I algebraic geometry

Major research achievements:

R. RUDNICKI has published 72 papers, most of them in well known mathematical and biological journals. He has MathSciNet Math. Rev. 157 citations by 109 authors, ISI gives 251 cited references. His scientific activity is connected with three fields of mathematics: differential equations, probability theory and biomathematics. In particular he is interested in the following subjects:

1. 
   ergodic properties of dynamical systems generated by partial differential equations
   
2. 
   asymptotic properties of Markov semigroups and their applications
   
3. 
   population dynamics and modelling of cell cycle.
   

Ad 1. He has given a general construction of invariant measures for dynamical systems generated by partial differential equations [11]. Measures of this type are constructed by means of stochastic processes and stochastic fields and have strong ergodic and analytic properties (positivity on open sets, mixing and so on). From these properties follow additional features of the dynamical systems: chaos in the sense of Auslander and Yorke and the existence of turbulent trajectories in the sense of Bass [12].

Ad 2. Markov operators are linear transformations from the space L1 into itself which preserve the set of densities. They appear in the ergodic theory and in the theory of Markov chains. Semigroups of Markov operators are often generated by partial differential equations and by partial differential equations with some perturbations. Rudnicki's results are applied to transport equations describing diffusion and coagulation-fragmentation processes and structured population models.

Ad 3. The most cited paper with M. C. Mackey [15] (cited 21 times in MSN MR and more than 50 found in the Internet) deals with the problem of asymptotic stability of a nonlinear partial differential delay equation describing cellular replication. This equation is difficult to study and Rudnicki developed completely new analytic and probabilistic methods to establish its asymptotic stability, playing an important role in mathematical models of the cell cycle and in a model of the electrical activity of neurons. With O. Arino [16] Rudnicki introduced a coagulation-fragmentation model of phytoplankton dynamics and showed with R. Wieczorek how to obtain this equation as a limit of some individual-based model.

He is an advisor to the Board of the European Society for Mathematical and Theoretical Biology. He has organized four conferences. Now he is president of the Scientific and Organizing Committees of VIII European Conference on Mathematical and Theoretical Biology}, Kraków, June 28 - July 2, 2011.(we expect about 600 participants).

He was awarded the Hugon Steinhaus Prize of the Polish Mathematical Society in 2010 which is the main prize of the society in applications of mathematics.