Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Plant communities of field margins : the effects of management and environmental factors on species composition and diversity

Agri-environmental schemes have so far resulted in only minor positive implications for the biodiversity of agricultural environments, in contrast to what has been expected. Land-use intensification has decreased landscape heterogeneity and the amount of semi-natural habitats. Field margins are uncultivated areas of permanent vegetation located adjacent to fields. Since the number of these habitats is high, investing in their quality may result in more diverse agricultural landscapes. Field margins can be considered as multifunctional habitats providing agronomic, environmental and wildlife services. This thesis aimed at examining the plant communities of different types of field margin habitats and the factors affecting their species diversity and composition. The importance of edaphic, spatial and management factors was studied on regional, landscape and habitat scales. Vegetation surveys were conducted on regional and landscape scales and a field experiment on cutting management was conducted on a habitat scale. In field margin plant communities, species appeared to be indicators of high or intermediate soil fertility and moist soil conditions. The plant species diversity found was rather low, compared with most species-rich agricultural habitats in Finland, such as dry meadows. Among regions, land-use history, main production line, natural species and human induced distribution, climate and edaphic factors were elements inducing differences in species composition. The lowest regional species diversity of field margins was related to intensive and long-term cereal production. Management by cutting and removal or grazing had a positive effect on plant species diversity. The positive effect of cutting and removal on species richness was also dependent on the adjacent source of colonizing species. Therefore, in species-poor habitats and landscapes, establishment of margins with diverse seed mixtures can be recommended for enhancing the development of species richness. However, seed mixtures should include only native species preferably local origin. Management by cutting once a year for 5 years did not result in a decline in dominance of a harmful weed species, Elymus repens, showing that E. repens probably needs cutting more frequently than once per year. Agri-environmental schemes should include long-term contracts with farmers for the establishment, and management by cutting and removal or grazing, of field margins that are several metres wide. In such schemes, the timing and frequency of management should be planned so as not to harm other taxa, such as the insects and birds that are dependent on these habitats. All accidental herbicide drifts to field margins should be avoided when spraying the cultivated area to minimize the negative effects of sprayings on vegetation. The harmful effects of herbicides can be avoided by organic farming methods.