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。单项是变量的乘积，单项理想是一组单项组合的集合。简单复形的一个众所周知的例子就是图。**单项理想是所有理想中最简单的一类理想。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。多亏了强大的工具，如“Groebner基”，研究单项理想的代数提供了对任何理想的代数性质的洞察。出于这些原因，单项理想是代数中的例子和反例的温床，在代数中，它们充当衡量一个人可以和不能期望的事情发生的标尺。**组合学是开发计算工具的领域，一直存在于数学界，*如果不总是突出的话。一些最深刻的数学争论可以归结为组合学。因此，随着数学发展到新的前沿，组合数学调整和更新其结构和工具以与之相适应。非常神奇的是，新技术的进步和发明似乎验证和加强了旧的技术，就好像这些结构只是躺在那里等待发现。**使用组合学理解理想的想法可以追溯到几十年前，但在过去的十年里，该领域重新活跃起来，出现了许多新的工具和数百篇新的论文。这项提议的一部分*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。**我研究的一个方向是调查其相关理想是“科恩-麦考利”的组合物体。*Cohen-Macaulay性质是一种微妙的性质，它存在于代数或组合结构中*确保“事物运行”，即使不是完美的。一旦一个物体是Cohen-Macaulay，它就会表现得很漂亮，复杂的计算也变得很容易。此外，一旦你理解了是什么造就了科恩-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对象进行分类是非常流行、重要且非常困难的。**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用一组不变量来描述它的方法，如“射影维度”、“贝蒂数”、“正则性”和“希尔伯特函数”。这个想法是，即使你可能难以描述一个理想*本身，它的分辨率也可以用更简单的物体来描述它。对决断的研究可以追溯到希尔伯特19世纪著名的协和定理。上面描述的Cohen-Macaulayness*的概念可以用分辨率来描述，并对其有很大的影响。关于一般分辨率和特别是组合分辨率的文献非常多。我最近的一些工作和我最近的研究*计划涉及到通过只画一个图或单纯复形来寻找不变量的新想法。**我提议的研究旨在产生一种方法，在不进行复杂的代数计算的情况下，“计算”单项理想的代数不变量，或者检查它们是否*是Cohen-Macaulay。这类结果是数学界最受欢迎的结果，因为它们简化了本应复杂的事情。因此，我希望我的研究成果能够产生很大的影响，并得到很多应用。

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.