Mathematics

1. Application of methods of scattering theory and methods of Krein spaces theory for the investigation of non-self-adjoint operators.

Supervisor: dr hab. Sergiusz Kużel, prof. AGH

Faculty of Applied Mathematics

Abstract: The steady interest in spectral theory of non-self-adjoint operators has increased significantly over the last twenty years. This is due to the recent progress in PT-symmetric quantum mechanics, see doi.org/10.1142/q0178. The basic idea of ​​the research problem is the spectral analysis of not self-adjoint operators (PT-symmetric hamiltonians) with emphasis on the use of the methods of Krein space theory and the scattering theory. The research problem is the continuation of the research projects: "Application of Krein spaces methods" and "Lax-Phillips scattering method and its applications", www.researchgate.net/profile/Sergii_Kuzel

Number of places: 1

 

2. Global and local irregularity of graphs.

Supervisor: prof. dr hab. Mariusz Woźniak

Faculty of Applied Mathematics

Abstract: The aim of the project is to obtain new results regarding broadly understood problems of distinguishing vertices of the graph. If we differentiate all vertices, we are talking about global irregularity; if only neighboring vertices, we are talking about local irregularities. This distinction in one way or another results from coloring the edge of the graph. Any such coloring assigns to the vertices of a graph palette of colors, i.e. the multiset of colors on incidental edges. These palettes are used to distinguish between them. If the coloring is general then we can consider sets as well as multisets. Usually, the colors are the numbers from the set {1, 2, . . . , k}, therefore, arithmetic operations on colors are also possible. Issues leading to the distinction of all graphs have been considered for over thirty years. Problems of coloring the edge in the way distinguishing only neighbors appeared only in the 21st century, gaining immediately large interest. Some new idea appeared recently and their systematic analysis seems necessary and very promising.

Research facilities: The project concerns theoretical research in mathematics, and the required research facilities are very modest (access to the Internet, printers, seminar and consultation rooms). The faculty has such facilities.

Number of places: 5

 

3. Arbitrarily partitionable graphs.

Supervisor: dr hab. Rafał Kalinowski

Faculty of Applied Mathematics

Abstract: The concept of arbitrarily partitionable graphs arised in the beginning of XXI century (independently in Paris and Kosice) and it was motivated by some applications in computer science. Rougly, the question concerns a possibility of a partition of a vertex set into connected subgraphs of prescribed cardinality. Afterwards, tens of papers on the topic appeared in significant journals of mathematics and theoretical computer science. This topic is currently investigated in several centerds in a dozen of countries, including United Kingdom, Hungary, China, Germany, the Nethrlands and Slovenia. Several natural modifications of the original concept of arbitrary partitionability were defined. Numerous interesting open problems are waiting for solutions. Some of them are very suitable for PhD dissertations.

Research facilities: Faculty of Applied Mathematics provides good conditions for scientific research of PhD students, like a desk with a connection to a computer and a printer, and also to the Internet with reach data bases and mathematical journals.

Number of places: 2

 

4. Inverse limits, attractors and rotation sets.

Supervisor: prof. dr hab. Piotr Oprocha

Faculty of Applied Mathematics

Abstract: The doctoral dissertation will deal with the applications of inverse limits to the construction of various interesting attractors. At the center of interest there will also be a possibility of embedding into the plane of a continuum with the map on it in such a way that the map can be extended to the entire plane. The main object of interest will be indecomposable continua. The shape of the rotation set for attractors in the two-dimensional torus will also be studied, as well as other dynamic properties of attractors obtained through the use of inverse limits..

Research facilities: The doctoral student will have access to standard mathematics tools, such as: computer, specialized journals and monographs, mathscinet and other reference databases. When necessary, also specialized mathematical software (mathematica, maple, mathlab) will be available.

Number of places: 1

 

5. Lovász Local Lemma and entropy compression method in graph colouring.

Supervisor: dr hab. Jakub Przybyło

Faculty of Applied Mathematics

Abstract: The Lovász Local Lemma is one of the principal tools of the probabilistic method, having been utilized within widely understood combinatorics to solve problems which frequently could not have been solved by means of other methods. The entropy compression method is a relatively new approach developed within research on a constructive version of the Lovász Local Lemma which often yields better results than its archetype and provides a randomized algorithm for finding a required object or structure. The objective of the research project shall be applications of the mentioned methods to solve various problems within the discipline of graph colourings and labellings.

Research facilities: The research objectives do not require using any specialistic equipment or device. The doctoral advisor possesses wide knowledge and experience concerning the proposed subject, confirmed by his grant projects and the object of his habilitation dissertation.

Number of places: 1

 

6. Problems of stability of some families of solutions of nonlinear nonlocal models of structured media.

Supervisor: dr hab. Vsevolod Vladimirov, prof. AGH

Faculty of Applied Mathematics

Abstract: Search for physically meaningful solutions (in particular travelling wave solutions) of non-linear partial differentia equatins describing structured media, such as block and granular media, multiphase media, etc. Study of spectra stability of the aforementioned solutions by analytical methods and numerical methods based on the Evans function technique. Numerical study of the dynamics and evolution of wave patterns.

Research facilities: personal computer

Number of places: 1

 

7. Numerical methods to interdiffusion models.

Supervisor: dr hab. Bogusław Bożek

Auxiliary supervisor: dr Lucjan Sapa

Faculty of Applied Mathematics

Abstract: The local mass conservation law for fluxes with the Darken drift term and the Vegard rule lead to the parabolic or the parabolic-elliptic systems of strongly coupled nonlinear partial differential equations with the nonlinear coupled initial-boundary conditions. Problems: a) theorems on existence, uniqueness and properties of weak solutions in the suitable Sobolev spaces and classical solutions, b) construction of numerical methods (difference schemes, Galerkin methods) and theorems concerned convergence and stability, c) the agreement between the theoretical results, numerical simulations and experimental data.

Number of places: 1

 

8. Boundary values of holomorphic functions.

Supervisor: dr. hab. Piotr Kot

Faculty of Applied Mathematics

Abstract: We consider the strictly pseudoconvex domains D in C^n (n>1) and holomorphic functions f on D with predetermined limits in non-tangent directions for nearly all points of the boundary D. If limits of the module f are equal to 1 a.e. on boundary D then such a function is called inner and is characterized by strong pathological behavior. In particular, if f is continuous at one point of boundary D, it must be constant. We will examine various pathological behaviors f near the boundary of D. We will examine the influence of zeros f on its possible pathological behavior. We consider also the construction of peak and maximum modulus sets.

Research facilities: Good access to literature is required to implement the project.

Number of places: 1

 

9. Symmetry breaking by colourings in graphs.

Supervisor: dr hab. Monika Pilśniak

Second supervisor: dr hab. Rafał Kalinowski

Faculty of Applied Mathematics

Abstract: Symmetry breaking in discrete structures (graphs, groups, posets, vector spaces etc.) has belonged to the mainstream of discrete mathematics for a dozen of years. This was caused by applications in theoretical computer science, and first of all, by the fact that in the middle of the first decade of XXI century excellent mathematicians joined the investigation of this topic. In 2015, Kalinowski and Pilśniak published a paper in European Journal of Combinatorics introducing a concept of distinguishing edge-colourings of graphs. They defined the distinguishing index of a graph as the least number of colours in an edge-colouring breaking all non-identity automorphisms. This idea almost immediately found the interest in several papers of recognized mathematicians in the United States, South Africa, Austria, Slovenia, Iran and United Kingdom. Many open problems could be attacked by our PhD students. For instance, a sharp upper bound of the distinguishing index for graphs of bounded minimum degree. Another question is wheater the distinguishing index of any countable regular graph is less or equal two with a finite number of exceptions.

Research facilities: Faculty of Applied Mathematics provides good conditions for scientific research of PhD students, like a desk with a connection to a computer and a printer, and also to the Internet with reach data bases and mathematical journals. The two supervisors are known experts in symmetry breaking in graphs and they provide suitable research problems to be solved in doctoral dissertations.

Number of places: 1

 

10. Information-Based Complexity of numerical problems.

Supervisor: dr hab. Paweł Przybyłowicz

Auxiliary supervisor: dr Paweł Morkisz

Faculty of Applied Mathematics

Abstract: Research topics cover: construction and analysis of efficient numerical methods for approximate solving of deterministic and stochastic differential equations, investigation of optimality of designed methods in the Information-Based Complexity framework, efficient implementation of defined methods in CUDA C together with Monte Carlo simulations on graphics processing unit (GPU), practical applications of defined methods to, for example, option pricing and simulation of real world engineering processes.

Research facilities: The doctoral student will have access to standard mathematics tools, such as: computer, specialized journals and monographs, reference databases (like, for example, MathSciNet). Moreover, specialized mathematical software (Mathematica, Maple, Mathlab, Statistica) is available when needed. Additionally, we provide access to efficient computing workstation powered by graphical processing units for conducting parallel numerical computations. Moreover, it is possible to run the code also on the AGH supercomputer infrastructure.

Number of places: 3

 

11. Resampling methods for generalizations of almost periodically processes.

Supervisor: dr hab. Anna Dudek

Faculty of Applied Mathematics

Abstract: Many complex physical and biological signals cannot be sufficiently characterized by the classical second-order spectra. Signals such as electrocardiograms, communications, astronomical and financial often present irregular cyclicity and nonstationarity in their statistical features and thus present challenges to current statistical tools. To model such phenomena generalizations of almost periodically processes are used. The main aim of this research task will be to consider nonstationary processes like spectrally correlated processes, oscillatory almost periodically correlated processes and generalized almost cycloctationary processes. To be able to construct confidence intervals for characteristics of these processes it is necessary to introduce new resampling methods or to adapt existing algorithms and to show their consistency in the considered problem.

Research facilities: personal computer

Number of places: 1

 

12. Total domination in graphs.

Supervisor: dr hab. Monika Pilśniak

Faculty of Applied Mathematics

Abstract: A set D of vertices in an isolate-free graph G is a total dominating set of G if every vertex is adjacent to a vertex in D. The total domination number, γt (G), of G is the minimum cardinality of a total dominating set of G. We note that γt (G) ≥ 2 for every isolate-free graph G. A basic monograph in this area is M.A. Henning and A. Yeo, Total Domination in Graphs, in: Springer Monographs in Mathematics, 2013. It is worth notice that Monika Pilśniak collaborates with Michael A. Henning. A paired dominating set in a graph G is a dominating set D that a perfect matching is included in the subgraph of G induced by D. Every paired dominating set is a total dominating set, so the total domination number is always at most the paired domination number. It was defined in T.W. Haynes, P. J. Slater: Paired-domination in graphs. Networks 32 (1998), 199–206. We shall consider open problems and intriguing conjectures of this topic. In particular we will research the minimum size of a non-isolating set of vertices in G whose removal changes the total domination number, called the total domination stability. It was introduced in D. Bauer, F. Harary, J. Nieminen, C. Suffel, Domination alternation sets in graphs, Discrete Math. 47 (1983) 153–161. We shall prove some general bounds for connected graphs, and we determine also the paired domination stability and the total domination stability for important classes of graphs.

Research facilities: Faculty of Applied Mathematics provides sufficient research facilities for a PhD student of mathematics (a desk with computer and printer with Internet access to reach scientific bases). Dr hab. Monika Pilśniak is a dynamically working scientist - in the last two years, she published more than 10 papers in international journals, indexed by ISI and included in the list A of Polish Ministry of Science and Higher Education, and 5 next submitted and available as preprints in the Internet. Recently, she started scientific cooperation with Michael A. Henning of University of Johannesburg, South Africa. Professor Henning is an undisputed expert in domination theory in graphs (above 450 published papers, and more than 2750 citations by MathSciNet). In the coming years, he will give lectures at Faculty of Applied Mathematics in AGH as a visiting professor.

Number of places: 1

 

13. Algebras of analytic functions.

Supervisor: dr hab. Krzysztof Rudol

Faculty of Applied Mathematics

Abstract: Analysis of joint spectra related to algebras of bounded analytic functions and their representations in operator algebras. The study of geometry of the spectra of these algebras and of their representing measures. Application of complex functions theory, measure theory, uniform algebras and functional analysis.

Research facilities: Possibilities of collaboration with experts from the Jagiellonian University in Cracow, availability of a well- supplied library of the faculty, professional periodicals, two seminars at the faculty devoted to applied functional analysis and function algebras, possibilities of applications in control theory represented at one of the faculties of AGH University of Science and Technology.

Number of places: 1

 

14. Discrete Mathematics.

Supervisor: dr hab. Wit Foryś

Faculty of Applied Mathematics

Abstract: In the frame of discrete mathematics the research includes discrete models and their combinatorial aspects including the combinatorics on words.

Research facilities: seminars of FAM; realization as a part of scientific projects is possible.

Number of places: 1

 

15. Infinite-dimensional Lie algebras and integrable nonlinear differential equations.

Supervisor: dr hab. Oleg Morozov

Faculty of Applied Mathematics

Abstract: Applications of the structure theory of infinite-dimensional Lie algebras to nonlocal geometry of integrable nonlinear partial differential equations, in particular, the analysis of relations between the structure properties of contact symmetry algebras and existence of Lax repre­sentations for PDEs. The study of deformations of cer­tain infinite-dimensional Lie algebras, their non-central extensions and asso­ciated integrable PDEs.

Research facilities: personal computer

Number of places: 1

 

16. Link invariants of finite order in 3-dimensional space.

Supervisor: dr. hab. Leonid Plakhta

Faculty of Applied Mathematics

Abstract: Invariants of finite order of knots were introduced by V.Vassiliev in 1990 year. The main results concerning properties of Vassiliev invariants and their relations to classical polynomial invariants were obtained in 90th of XX century and at the beginning of XXI century. However the main questions of whether these invariants are more powerfull then the classical polynomial invariants and whether they can distinguish all knots remain without answer. In the case of links, there is a variety of definitions for link invariants of finite order, dependent on the singularities of links which are allowed and on an equivalence relation given by isotopy. One of such concepts is the Kirk-Livingston invariant for 2-component links. In the proposed project, we focus mainly on the following topics and problems.

1) The study of algebraic structure of groups of Kirk-Livingstone invariants;

2) Constuctions of Kirk-Livingstone type invariants of links based on known classical algebraic and geometric invariants of them;

3) Understanding the geometric properties of some link invariants of finite order and finding their relations with classical link invariants.

The advisor has made several contributions to the study of knot and link invariants of finite order and to the related topics in knot theory. The results on this subject were presented in his habilitation dissertation titled „Knot invariants and surfaces in 3-dimensional space “ and were published in mathematical journals.

1. L.Plachta, C_n-moves, braid commutators and Vassiliev knotInvariants, J. Knot Theory Ramifications, 13, No.6, 2004,09-828.

2. L.Plachta, Essential tori admitting standard tiling, Fundamenta Math., 189, 2006, 195-206.

3. L.Plachta, Knots, satellite operarions and invariants of finite order, J. Knot Theory Ramifications, 15, No.8, 2006, P.1061-1077.

4. L.Plachta , Genera, band sums of knots and invariants of finite order, Topology Appl., 2007, 154, 2880-2887.

5. L.Plachta, Invariants of knots, surfaces in R3 and foliations, Ukr. Math. Zh., 59, 2007, No.9, P.1239-1252

6. L.Plachta, Notes on tiled incompressible tori, Cent. Eur. J. Math., 10(6), 2012, pp.2200-2210

Liczba miejsc: 1

Number of places: 1

 

17. Generalized symmetries and reduction of nonlinear evolution type differential equations.

Supervisor: dr hab. Ivan Tsyfra

Faculty of Applied Mathematics

Abstract: We will study the generalizations of classical Lie symmetry, namely conditional point symmetry and Lie-Bäcklund symmetry and their applications for reducing nonlinear evolution equations. By integrating the reduced systems of ordinary differential equations one can obtain solutions of partial differential equations under study. Mainly we will investigate nonlinea diffusion equations and we also show when the generalized symmetry method enables us to find solutions which can’t be obtained in the framework of classical approach. We will also consider applying of the method to integrable equations. The research is based on the articles

Fushchich W.I., Tsifra I.M. on a reduction and solutions of nonlinear wave equations with broken symmetry, J. Phys. A: Math. Gen. 1987, v.20, no. 2, L45-L48

Zhdanov R.Z., Tsyfra I.M. and Popovych R.O.A precise definition of reduction of partial differential equations, J. Math. Anal. Appl. 1999, v.238, 101—123

Tsyfra I. M. Symmetry reduction of nonlinear differential equations, Proceedings of Institute of Mathematics of NAS of Ukraine 2004, v.50, 266-270

Number of places: 1

 

18. Weighted shifts on directed semi-trees.

Supervisor: prof. dr hab. Petru Cojuhari

Auxiliary supervisor: dr Jerzy Stochel

Faculty of Applied Mathematics

Abstract: The basic properties of weighted shifts on direct semi-trees should be studied. It is also planned to describe weighted shifts and their adjoints on directed semi-trees. Hyponormality, cohyponormality and subnormality of such operators should be characterized in terms of their weights.

Number of places: 1

 

19. Continuous dependence in a problem of convergence of random iteration.

Supervisor: dr hab. Rafał Kapica

Faculty of Applied Mathematics

Abstract: Markov operators acting on measures play an important role in describing the evolution of distributions for stochastic dynamic systems. One of the most important and well-researched properties of such operators is asymptotic stability, which means the existence of (a unique) and attractive invariant measure. The research goal is to analyze an invariant measure depending on changes of operator parameters in the context of applications to the iterates of random-valued vector functions and linear iterative equations of infinite order including refinement equations.

Number of places: 1