Mathematical structures and their physical applicationsAnnotation
The subject of the research activity implemented within the research plan is the study of structures of algebra, geometry and analysis and their applications, mainly in physics. This is a very topical and very broad subject belonging to a basic research in mathematics and in mathematical physics, and the research plan concentrates on some of its areas and directions. The most important of them are differential geometry, algebraic topology, quantum gravity, category theory, algebraic number theory, the theory of semigroups and automata, and differential and difference equations. These areas are deeply interrelated. For instance, algebra, analysis, geometry and topology provide mathematical tools for quantum gravity and they have decisively influenced the recent progress in algebraic number theory. On the other hand, they use the language of category theory. Also, mathematical physics in general and quantum gravity in particular provide intuitive understanding of some complicated structures of algebra, geometry and analysis and their mutual relations. The general research plan objective is to support basic research in mathematics and mathematical physics at the Masaryk University. These are fields in which the Czech Republic in general and the Masaryk University in particular are keeping a high-level international standard. The aim is to provide necessary conditions for a highly competitive research in these fields at the Masaryk University, which also establish a fruitful background for PhD students and stimulate PhD projects in mainstream topics. The concrete research plan objective is to get new and significant results in all areas of the subject of the research plan contributing to those areas on an international level. These results are communicated at international conferences and published in leading international journals. Some parts of the research are intended to be completed, written down in a monographic form and published by a reputable publishing house.