Research work 

Department of Mathematics

Department of Physics

Department of Mathematics Faculty of Mathematics and Applied Physics

Areas of research and scientific activity

Algebra, Geometry and Topology

Main research interests:

• Braid groups and Artin groups,
• Mapping class group of a surface,
• Branched coverings of a disk and a surface and their automorphisms,
• Topological, diffeomorphic and symplectic structure of complex algebraic surfaces,
• Geometric group theory, groups and spaces of non-positive curvature,
• Affine algebraic geometry,
• Jacobian conjecture,
• Homogeneous forms and their automorphisms,
• Representing forms of higher degree as a sum of powers of linear forms,
• Automorphisms of higher degree forms,
• Affine submanifolds of higher codimensions,
• Classifing submanifolds with planar geodesics.
  

Among others, the results have been published in:

• Inventiones Mathematicae, Journal of Differential Geometry, Mathematische Annalen, Journal of Algebra, Israel Journal of Mathematics,  Journal d'Analyse, Contemporary Mathematics, Geometry and Topology, Indagationes Mathematicae, Acta Mathematica Hungarica, Geometriae Dedicata, Linear Algebra and its Applications.
  

Collaboration:

• Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Izrael,
• Bayreuth University, Germany, 
• Columbia University, New York, USA,
• California University, Davis, USA,
• Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland,
• Institute of Mathematics, Jagiellonian University in Kraków, Poland,
• Institute of Mathematics, University of Silesia, Katowice, Poland.

Applied Mathematics

Main research interests:

• Boundary value problems of PDEs with discontinuous coefficients,
• Solid, fracture and computational mechanics: fracture criteria, fracture pulse, hydraulic fracture
• Fluid dynamics,
• Singular integral equations with fixed point singularity, hypersingular equations,
• Functional equations of the Wiener-Hopf type,
• Asymptotic analysis of BVP in non-smooth domains,
• Numerical methods: numerical simulation of seismic and aseismic events,
• Applications to Solid Mechanics: mathematical modelling of thin interphases in composites, singular fields in domains with defects, waves in complex structures, plastic and viscoplastic flow of homogeneous materials,
• Industrial Mathematics: metal forming processes, crack propagation, fracture, damage
• Micromechanics,
• (Multi-) wedge singular points,
• Boundary Element Method for transitional processes and large systems.

Among others, the results have been published in:

• Differential Equations, Demonstratio Mathematica, European Journal of Applied Mathematics, Journal of Mathematical Analysis and Application, Applicable Analysis, ZAMM – Journal of Applied Mathematics and Mechanics, Quarterly Journal of Mechanics and Applied Mathematics, Applied Mathematics and Mechanics, Journal of Mathematics and Applications,
• Archives of Mechanics, Mechanics of Composite Materials, International Journal of Solids and Structures, European Journal of Mechanics, International Journal of Mechanical Sciences, Archives of Applied Mechanics, Communications in Numerical Methods in Engineering, Journal of the Russian Academy of Sciences, Composite Structures, Waves in Random and Complex Media, Journals of Physics and Mechanics of Solids,
• Engineering Transaction, International Journal of Damage Mechanics, International Journal of Fracture, Metal Science and Heat Treatment, Archives of Metallurgy and Materials, Journal of Adhesive and Interface, Materials Science Forum, Engineering Fracture Mechanics, Journal of Geophysical Research, International Journal of Rock Mechanics and Mining Science, Journal of Mining Science.

Collaboration:

• Division of Applied Mathematics, Department of Mathematical Sciences, University of Liverpool, UK,
• Institute of Applied Analysis and Numerical Simulation, Stuttgart University, Germany,
• Department of Mathematics, University of Southern Mississippi, USA,
• Institute of Applied Mechanics, Erlangen-Nuremberg University, Germany,
• Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia,
• Department of Structural and Mechanical Engineering, Trento University, Italy,
• Department of Mechanical Engineering, Aveiro University, Portugal,
• Faculty for Mechanical Engineering, University of Maribor, Slovenia,
• Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, Moscow, Russia,
• All-Russian Institute for Rock Mechanics and Surveying, St-Petersburg, Russia,
• St-Petersburg State Marine Technical University, St-Petersburg, Russia,
• International Seismic Systems Integrated, Stellenbosch, South Africa.

Combinatorial Analysis

• Independent and dominating sets  and its generalizations (for instance: independent and dominating by monochromatic paths sets) in graphs and in their products,
• Counting special subsets in graphs, characterization of extremal graphs – special kinds of parameters of independence and domination,
• Applications of Fibonacci numbers in graph arithmetics.

Among others, the results have been published in:

• Ars Combinatoria, Discussiones Mathematicae - Graph Theory, Australasian Journal of Combinatorics, Discrete Mathematics, Combinatorica, Journal of Mathematics and Applications.

Collaboration:

• Institute of Mathematics, University of Technology in Szczecin, Poland,
• Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland,
• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Poland.

Complex Analysis

Main scientific researches:

• Geometric function theory of univalent and multivalent functions,
• Convolutions and neighborhoods in some classes of analytic functions,
• Subordinations, differential subordination and majorization of functions,
• Integral transforms,
• Conformal and quasiconformal mappings on a complex plane,
• Analytic functions of several complex variables,
• Extremal problems (e.g. estimates of coefficients, growth and distortion theorems, etc.) in several classes of analytic functions,
• Extremal and critical points of set of  functions,
• Strongly starlike, spirallike  and strongly convex domains,
• E-starlikeness with respect to inner point,
• Harmonic univalent functions,
• Convex harmonic mappings.

Among others, the results have been published in:

• Complex Variables  Theory and Applications, Duke Journal of  Mathematics, Annales Universitatis Mariae Curie-Sklodowska, Annales Polonici Mathematici, Mathematica Japonica, Journal of Computational and Applied  Mathematics, Integral Transforms & Special  Functions, Indian Journal of Pure and Applied Mathematics, Serdica Mathematical Journal, Israel Journal of Mathematics, Rocky Mountain Journal of Mathematics, Journal of the Australian Mathematical  Society, Journal of the Korean Mathematical  Society, International  Journal of  Mathematics and Mathematical  Sciences, Journal of Applied Analysis, Commentiones  Mathematicae Universitatis Carolinae, Annales Academiae Scientarum Fennicae Mathematica, Journal of Mathematical Analysis and Applications, Proceedings of the American Mathematical Society, Revue Roumaine de Mathematiques Pures et Appliquees, Mathematica (Cluj-Napoca),  Research Institute for Mathematical Sciences, Kodai Mathematical Journal.

Collaboration:

• Department of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria,
• University of Helsinki, Finland,
• University of Jyvaskyla, Finland,
• University of Joensu, Finland,
• Universidad Autonoma Metropolitana, Mexico, Mexico,
• Nihon University, Tokyo, Japan,
• Kinki University, Higashi-Osaka, Japan,
• University of Victoria, Victoria, Canada,
• Norwegian University of Science and Technology, Trondheim, Norway,
• Babes-Bolyai University, Cluj-Napoca, Romania,
• University of Padova, Italy,
• Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland,
• Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland,
• Institute of Mathematics, The John Paul II Catholic University of Lublin, Poland,
• Lublin University of Technology, Lublin, Poland,
• Institute of Mathematics, Technical University of Łódź, Poland,
• Faculty of Mathematics and Computer Science, University of Łódź, Poland,

• Institute of Mathematics, Jagiellonian University in Kraków, Poland,

• Institute of Mathematics, University of Silesia, Katowice, Poland.

Differential Equations and Functional Analysis

Main scientific researches:

• Measure of noncompactness,
• Nonlinear differential equations,
• Partial differentia equations,
• Operator – functional equations,
• Nonlinear integral equations of Volterra, Hammerstein and Urysohn type,
• Quadratic integral equations and integral equations of functional order,
• Operator theory with a special emphasis on integral operators,
• Nonlinear stochastic differential and integral equations,
• Infinite systems of differential and integral equations,
• Complemented subspaces of Banach and quasi-Banach spaces,
• Bases in Banach and quasi-Banach spaces,
• Geometry of normed spaces, 
• Metric fixed point theory,
• Nonlinear ergodic theory,
• Fixed points of lipschitzian mappings in Banach spaces,
• Fixed points of rotative mappings in Banach spaces.

Among others, the results have been published in:

• Proceedings of American Mathematical Society, Journal of Nonlinear Analysis, Journal of Mathematical Analysis and Applications, Stochastic Analysis and Applications, Proceedings of Royal Irish Academy, Mathematica Scandinavica, Applied Mathematics Letters, Dynamic Systems and Applications, Mathematical and Computer Modelling, Computers and Mathematics with Applications, Canadian Mathematical Bulletin, Bulletin of the Australian Mathematical Society, Journal of the Australian Mathematical Society, International Journal of Mathematics and Mathematical Sciences, Glasgow Mathematical Journal, Proceedings of Royal Society of Edingbourgh, Bolletino Unione Matematica Italiana, Studia Mathematica, Colloquium Mathematicum, Mathematica Japonica, Commentiones  Mathematicae Universitatis Carolinae, Annales Universitatis Mariae Curie-Sklodowska, Indian Journal of Pure and Applied Mathematics,

Collaboration:

• Indiana University, Bloomington, USA,
• National University of Ireland, Galway, Ireland,
• Northwest Research Institute, Port angeles, Washington, USA,
• Department of Mathematics, Florida Institute of Technology, Melbourne, USA,
• Universidad de Sevilla, Sevilla, Spain,
• Universidad de Santander, Santander, Spain,
• Universidad de Extramadura, Badajoz, Spain,
• Universiada de Granada, Granada, Spain,
• Universiada de Oviedo, Oviedo, Spain,
• Universidad de Las Plamas, Las Palmas de Gran Canaria, Spain,
• Universidad de La Laguna, La Laguna (Tenerife), Spain,
• Universidad de Los Andes, Merida, Venezuela,
• Alexandria University, Alexandria, Egipt,
• Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Izrael,
• Faculty of Mathematics and Information Technology, Warsaw University of Technology, Warszawa, Poland,
• Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland,
• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland,
• Institute of Mathematics, Warsaw University, Poland,
• Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland,
• Institute of Mathematics, Nicolaus Copernicus University, Toruń, Poland

Mathematics Education

Main scientific researches:

• Researches on mathematical activity from two perspectives: classroom practice and as research object in didactics,
• Problem solving in the field of Mathematics education,
• Theoretical and practical aspects of the interaction in teaching Mathematics at school,
• Creativity in Mathematics education and the education of gifted students,
• Teacher’s intervention vs. mathematical activity and creativity of gifted pupils.
• Problems and background for further investigation of the phenomenon „to see” - training and development of „mathematical vision” of pupils and students in learning Mathematics,
• Mathematical culture and the awareness of seeing in Mathematics education,
• Spatial vision training in the process of science and Mathematics teaching.
• New trends in Mathematics education as a chance to raise its level,
• Contemporary tendency in Mathematics background for economists,
• The Bologna Process and the modernization of the mathematics education in Poland.

Among others, the results have been published in:

• Teaching of Mathematics (Polish), Mathematics at School (Russian), proceedings of conferences on Mathematics Education.

Collaboration:

• University of Rousse, Bulgaria,
• University of Hradec Kralove, Czech Republic,
• Auckland University of Technology, New Zeland,
• Herzen State Pedagogical University, Sankt Petersburg, Russia,
• Pedagogical University of Kharkov, Ukraine,
• Institute of Mathematics, Pedagogical Academy, Kraków, Poland,
• Institute of Mathematics, Swietokrzyska Academy, Kielce, Poland.

Department of Physics Faculty of Mathematics and Applied Physics

Areas of research and scientific activity

Main research interests:

• Application of the light diffraction patterns for the examination of mechanical properties of fibers;
• experimental studies of molecular dynamics of materials with glassy phase;
• experimental studies of properties of ferroelectrics and dielectrics;
• experimental studies of structure and physical properties of selected biological and synthetic biopolymers;
• interaction of sonic and ultrasonic waves with dispersions;
• kinetics of phonons and their interaction with low-dimensional conducting structures in semiconductors, propagation of acoustic waves, and mechanical properties of crystalline solids;
• mathematical methods in quantum mechanics and the theory of magnetism;
• muonic atoms and molecules in crystalline solids.
• spintronics;
• theory of patterns propagation.