Tatyana Turova och Erik Wahlén har tilldelats medel från Vetenskapsrådets utlysning för projektbidrag inom naturvetenskap och teknik för sina forskningsprojekt "Fasövergångar i modeller för statistisk mekanik på slumpgrafer" och "Icke-linjära vattenvågor och icke-lokala modellekvationer."
Tatyana Turova får 700 000 kronor under fyra år och Erik Wahlén får 750 000 kronor under fyra år.
2015 tilldelades Erik Wahlén även ett ERC Starting Grant på ca 1 miljon Euro av European Reseach Council. Läs mer om det bidraget här
Tatyana Turovas webbsida: http://www.maths.lth.se/matstat/staff/tatyana/
Erik Wahléns webbsida: http://www.maths.lu.se/staff/erik-wahlen/
Fasövergångar i modeller för statistisk mekanik på slumpgrafer/ Phase Transitions in the Models of Statistical Physics on Random Graphs, Tatyana Turova
The overall purpose of the project is the development of the theory of large systems defined on random structures, with a particular focus on phase transitions. We construct and study probability spaces where measures may couple space, time and matter. The specific goal of the project is to describe the phenomena of phase transitions in the following models: monotone and non-monotone bootstrap percolation processes on distance random graphs, continuum percolation on randomly grown trees, Ising model of quantum gravity on planar random triangulations. The research in this area uses a broad spectra of mathematical methods, in particular from probability, as, e.g., branching processes, Markov processes, diffusion processes, martingale theory, and from statistical mechanics. In turn, new models require new methods of investigation, hence the project should result in development of probability theory in general. The motivation for this study comes from mathematics, theoretical physics and neurobiology. Thus besides a mathematical interest on its own, the theoretical results of the project should be of use for the natural sciences.
Icke-linjära vattenvågor och icke-lokala modellekvationer, Erik Wahlén
The study of water waves is mathematically challenging, since they are described by a system of nonlinear partial differential equations in a domain with a free boundary. The aim of this project is to construct and study different types of solutions to these equations as well as to simplified nonlocal model equations. In particular, we will study travelling waves, that is, waves of permanent form which travel with constant speed. We will investigate singular travelling wave solutions of nonlocal model equations as well as three-dimensional solitary waves of the full water wave problem. We will also investigate stability properties of such waves, which is important for determining their physical relevance. In addition, we will investigate modulating pulses (or moving breathers). These consist of a permanent localised envelope which modulates an underlying periodic wave train which moves relative to the envelope. Such solutions have only been constructed for simpler model equations. Our aim is to prove the existence of modulating pulses in the full water wave problem. The analysis relies on bifurcation theory, inverse spectral theory, variational methods and infinite-dimensional dynamical systems theory. The questions that we plan to study will require the development of new mathematical tools in these areas.