Combinatorial investigations in commutative algebra

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The goal of this proposal is to explore connections between algebraic objects called "monomial ideals" and geometric ones called "simplicial complexes". A monomial is a product of variables, and a monomial ideal is a collection of combinations of a set of monomials. A well-known example of a simplicial complex is a graph. Monomial ideals are the simplest class of ideals to study among all ideals. Over the past decades many combinatorial tools have been developed to capture the behaviours of such ideals. Thanks to powerful tools such as "Groebner Bases", studying the algebra of monomial ideals provides insight into algebraic properties of any ideal. For these reasons, monomial ideals are the breeding ground for examples and counterexamples in Algebra, where they serve as a measuring stick for what one can and cannot expect to happen for a general ideal. The field of Combinatorics, which develops counting tools, has always been present in the mathematical world, if not always prominent. Some of the deepest mathematical arguments reduce to Combinatorics. Therefore, as Mathematics progresses to new frontiers, Combinatorics adjusts and updates its structures and tools to move along with it. It is quite magical that the progress and invention of new techniques seems to validate and strengthen the old ones, as if these structures were simply lying there waiting to be discovered. The idea of using Combinatorics to understand ideals goes back several decades, but the last ten years has seen renewed activity in the area, with many new tools and hundreds of new papers. Part of this proposal is to re-evaluate some of the older techniques in a more modern setting, and to use the findings to strengthen our latest tools. One direction of my research is investigating combinatorial objects whose related ideal is "Cohen-Macaulay". The Cohen-Macaulay property is a subtle property whose presence in an algebraic or combinatorial structure ensures that ``things work'', even if not perfectly. Once an object is Cohen-Macaulay, it behaves beautifully and complex calculations become easy. Moreover, once you understand what makes an object Cohen-Macaulay, you have inside knowledge of the structure of that object. The classification of Cohen-Macaulay objects using algebraic, geometric, or combinatorial language is popular, important, and very difficult. A related concept of interest to me is the "resolution" of monomial ideals. The resolution of an algebraic object is a way to describe it using a set of invariants such as "projective dimension", "Betti numbers", "regularity" and "Hilbert functions". The idea is that even if you might have difficulty describing an ideal itself, its resolution describes it in terms of simpler objects. The study of resolutions goes back to Hilbert's celebrated Syzygy Theorem from the nineteenth century. The concept of Cohen-Macaulayness described above can be described by and has a great impact on resolutions. The literature on resolutions in general and combinatorial resolutions in particular is vast. Some of my recent work and my immediate research plans concern new ideas to find invariants by only drawing a graph or simplicial complex. My proposed research aims to produce ways to "count" algebraic invariants of monomial ideals, or check if they are Cohen-Macaulay, without doing complicated algebraic calculations. Such results are the most sought-after in Mathematics, since they simplify what is supposed to be complicated. I therefore expect high impact and many applications for the results of my research.