I am interested in branching structures and their applications. Possible projects are:
Random tree structures - The purpose of this project is to develop new probabilistic techniques to provide a thorough description of various aspects of (random) trees in different random settings. For example, describing the structure of typically large (random) trees, and more generally large graphs. Beyond the purely probabilistic or combinatorial aspects, the goal also is to study several other models that come from:
- Biology to understand the spread of epidemics;
- Theoretical computer science in the analysis of algorithms and study of data structures;
- Statistical physics such as percolation.
Probabilistic aspects of evolutionary biology - This project is devoted to the study of interactions between organisms and their environment which influence their reproductive success and contribute to genotype and phenotype variation which is one of the main questions in evolutionary ecology and population genetics. The understanding of such evolutionary-ecological processes is very complicated and it requires the substantial use of mathematical models and methods. These include branching processes, superprocesses, fragmentation-coalescence processes, Lévy processes, stochastic partial differential equations, etc. Typical questions related to this subject of research might concern the growth rate of interacting populations, the probability a certain sub-population survives or perishes, genealogies of certain individuals in some population, the effects of mutation and recombination of genetic material, reproduction mechanisms, competition, cooperation and selection.
I am interested in optimal control (theory and applications):
Many real life phenomena are described by ordinary differential equations. Others can be approximated by Markov chains or more complicated random processes. If we can affect dynamically this or that parameter then we deal with a controlled dynamical system. Generally speaking, the problem is to find the best, optimal control. Remember, many problems are multiple objective. The aim of such a project for students can be: to justify the problem statement, to study optimisation methods like dynamic programming and Pontryagin's maximum principle, to investigate analytically (or numerically) a particular version of the problem formulated, to undertake computer simulations, to compare different mathematical models describing the same real life phenomenon. Possible projects are:
- Convex Analytic Approach to certain controlled Markov chains and jump processes;
- Analysis of communication networks;
- Approximations of controlled birth-and-death processes.
I am interested in probability theory and its applications. In particular, coalescent theory, branching processes, random trees, measure valued processes; stochastic modelling in biology, data-driven computations; condensation in stochastic models; exchangeability, extendibility of finite exchangeable sequences. Possible projects are:
- Finer properties of coalescent processes and branching processes;
- Stochastic modelling and statistical inference for biological experiments;
- Stochastic modelling of diploid populations;
- Exchangeable stochastic networks and its applications;
- Condensation phenomena in complex stochastic networks.