Combinatorics of generalized continuum and large cardinals

The proposed work shall advance our knowledge in the field of set theory, which is a subfield of mathematical logic. More precisely, the proposed research aims at studying cardinal characteristics, relations between them, combinatorial consequences of these relations, and restrictions that ZFC imposes on these constellations. This is an area where independence from the usual axioms of mathematics (the Zermelo Fraenkel axiom system together with the axiom of choice, abbreviated ZFC) often arises. Therefore one of the main parts of the proposed work will be to find suitable forcing techniques and possibly develop new ones. This branch of set theory has two directions. The "classical" part, cardinal characteristics of the continuum, is nowadays a well-developed branch of set theory with applications in many mathematical fields. This name reflects their importance for the study of the real line, but in the present context it is more natural to call them cardinal characteristics at w, the cardinality of the set of natural numbers. Cardinal characteristics allow for a concise description of the premises beyond ZFC in an independence result: for example, a large part of the most important problems of set-theoretic topology (e.g., the existence of Michael spaces, the existence of countable nonmetrizable topological groups with the Frechet-Urysohn property, etc.) can be (sometimes rather simply) solved under certain (in)equalities between cardinal characteristics. One of the more recent directions here is to investigate the interactions of inequalities between cardinal characteristics and projectiveness of certain objects whose construction heavily relies on the Axiom of Choice (e.g., wellorders, maximal almost disjoint families, etc.). The study of cardinal characteristics at arbitrary uncountable cardinals is the newer direction, whose beginning may be traced back to the work of Cummings and Shelah on the global behavior of the bounding and dominating numbers, emerging from the theory of cardinal characteristics at w. Since then, only a dozen of papers on this topic were written. The topic often requires some new techniques and approaches and is still not well understood. By a cardinal characteristic we mean here a class-function whose domain and range consist of cardinals. Given a cardinal characteristic f, the main points here are: the possible values of f(k) at a cardinal k; the behavior of the function f on classes of regular cardinals; the consistency strength of the inequality f(k)>k+ for a measurable k, the internal consistency (=consistency in an inner model) of a specific global behavior of f. One of the main goals here will be to "bring`` more forcings to the large cardinal theory from the set theory of reals, with special emphases on tree forcings. The latter often allow for extending elementary embeddings assuming only certain degree of strongness of the cardinal in question. We plan to try to introduce a variant of Axiom A of Baumgartner suitable for uncountable cardinals.