program: Comparative study of the curriculum of mathematics in various countries, and its ties to the history of mathematics. Research of productivity in mathematics. Development of instruction of the different subfields of mathematics. Research of students' mathematical thinking processes and methods for making these processes more successful. Methodology of talent management and of training for mathematical competitions. Methodology of instruction of mathematics at university level and comparative analysis of its content and organisational framework. Application of multimedia in the instruction of mathematics.
Differential geometry and its applications
Leader: Dr. Csaba Vincze, associate professor Email: csvincze at science.unideb.hu
Description of program: Local and global properties of Riemannian and Finsler manifolds. Study of manifolds with special metrics. Investigation of curvature properties. Linear and nonlinear connection theory. Investigation their holonomy groups. Metrization problems. Invariant metrics and invariant connections with respect to Lie transformation groups. Higher order connections (conform, projective connections and generalisations). Conformal and projective change of Finsler metrics and their invariants. Special Finsler spaces, spaces of scalar or constant curvature. Berwald, Landsberg, P*spaces, and the study of their generalisations. Finsler metrics of Kropina and Randers type. The conditions for the vanishing of Douglas' tensor. Geometric theory of partial differential systems. Foliations and their tangent distributions. Web geometry. SpencerGoldschmidt version of the CartanKaehler theory of overdetermined partial differential systems, its application to differential geometry and to the inverse problem of calculus of variations. Theory of affine symmetric spaces and of reductive homogeneous manifolds. Application to metric differential geometry. Automorphism groups of second order differential equations. Algebraic theory of Lie triple systems and the analytic theory of the corresponding affine symmetric spaces. Generalisations. Topological and differentiable algebraic systems, applications to geometry. Topological geometry. Differentiable nonassociative algebra. Lie theory of Moufang and Bol loops.
Diophantine and constructive number theory
Leader: Prof. Dr. Kálmán Győry, member of the Hungarian Academy of Sciences Email: gyory at science.unideb.hu
Description of program: Finiteness theorems for Diophantine equations. Applications of the ThueSiegelSchmidt method to Diophantine equations, including decomposable form equations, unit equations. Quantitative results, bounds for the number of solutions. Effective finiteness theorems for Diophantine equations. Extensions and improvements of some known effective results, combining Baker's method with some other techniques. Constructive algebraic number theory. Development, analysis and implementation of algorithms for determining arithmetic invariants of algebraic number fields and elliptic curves. Numerical resolution of Diophantine equations. Development, analysis and implementation of algorithms for numerical resolution of equations. Recurrence sequences. Investigation of Diophantine and arithmetical properties of linear recurrence sequences.
Explicit methods in algebraic number theory
Leader: Prof. Dr. István Gaál, full professor Email: gaal.istvan at unideb.hu
Description of program:
Functional analysis
Leader: Prof. Dr. György Gát, full professor Email: gat.gyorgy at science.unideb.hu
Description of programs:
Ring theory: group algebras and unit groups
Leader: Prof. Dr. Ákos Pintér, full professor Email: apinter at science.unideb.hu
Description of program: The grouptheoretical properties of the group of units and its unitary subgroups in group ring and in crossed algebra. Description of involutions in group rings and the study of corresponding unitary subgroups of the group of units. Determination of the generating system of the group of units and its unitary subgroups. Computation of the nilpotency class and the exponent of the group of units and its unitary subgroups in modular algebra. The Lie properties of the group algebra and crossed products. Application of Lie properties for studying the group of units in group algebra. Determination of formal languages and codes possessing special algebraic, combinatorial and algorithmic properties. Description of composition of automata and sequential machines by means of group algebraic methods.
Mathematical analysis, functional equations and inequalities
Leader: Prof. Dr. Zsolt Páles, member of the Hungarian Academy of Sciences Email: pales at science.unideb.hu
Description of program: Classes of interval filling sequences, possibilities of generalisations and open problems. Algorithms, functions additive with respect to an algorithm. Functional equations and functional inequalities. Problems connected with classical functional equations. Equations of Abel type and of sum form, the corresponding inequalities. Generalisations of mean values. Regularity properties of solutions of functional equations. Boundness, integrability and continuity of measurable solutions. Lipschitz property, differentiability and analicity of solutions. Functional equations in probability theory. Characterisations of probability distributions by help of functional equations. Functional equations in the spectral theory of probability fields. Functional equations of sum form and their applications. Methods of solution for equations of sum form. Structure theorems for functional equations of sum form of (2,2) type. Applications in the theory of information. Mean values, inequalities. Inequalities for deviation means. Inequalities for the powers of linear operators, connection with differential inequalities. Special solutions of functional equations. Nonnegative solutions. Special additive functions and derivations. Convex quadratic functions. Extremum problems. The theory of DubovitskiiMiljutin and its applications. The clark derivate and its applications for the solutions of extremum problems with constraints. Covering properties of relator spaces. Compact, Lindelöf, Lebesgue and paracompact relators. Functional equations on topological groups. Polynomials and exponential polynomials on topological groups. Fourier transformation and its applications, meanperiodic functions on Abelian groups.
Computational science and its applications
Leader: Prof. Dr. Lajos Hajdu. full professor, Email: hajdul at science.unideb.hu
Description of program: Logic aspects of the artificial intelligence. Automated reasoning, the inference technique in expert systems using mathematical logics. System control theory. Control of dynamic and stochastic systems. Performance evaluation of computer systems. Patter recognition. Decisiontheoretic and syntactic approach, statistical and structural methods in image analysis. Problems of computer graphics. Construction, visibility and visualisation algorithms of 3D objects in computer graphics.
Probability theory, mathematical statistics and applied mathematics
Leader: Prof. Dr. Attila Bérczes, full professor, Email: berczesa at science.unideb.hu
Description of program:
Stochastic differential equations. Solution of statistical problems in connection with stochastic differential equation. Determination of RadonNykodim derivates of statistics concerning parameter estimation. Applications of stochastic models. Control of stochastic systems and optimisation problems in real computer systems. Mathematical models for economic processes. Banach space valued random variables. Sequences of random variables with multidimensional indices, laws of large numbers, martingales and their generalisations, statistical distances. Mixtures of probability measures. Linear approximation in convex metric spaces and the application in the mixture theory of probability. Statistical problems for unstable and nonstationary systems. Investigation of unstable and nearly nonstationary ARMA models. Parameter estimation and hypothesis testing problems for nearly nonstationary autoregressive processes. The theory of semimartingales. Limit theorems and probability measures on Lie groups. Infinitely divisible, stable and Gaussian measures on topological groups, especially on nilpotent Lie groupscases. Central limit theorems. Queuing problems in computer science and reliability theory. Stochastic models of large queuing systems. Explicit solutions for systemparameters in different special cases. Nonlinear identification of time series. Interpretation of nonlinear stochastic models. Investigation of bilinear stochastic processes. statistical and identification type results, simulations.
Information Academic staff of Doctoral School
Contact: Dr. Magda Várterész associate professor, Secretary of Doctoral School Institute of Mathematics, University of Debrecen P.O. Box 400, H4002 Debrecen, HUNGARY 47.556274,21.619455 +3652 / 512900 / 75112 phd.titkar at science.unideb.hu
