Combinatorial investigations in commutative algebra

这个提议的目的是探索被称为“单项理想”的代数对象和被称为“单纯复形”的几何对象之间的联系。单项式是变量的乘积，而单项式理想是一组单项式的组合的集合。单纯复形的一个著名例子是图。单名理想是所有理想中最容易研究的一类理想。在过去的几十年里，许多组合工具已经开发出来，以捕捉这些理想的行为。由于强大的工具，如“Groebner基地”，研究单项式理想的代数提供洞察任何理想的代数性质。由于这些原因，单项理想是代数中例子和反例的温床，在那里它们作为衡量一个一般理想可以和不可以期望发生什么的标尺。组合数学领域，开发计数工具，一直存在于数学世界中，如果不总是突出的。一些最深刻的数学论证归结为组合数学。因此，随着数学向新的领域发展，组合数学也在调整和更新其结构和工具，以便与之一起沿着前进。新技术的进步和发明似乎验证和加强了旧技术，这是相当神奇的，就好像这些结构只是躺在那里等待被发现。使用组合数学来理解理想的想法可以追溯到几十年前，但在过去的十年里，该领域出现了新的活动，有许多新的工具和数百篇新论文。这个提议的一部分是在一个更现代的环境中重新评估一些旧的技术，并利用这些发现来加强我们的最新工具。我的研究方向之一是研究与“Cohen-Macaulay”理想相关的组合对象。Cohen-Macaulay性质是一种微妙的性质，它在代数或组合结构中的存在确保了“事物工作”，即使不是完美的。一旦一个对象是科恩-麦考利，它的行为很漂亮，复杂的计算变得容易。此外，一旦你理解了是什么使一个对象成为科恩-麦考利，你就有了关于该对象结构的内部知识。使用代数、几何或组合语言对Cohen-Macaulay对象进行分类是流行的、重要的，也是非常困难的。我感兴趣的一个相关概念是单项理想的“分解”。代数对象的分解是一种使用一组不变量来描述它的方法，例如“投影维数”，“贝蒂数”，“正则性”和“希尔伯特函数”。这个想法是，即使你可能很难描述一个理想本身，它的解决方案描述了它的更简单的对象。对归结的研究可以追溯到希尔伯特在世纪提出的著名的合朔定理。上述Cohen-Macaulayness的概念可以用分辨率来描述，并且对分辨率有很大的影响。关于一般决议和特别是组合决议的文献是大量的。我最近的一些工作和我近期的研究计划涉及新的想法，只画一个图或单纯形复杂找到不变量。我提出的研究目的是产生方法来“计数”单项理想的代数不变量，或者检查它们是否是科恩-麦考利，而不需要进行复杂的代数计算。这样的结果在数学中是最受欢迎的，因为它们简化了本来应该复杂的东西。因此，我期望我的研究成果具有很高的影响力和许多应用。

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.