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.