Combinatorial investigations in commutative algebra

这个提议的目标是探索被称为“单项式理想”的代数对象和被称为“简单复合体”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。一个众所周知的简单复合体的例子是图。单项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，已经开发了许多组合工具来捕捉这些理想的行为。由于强大的工具，如“格罗布纳基”，研究单项式理想的代数提供了洞察任何理想的代数性质。由于这些原因，单项式理想是代数中例子和反例子的滋生地，在代数中，它们作为衡量一般理想能发生什么和不能发生什么的标尺。开发计数工具的组合学领域一直存在于数学界，如果不是一直很突出的话。一些最深奥的数学论证可以归结为组合学。因此，随着数学发展到新的领域，组合学调整和更新其结构和工具，以配合它的发展。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。使用组合学来理解理想的想法可以追溯到几十年前，但最近十年，该领域出现了新的活动，出现了许多新工具和数百篇新论文。这个建议的一部分是在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最新的工具。我的研究方向之一是研究相关理想为“科恩-麦考利”的组合对象。Cohen-Macaulay性质是一种微妙的性质，它存在于代数或组合结构中，即使不完美，也能确保“事物正常工作”。一旦一个对象是科恩-麦考利，它的行为就会变得很漂亮，复杂的计算也变得很容易。此外，一旦你理解了是什么使一个物体成为科恩-麦考利，你就对这个物体的结构有了深入的了解。使用代数、几何或组合语言对Cohen-Macaulay对象进行分类是流行的、重要的和非常困难的。我感兴趣的一个相关概念是单体理想的“解决”。代数对象的解析是用一组不变量如“射影维数”、“贝蒂数”、“正则性”和“希尔伯特函数”来描述它的一种方法。这个想法是，即使你可能难以描述一个理想本身，它的解析度用更简单的对象来描述它。对决心的研究可以追溯到19世纪希尔伯特著名的Syzygy定理。上面描述的Cohen-Macaulayness概念可以用resolution来描述，并且对resolution有很大的影响。一般的决议，特别是组合决议的文献是巨大的。我最近的一些工作和我近期的研究计划涉及到仅通过绘制图形或简单复合体来寻找不变量的新想法。我提出的研究目标是产生“计数”单项式理想的代数不变量的方法，或者检查它们是否符合Cohen-Macaulay，而不需要进行复杂的代数计算。这样的结果是数学中最受欢迎的，因为它们简化了本应复杂的东西。因此，我期望我的研究结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" andgeometric 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 movealong 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 structureensures that ``things work'', even if not perfectly. Once an object is Cohen-Macaulay, it behaves beautifullyand 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 algebraicobject 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.