Executive Summary : | The natural density function has been extensively studied from various perspectives, including number theory, combinatorial view points, and Levy groups. The ideal generated by natural density functions has been a significant study in set theory and the theory of forcing, which has subsequently influenced the study of more general classes of ideals. In 2015, the notion of natural density was extended to simple density functions and associated ideals, with the aim of showing the existence of uncountably many distinct analytic P-ideals (nice ideals in set theory). In 2018, the author introduced the more general idea of moduler simple density functions and associated ideals, establishing that there are ideals that cannot be generated by simple density functions. This project aims to shed light on this area and explore whether different aspects of simple density ideals already established hold for the larger class of moduler simple density ideals. Several aspects of simple density ideals remain unsolved, and if successfully established, could extend some of the best known results of mathematics. The task is challenging but there is a lot of scope in these investigations, with open questions and small advancements likely bringing researchers from set theory, analysis, number theory, and combinatorics to this particular topic. The research on these density functions and general ideals is fundamental and multidimensional, making it most appropriate to place it in the area of "mathematical foundations." As the author's expertise lies mainly in set theoretic topology and set theory, they plan to work on these aspects and seek the help of number theorists for other problems. |