**Institute of Mathematics, Lodz University of Technology I-73 **

**Head of the unit: **

dr hab. Katarzyna Szymańska-Dębowska

**Potential supervisors: **

prof. dr hab. inż. Jacek Banasiak

dr hab. Marek Galewski

prof. dr hab. Wojciech Kryszewski

prof. dr hab. Urszula Ledzewicz

dr hab. Katarzyna Szymańska-Dębowska

**Contact person:**

**Scope of activities: **

Main areas of interest and directions of scientific research: differential equations, topological methods, nonlinear analysis, dynamical systems, applications of differential equations

**Present activities: **

Our research concerns the broadly understood theory of evolutionary processes occurring in exact and natural sciences. This includes analysis of stationary states of these processes, controlling them and optimizing using various quality criteria. Such problems are generally described by ordinary or partial differential equations, functional differential equations, integral equations, and difference equations (for discrete processes) under the presence of local or non-local boundary conditions. We are interested in fundamental problems in functional analysis, operator theory, operator semigroup theory, calculus of variations and variational methods, theory of dynamical systems, fixed point theory and topology. In particular, our research pertains to questions of existence, multiplicity, structure, asymptotic behaviour and stability of solutions, and other qualitative problems. We are also interested in the applications and quantitative aspects of the models. Specifically, we model and study dynamical processes in medicine (e.g., epidemiological and transmission models, drug therapy models) and in biology (e.g., the description of fragmentation, coagulation, population and ecological processes).

**Future activities: **

Continuation of the conducted research, extending the scope of the studied problems.

**Publications/patents, awards, projects:**

Monographs

- Analytic Methods for Coagulation-Fragmentation Models, series Chapman & Hall/CRC Monographs and Research Notes in Mathematics, CRC Press (Taylor &Francis Group, 2019).

Articles

- J. Banasiak and A. Błoch, Telegraph systems on networks and port-Hamiltonians. II. Network realizability, Networks & Heterogeneous Media, Vol. 17, 2022, 73-99, DOI: 10.3934/nhm.2021024
- W.A. Woldegerima, R. Ouifki and J. Banasiak, Mathematical analysis of the impact of transmission-blocking drugs on the population dynamics of malaria, Applied Mathematics and Computation, Volume 400, 2021, Article number 126005, DOI 10.1016/j.amc.2021.126005,
- J. Banasiak and W. Lamb, Growth-fragmentation-coagulation equations with unbounded coagulation kernels, Phil. Trans. R. Soc. A 378, 2020, Article number 20190612, DOI 10.1098/rsta.2019.06122020,
- J. Banasiak, M.S. Seuneu Tchamga and K. Szymańska-Dȩbowska, Canard solutions in equations with backward bifurcations of the quasi-steady state manifold, Journal of Mathematical Analysis and Applications, 471(1-2), 2019, pp. 776-795,
- G. A. Ngwa, M. I.Teboh-Ewungkem, Y.Dumont, R. Ouifki and J. Banasiak, On a three-stage structured model for the dynamics of malaria transmission with human treatment, adult vector demographics and one aquatic stage, Journal of Theoretical Biology, 481(7-8), 2019, 202-222, DOI 10.1016/j.jtbi.2018.12.043,
- J. Banasiak, L. Joel and S. Shindin, Discrete growth-decay-fragmentation equation: well-posedness and long term dynamics, Journal of Evolution Equations, 2019, 19, 771–802, DOI 10.1007/s00028-019-00499-4,
- U. Ledzewicz, H. Schättler, On the Role of Pharmacometrics in Mathematical Models for Cancer Treatments, Discrete and Continuous Dynamical Systems - Series B, 26(1), pp. 483-499, 2021, DOI: 10.3934/dcdsb.20202213 (2021),
- 8. U. Ledzewicz, H. Schättler, On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies, Journal of Optimization Theory and Applications (JOTA), 87, pp. 305-335, 2020.
- M. Leszczynski, U. Ledzewicz, H. Schaettler, “Optimal Control for a Mathematical Model for Anti-Angiogenic Treatment with Michaelis Menten Pharmacodynamics” Discrete and Continuous Dynamical Systems, Series B, 24 (5), pp.2315-2334, 2019.
- H. Moore, L. Strauss and U. Ledzewicz, Optimization of Combination Therapy for Chronic Myeloid Leukemia with Dosing Constraints, Journal of Mathematical Biology, 77(5), pp.1533-1561, 2018, doi: 10.1007/s00285-018-1262-6.
- U. Ledzewicz, S. Wang, H. Schaettler, N. André, A.M. Heng and E. Pasquier, On Drug Resistance and Metronomic Chemotherapy: A Mathematical Modeling and Optimal Control Approach, Mathematical Biosciences and Engineering (MBE), 14(1), pp. 217-235, 2017, doi:10.3934/mbe.2017014.
- Ćwiszewski A., Gabor G., Kryszewski W., Invariance and strict invariance for nonlinear evolution problems with applications, Nonlinear Analysis, Theory, Methods and Applications (140 pkt.), 218, art. 112756 (2022);
- Kryszewski, W., Siemianowski, J., Constrined semilinear elliptic systems on R N , Advances in Differential Equations, 2021, 26(9-10), pp. 459–504
- Kryszewski, W., Maciejewski, M., Degree for weakly upper semicontinuous perturbationsof quasi-m-accretive operators: Perturbations of accretive operators, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2021, 379 (2191).
- Ćwiszewski A., Kryszewski W., Bifurcation from infinity for elliptic problems on R^N Calc. Var. and Part. Diff. Eq. 2019-02
- Kryszewski W., Siemianowski J., The Bolzano mean-value theorem and partial differential equations. J. Math. Anal. Appl., Vol. 457 (2018)
- Michał Bełdzinski, Marek Galewski, Igor Kossowski, Dependence on parameters for nonlinear equations—Abstract principles and applications, Mathematical Methods in the Applied Sciences, 45 (3) (2022), 1668–1686
- Marek Galewski, On variational nonlinear equations with monotone operators, Advances in Nonlinear AnalysiS, 10 (2021), 289–300, doi.org/10.1515/anona-2020-0102
- Michał Bełdzinski, Marek Galewski, Nash–type equilibria for systems of non-potential equations, Applied Mathematics and Computation, vol.385 (2020)
- Jean Mawhin, Ewa Skrzypek, Katarzyna Szymańska-Dębowska, Du Bois–Reymond Type Lemma and Its Application to Dirichlet Problem with the p(t)–Laplacian on a Bounded Time Scale, Entropy, 23 (10), 1352, 21 pp.
- Marek Matyjasik, Katarzyna Szymańska-Dębowska, Solvability for nonlocal boundary value problems with generalized p-Laplacian on an unbounded domain, Forum Mathematicum, 33 (5) (2021), 1321-1330
- Katarzyna Szymańska-Dębowska, Mirosława Zima, A topological degree approach to a nonlocal Neumann problem for a system at resonance, Journal of Fixed Point Theory and Applications, 21 (2) (2019), 1-14
- Jean Mawhin, Katarzyna Szymańska-Dębowska, Convexity, topology and nonlinear differential systems with nonlocal boundary conditions: a survey, Rend. Istit. Mat. Univ. Trieste, 51 (2019), 125–166

Project: NCN OPUS ‘Mathematics of multiscale models in life and social sciences’, Jacek Banasiak (PI)

Project: NCN MINIATURA 2, 2018/02/X/ST1/02082, Katarzyna Szymańska-Dębowska

Project: NCN OPUS 2013/09/B/ST1/01963 “Topological methods in the study of dynamics of nonlinear evolution equations”, Wojciech Kryszewski (leader)

**Keywords:**

differential equations, topological methods, nonlinear analysis, dynamical systems, applications of differential equations, integro-differential equations, mathematical epidemiogy

**List of internship proposal in this research team:**

Scientific cooperation in a selected area of research or related topics.

**Institute of Mathematics, Lodz University of Technology I-73 **

**Head of the unit: **

Prof. dr hab. Marek Balcerzak

**Potential promoters: **

Dr hab. Włodzimierz Fechner

**Contact person:**

Dr hab. Włodzimierz Fechner, tel: 42-631-3854, wlodzimierz.fechner@p.lodz.pl

**Scope of activities: **

Main areas of interest and directions of scientific research:

Classical functional inequalities for different classes of functions

Convexity and its generalizations

Applications of functional equations and inequalities, particularly in areas such as game theory and fuzzy reasoning graphic material

**Present activities: **

The scope of our research activity includes contemporary problems in the theory of functional equations and inequalities. We place emphasis on the application of the obtained results. We study Sincov's multiplicative inequality and similar inequalities and their relationship to some problems in game theory. Our results describe a form of solutions, which in turn leads to answers to questions about the Nash equilibrium points in the studied stochastic two-player or multiplayer games. We are also interested in fuzzy implications, where Sincov's functional equation appears on a truncated domain.

**Future activities: **

Continuation of the conducted research, extending the scope of the studied problems.

**Publications/patents, awards, projects:**

- W. Fechner, Z. Páles, Convexity properties of functions defined on metric Abelian groups, Aequationes Mathematicae 95/3 (2021), 449-481.
- A. Bartoszewicz, W. Fechner, A. Świątczak, A. Widz, On the c 0 -equivalence and permutations of series, Annals of Functional Analysis 12/2 (2021), 1-10.
- W. Fechner, Z. Páles, Convexity of sets in metric Abelian groups, Forum Mathematicum 32/6 (2020), 1477-1486
- W. Fechner, Richard’s inequality, Cauchy–Schwarz’s inequality, and approximate solutions of Sincov’s equation, Proceedings of the American Mathematical Society 147/9 (2020), 3955-3960.
- W. Fechner, Á. Száz, Composition iterates, Cauchy, translation, and Sincov inclusions, Acta Universitatis Sapientiae, Mathematica 12/1 (2020), 54-84.

Graduate of dr hab. W. Fechner, Maria Słomian, MA is the winner of the Marian Rejewski competition for the award of the director of the Institute of Mathematics for the best master's thesis in the field of Applied Mathematics in the 2020/21 academic year. The work is part of the presented research project.

**Keywords:**

functional inequality, functional equation, convexity, stochastic binary game, fuzzy implication

**List of internship proposal in this research team:**

Scientific cooperation in a selected area of research or related topics.

**Institute of Mathematics., Lodz University of Technology I-73 **

**Head of the unit: **

dr hab. Grzegorz Andrzejczak, prof. uczelni

**Potential promoters: **

dr hab. Szymon Głąb, prof. uczelni

dr hab. Przemysław Gordinowicz

**Contact person:**

dr hab. Grzegorz Andrzejczak, prof. uczelni, grzegorz.andrzejczak@p.lodz.pl

**Scope of activities: **

Graph theory: adjacency properties, graph colorings (incl. list colorings), graphical polynomial, graph searching games, topological graphs

Fraïssé theory

**Present activities: **

- The polynomial method for list colorings of planar graphs
- Graph searching through relative distance queries
- Graphical, agent based models
- Fraïssé limits and inverse Fraïssé limits of finite structures

**Future activities: **

Continuation of the conducted research

**Publications/patents, awards, projects:**

- Przemysław Gordinowicz, Paweł Twardowski, The polynomial method for list-colouring extendability of outerplanar graphs, Ars Mathematica Contemporanea, 21 (2) (2021), art. P2.08.
- Szymon Głąb, Przemysław Gordinowicz, Filip Strobin, Dense free subgroups of automorphism groups of homogeneous partially ordered sets, Forum Mathematicum, 31 (1) (2019), 215-240.
- Bartłomiej Bosek, Przemysław Gordinowicz, Jarosław Grytczuk, Nicolas Nisse, Joanna Sokół, Małgorzata Śleszyńska-Nowak, Localization game on geometric and planar graphs, Discrete Applied Mathematics, 251 (2018), 30-39.
- T. Gavenčiak, Przemysław Gordinowicz, V. Jelínek, P. Klavík, J. Kratochvíl, Cops and Robbers on intersection graphs, European Journal of Combinatorics, 72 (2018), 45-69.
- Przemysław Gordinowicz, The 2-surviving rate of planar graphs with average degree lower than 9/2, Journal of Graph Theory, 89 (2018), 341-349.
- Szymon Głąb, Michał Pawlikowski, An inverse Fraïssé limit for finite posets and duality for posets and lattices, preprint: arxiv.org/abs/2201.10302

**Keywords:**

graphs, graph colorings, graph searching games, adjacency properties, Fraïssé limit

**Institute of Mathematics, Lodz University of Technology I-73 **

**Head of the unit: **

Professor Lesław Gajek, PhD, DSc

**Potential promoters: **

Marek Kałuszka, PhD, DSc, TUL prof.

Andrzej Okolewski, PhD, DSc, TUL prof.

**Contact person:**

Professor Lesław Gajek, PhD, DSc, tel: 42-631-36-34, leslaw.gajek@p.lodz.pl

**Scope of activities: **

Research activities are carried out in two directions:

- The first one concerns modeling of insurer’s solvency in switching models, evaluation of ruin probability, time to ruin, deficit size, etc., as well as estimation of model’s parameters.
- The second direction is related to interest rate risk in long-term insurance, methods of pricing and hedging against this risk. graphic material

**Present activities: **

Ad. 1. Both numerical and theoretical approaches are being developed to investigate switching models. The main theoretical tool is the risk operator approach introduced in Gajek (2005). Based on the properties of the risk operator, we evaluate ruin probability, time to ruin, deficit size, etc.

Ad. 2. New evaluations of the VaR measure of the change in portfolio surplus due to the change of the term structure of interest rates are investigated. The bounds are supposed to be products of two terms: the one depending on the portfolio structure and the second depending on the interest rate change measure. The measure is related to the Lp-distance in the space of financial instruments. We investigate its relationship with other measures, like M2. Explicit formulas for some life insurance products are obtained. Optimal streams of assets are investigated under several restrictions imposed on e.g. duration, M2 or other parameters of the asset flow. The influence of the lack of short sale on the efficiency of the interest rate risk protection is investigated.

**Future activities: **

We plan to investigate the risk switching models in the uncertain environment when the switch’s parameters are partly known and partly unknown. This approach will be also useful when life-insurance products are analysed from a long time perspective and we plan to explore it in this context as well.

**Publications/patents, awards, projects: **

- Lesław Gajek, On the deficit distribution when the ruin occurs-discrete time model, Insurance: Mathematics and Economics 36 (2005), 13-24
- Lesław Gajek, Marcin Rudź, Banach Contraction Principle and ruin probabilities in regime-switching models, Insurance: Mathematics & Economics 80 (2018) 45–53
- Lesław Gajek, Elżbieta Krajewska, Balance-sheet interest rate risk: a weighted Lp approach, Journal of Risk 21 (2018) 91-104
- Lesław Gajek, Marcin Rudź, Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model, Methodology and Computing in Applied Probability, 22 (2020) 1493-1506
- Lesław Gajek, Łukasz Kuciński, Complete discounted cash flow valuation, Insurance: Mathematics & Economics 73 (2017) 1-19
- Lesław Gajek, Marcin Rudź, Sharp approximations of ruin probabilities in the discrete time models, Scand. Actuar. J. 5 (2013) 352-382
- Lesław Gajek, Elżbieta Krajewska, A new immunization inequality for random streams of assets, liabilities and interest rates, Insurance: Mathematics and Economics 53 (2013) 624-631
- Marek Kałuszka, Andrzej Okolewski, A note on multiple life premiums for dependent lifetimes, Insurance: Mathematics and Economics 57 (2014), 25-30

**Keywords:**

insurer’s solvency; risk switching models; ruin probability; time to ruin; deficit size; estimation of model’s parameters; term structure of interest rates; VaR ; M2-measure; portfolio optimization

**List of internship proposal in this research team:**

Risk management in insurance.

**Institute of Mathematics, Lodz University of Technology I-73**

**Head of the unit: **

Prof. Jacek Jachymski

**Potential promoters: **

PhD, DSc Filip Strobin, PhD, DSc, TUL Prof.

Prof. Jacek Jachymski, PhD, DSc

**Contact person:**

PhD, DSc Filip Strobin, PhD, DSc, TUL Prof., filip.strobin@p.lodz.pl

**Scope of activities: **

Iterated function systems and their generalizations, fractals, attractors, semiattractors, fuzzy attractors, algorithms generating images of attractors (e.g., deterministic chaos game), invariant measures, Markov operators, semimetric spaces, transformations of functions of distance type, metric fixed point theory, selected topics in functional analysis.

**Present activities: **

Analysis of deterministic chaos game, existence of idempotent invariant measures, compactness criteria in metric and Banach spaces; properties between compactness and completeness.

**Future activities: **

Attractors of expansive iterated function systems, chaos game for fuzzy version of iterated function systems, porosity and Baire category of family of attractors, around the Josefson-Nissenzweig theorem; Banach limits, Hahn-Banach type theorems.

**Publications/patents, awards, projects** (for the last 2 years):

- Krzysztof Leśniak, Nina Snigireva, Filip Strobin, A fractal triangle arising in the AIMD dynamics, Proceedings of the conference Contemporary Mathematics in Kielce 2020 (2021), 179-195
- R. D. da Cunha, E. R. Oliveira, Filip Strobin, A multiresolution algorithm to generate images of generalized fuzzy fractal attractors, Numer. Algorithms (100 pkt.) 86 (2021), 223-256
- Filip Strobin, Contractive iterated function systems enriched with nonexpansive maps, Results Math (100 pkt.) 76, art. nr 153 (2021), 30 pp.
- Filip Strobin, On the existence of the Hutchinson measure for generalized iterated function systems, Qual. Theory Dyn. Syst. (100 pkt.), 19 (3), art. nr 85 (2020), 21 pp.
- T. Banakh, M. Nowak, Filip Strobin, Embedding fractals in Banach, Hilbert or Euclidean spaces, J. Fractal Geom. (70 pkt.) 7 (2020), 351–386
- Filip Strobin, J. Swaczyna, Connectedness of attractors of a certain family of IFSs, J. Fractal Geom. (70 pkt.) 7 (2020), 219–231
- R. D. da Cunha, E. R. Oliveira, Filip Strobin, A multiresolution algorithm to approximate the Hutchinson measure for IFS and GIFS, Commun. Nonlinear Sci. Numer. Simul. 91 (2020), 105423, 22 pp.
- K. Leśniak, N. Snigireva, Filip Strobin, Weakly contractive iterated function systems and beyond: a manual, J. Differ. Equ. Appl. (70 pkt.), 26, art. nr 8 (2020), 1114-1173 (paper awarded in 2020 JDEA Best Paper)

- Jacek Jachymski, Filip Turoboś, On functions preserving regular semimetrics and quasimetrics satisfying the relaxed polygonal inequality, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas RACSAM (100 pkt.), 114 (3), art. 159 (2020)

**Keywords:**

Iterated function systems, fractals, attractors, invariant measures, algorithms generating images of attractors, spaces with the distance-type function, Banach spaces, completely continuous mappings, renorming and remetrization.

**List of internship proposal in this research team:**

Cooperation in investigations of undertaken problems of iterated function systems theory, e.g.

- existence and the structure of attractors of wide classes of iterated function systems;
- analysis of algorithms generating images of attractors of iterated function systems;
- analysis of size and structure of family of attractors of iterated function systems.