Nicolaus Copernicus University in Toruń |

# Scientific interests

- Qualitative theory of (nonautonomous) ordinary differential equations and difference equations (periodic and homoclinic solutions).
- Topological methods in nonlinear analysis (topological degree, fixed point index, fixed point theorems, Fredholm theory).
- Algebraic topology: homology and cohomology theories.
- Differential inclusions and nonsmooth analysis.
- Applications of K-theory in analysis.
- Numerical discretization theory.

# Publications

- R. Skiba, On the Lefschetz fixed point theorem for multivalued weighted mappings, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 40, 2001, 201-214.
- R. Skiba, Topological essentiality for multivalued weighted mappings, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 41, 2002, 131-145.
- J. Pejsachowicz, R. Skiba, Fixed point theory of multivalued weighted maps, Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 217-263.
- R. Skiba, Graph-approximation of multivalued weighted maps, Topol. Methods Nonlinear Anal., 29, 2007, 119-161.
- R. Skiba, Fixed points of multivalued weighted maps, Lecture Notes in Nonlinear Analysis, vol. 9, Juliusz Schauder Center for Nonlinear Studies,Nicolaus Copernicus University, Toruń 2007, 1-148.
- R. Skiba, M. Ślosarski, On a generalization of absolute neighborhood retratcs, Topology and its Applications, 156 (2009) 697-709.
- W. Kryszewski, R. Skiba, A cohomological index of Fuller type for set-valued dynamical systems, Nonlinear Analysis, Theory, Methods & Applications, 75 (2012), 684-716.
- J. Pejsachowicz, R. Skiba, Global bifurcation of homoclinic trajectories of discrete dynamical systems, Central European Journal of Mathematics, 10(6) 2012, 2088-2109.
- J. Pejsachowicz, R. Skiba, Topology and homoclinic trajectories of discrete dynamical systems, Discrete Contin. Dyn. Syst. S, 6(4), 2013, 1077-1094.
- R. Skiba, Applications of weighted maps to periodic problems of autonomous differential equations, Topol. Methods Nonlinear Anal., 42(1) 2013, 9-49.
- R. Skiba, A cohomological index of Fuller type for parameterized set-valued maps in normed spaces, Central European Journal of Mathematics, 12(8) 2014, 1164-1197.
- R. Skiba, N. Waterstraat, The Index Bundle and Multiparameter Bifurcation for Discrete Dynamical Systems, DCDS-A, 37(11), 2017, 5603-5629 (arXiv:1612.05837).
- C. Potzsche, R. Skiba, Global Continuation of Homoclinic Solutions, Zeitschrift fur Analysis und ihre Anwendungen, Volume 37 (2018), 159-187,DOI: 10.4171/ZAA/1608.
- C. Potzsche, R. Skiba, A Continuation Principle for Fredholm maps I:Theory and Basics, accepted in Mathematische Nachrichten.
- C. Potzsche, R. Skiba, A Continuation Principle for Fredholm maps II: Application to homoclinic solutions, accepted in Mathematische Nachrichten.
- R. Skiba, On (co)homological classes and new homotopical invariants with applications to differential equations (in preperation).
- R. Skiba,N. Waterstraat, Fredholm theory of discrete dynamical systems and its applications to bifurcation theory (in preperation).
- R. Skiba, A new statement of the Ważewski Theorem for differential equations (in preperation).

# Curriculum vitae

**EDUCATIONAL BACKGROUND:**

- Master of Mathematics (master thesis: “The Lefschetz fixed point Theorem and topological essentiality for multivalued weighted maps”), Nicolaus Copernicus University in Toruń, 2001, Poland.
- Doctor of Mathematics (dissertation: “Fixed points of multivalued weighted maps”), Nicolaus Copernicus University in Toruń, 2006, Poland.

**EMPLOYMENT HISTORY:**

- 2005-2006: Assistant, Nicolaus Copernicus University in Toruń, Faculty of Mathematics and Computer Science.
- 2006- : Assistant Professor (Adiunkt), Nicolaus Copernicus University in Toruń, Faculty of Mathematics and Computer Science, Toruń, Poland.

**HONORS AND AWARDS:**

- Scholarship of the Minister of Science and High Education, 1999/2000.
- Scholarship of the Minister of Science and High Education, 200/2001.
- The best student of the Faculty of Mathematics and Computer Science from the Nicolaus Copernicus University in Toruń, 1999/2000.
- The best graduate of the Faculty of Mathematics and Computer Science from the Nicolaus Copernicus University in Toruń, 2000/2001.
- Award of the Rector of the Nicolaus Copernicus University in Toruń, 2007.
- Award of the Rector of the Nicolaus Copernicus University in Toruń, 2015.
- Medal of the National Education Commission, 2018.
- Award of the Rector of the Nicolaus Copernicus University in Toruń, 2019.

**SCIENTIFIC GRANTS:**

- 2003-2006: Participation in grant of State Committee for Scientific Research, KBN 2 P03A 015 25, “Topological fixed point theory and its applications in nonlinear analysis”.
- 2007: Grant of Nicolaus Copernicus University, “Numerical methods in the study of homotopical invariants for weighted maps”.
- 2009-2012: Participation in grant of Ministry of Science and Higher Education, MNiSW N N201 395137, “Topological invariants in nonlinear analysis”.
- 2014-2017: Participation in grant of National Science Center (Poland), NCN 2013/09/B/ST1/01963, “Dynamics of nonlinear evolutionary equations - topological approach”.
- 2017-2018: Grant of National Science Center (Poland), NCN DEC-2017/01/X/ST1/00889, “Continuations and global bifurcations of homoclinic trajectories for nonlinear problems”.

**SELECTED RESEARCH TALKS:**

1. Marina Franca (Italy), 1-5.09.2003, “Mathematical Foundation of Turbolent Viscous Flows”, Centro Internazionale Matematico Estivo, workshop.

2. Cetraro (Italy), 19-29.06.2004, “Nonlinear and Optimal Control Theory”, Centro Internazionale Matematico Estivo, workshop.

3. Będlewo (Poland), Banach Center of the Institute of Mathematics of the Polish Academy of Sciences, 26-30.06.2006, Jean Leray Centennial Conference, Topological Methods in Nonlinear Problems, talk: “Graph-approximation of multivalued weighted maps”.

4. Będlewo (Poland), Banach Center of the Institute of Mathematics of the Polish Academy of Sciences, 22-28.07.2007, Topological Theory of Fixed and Periodic Points TTFPP, talk: “Fixed points of multivalued weighted maps”.

5. Toruń (Poland), Nicolaus Copernicus University in Toruń, 10-14.09,2007, Fifth Symposium on Nonlinear Analysis, talk: “On the Lefschetz fixed point theorem for multivalued weighted carriers”.

6. Hannover (Germany), 1-5.09.2008, Leibniz Universität, Summer School Functional Analytic Methods in Partial Differential Equations, workshop.

7. Brno (Czech Republic), 20–24.06.2009, Masaryk University, Conference on Differential Equations and Their Applications (EQUADIFF 12), talk: “A cohomological index of Fuller type for dynamical systems”.

8. Dresden (Germany), 25-28.05.2010, The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden University of Technology, talk: “The Fuller index and Bifurcation points”.

9. Będlewo (Poland), Banach Center of the Institute of Mathematics of the Polish Academy of Sciences, 24-30.06.2012, Dynamics, Topology and Computations DyToComp12, poster: “Global bifurcation of homoclinic trajectories of discrete dynamical systems”.

10. Białystok (Poland), 21-26.07.2013, PODE 2013, Progress on Difference equations, talk: “Homoclinic trajectories of discrete dynamical systems”.

11. Madrid (Spain), 7-11.07.2014, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, talk: “Cohomological index of Fuller type and its applications”.

12. Poznań (Poland), 17-20.09.2014, Joint Meeting of the German Mathematical Society (DMV) and the Polish Mathematical Society (PTM), talk: “Homoclinic trajectories of discrete dynamical systems”.

13. Szeged (Hungary), 30.06-05.07.2015, 10th Colloquium on the Qualitative Theory of Differential Equations.

14. Berlin (Germany), Free University of Berlin, “Patterns of Dynamics”, 25-29.07.2016, talk: “Homotopy invariants in bifurcation theory for dynamical systems”.

15. Mogilev (Belarus), the Belarusian-Russian University, International scientific and practical seminar of the team supervisors: 23.02.2017, talk: ”A topological degree of Fuller type for periodic orbits”.

16. Amadora (Portugal), 05-09.06.2017, International Conference on Diferential & Diference Equations and Applications, talk: “Homoclinic solutions for discrete dynamical systems”.

17. Klagenfurt (Austria), The Alpen-Adria-Universität Klagenfurt, Institut fur Mathematik, 1-8.10.2017, talk: “Periodic solutions of differential equations without uniqueness”, research visit.

18. Klagenfurt (Austria), The Alpen-Adria Universitat Klagenfurt, Institut fur Mathematik, 04-19.09.2018, , research visit.

19. Canterbury (United Kingdom), University of Kent, School of Mathematics, Statistics & Actuarial Science, 05-14.06.2018, research visit.

20. Halle (Germany), Martin-Luther-Universitat Halle-Wittenberg, Faculty of Natural Sciences II - Chemistry, Physics and Mathematics, 02-10.06.2019, talk: ”Periodic solutions of differential equations without uniqueness”, research visit.

21. AGH University of Science and Technology, Kraków (Poland), "Dynamics, Equations and Applications", 16-20.09.2019 r., talk: "A Continuation Principle for Fredholm maps and its application to differential equations".

**PROFESSIONAL ACTIVITY:**

- Member of the Organizing Committee, Conference: V Symposium on Nonlinear Analysis, 10-14.09.2007, Nicolaus Copernicus University in Toruń, Poland.
- Member of the Organizing Committee, Conference: VI Symposium on Nonlinear Analysis, 7-9.09.2011, Nicolaus Copernicus University in Toruń, Poland.
- Member of the Organizing Committee, Conference: East - West Conference on Mathematics Education, 04–08.08.2014, Warsaw, Poland.
- Member of the Organizing Committee, Conference: VII Symposium on Nonlinear Analysis, 14-18.09.2015, Nicolaus Copernicus University in Toruń, Poland.
- Member of the Organizing Committee, Conference: East West Conference on Mathematics Education 2015, 29.06-3.07.2015, Warsaw, Poland.
- Organizer, International Mathematics Competition for Students: Mirror of the William Lowell Putnam Mathematical Competition, Nicolaus Copernicus University in Toruń, 2014, 2015, 2016, 2017, 2018, Toruń, Poland.
- Member of the Organizing Committee, Conference: East - West Conference on Mathematics Education EWCOME 2017, 24-26.07.2017, Warsaw, Poland.
- Member of the Organizing Committee, Conference: East - West Conference on Mathematics Education EWCOME 2018, 22-23.09.2018, Warsaw, Poland.
- Member of the Organizing Committee, Conference: “Special Workshop on Nonlinear Analysis, Juliusz Schauder Medal Awarding Ceremony, Toruń, 21-23.05.2018”, Toruń, Poland.