The Universe under Very Large Cardinals Hypotheses

My proposal is a 24 months period of research at the TU, specifically at the Institute of Discrete Mathematics and Geometry. The main topic will be Set Theory, with Algebra and Topology as secondary topics. The main goal of the project is an analysis of a new strain of large cardinal numbers, stronger than all the known ones, in terms of consistency, both internal and external. The research will follow three main paths: the most natural questions to answer will be about the relationship of this novel cardinal numbers between each other. They are naturally composed in a definite hierarchy, but is this a strict hierarchy? Does every large cardinal strongly imply the ones below? In the past inverse limits of elementary embeddings solved the problem for weaker cardinal numbers, but we will need more sophisticated techniques, like a generalized limit of familites of inverse limits. The introduction of new large cardinal numbers hypotheses is always a risk, therefore is important to settle as soon as possible how the sets behave under them. There is a large literature about the consistency of large cardinal numbers with combinatorial properties, and new results will certainly be favourably welcomed by the community. Once established the problems above, the next logical step is to finally detect the implications of the very large cardinal numbers on "practical" mathematics, outside Set Theory. Similar hypotheses had a surprising connection with Free Algebra and Braid Groups, therefore it is natural to search for results in that field, while weaker hypotheses proved consistency results in Topology, via collapse forcing. The experience of the co- applicant in both the fields of Set Theory and Algebra will certainly be beneficial for the research, and the knowledge of the entire group on real forcing will be extremely important for establishing new results, that will surely be a basis for future research. This is conceived as part of a much general effort by the mathematical community to understand the boundaries of large cardinal numbers and their effectiveness.