Combinatorial investigations in commutative algebra

交换代数中的组合研究

• 批准号: RGPIN-2014-04392
• 负责人: Faridi, Sara
• 金额: $ 1.31万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676848
• 关键词: Combinatorial; investigations; commutative; algebra

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.

该提案的目标是探索称为“单项理想”的代数对象和称为“单纯复形”的几何对象之间的联系。单项式是变量的乘积，单项式理想*是一组单项式的组合的集合。单纯复形的一个众所周知的例子*是图。 **单项式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种理想的行为。 借助“Groebner Bases”等强大的工具，研究单项式理想的代数可以深入了解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*和反例的滋生地，它们充当了人们可以期望和不能期望*在一般理想中发生什么的衡量标准。**开发计数工具的组合学领域一直存在于数学世界中，*即使不是总是很突出。一些最深刻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随之发展*。新技术的进步和发明似乎验证并加强了旧技术，这是非常神奇的，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，但过去十年*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。该提案的一部分是在更现代的环境中重新评估一些较旧的技术，并利用这些发现来增强我们最新的工具。**我研究的一个方向是调查其相关理想是“Cohen-Macaulay”的组合对象。 *科恩-麦考利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“事物正常运转”，即使不是完美的。一旦一个对象成为 Cohen-Macaulay，它就会表现得非常漂亮*并且复杂的计算也会变得容易。此外，一旦您了解了 Cohen-Macaulay 对象的构成因素，*您就对该对象的结构有了内部了解。使用*代数、几何或组合语言对科恩-麦考利对象进行分类是流行的、重要的，但非常困难。**我感兴趣的一个相关概念是单项式理想的“分辨率”。 代数*对象的解析是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，即使你可能难以描述一个理想本身，它的解决方案也可以用更简单的对象来描述它。 对决议的研究可以追溯到十九世纪希尔伯特著名的 Syzygy 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非常多。我最近的一些工作和我近期的研究*计划涉及通过仅绘制图形或单纯复形来找到不变量的新想法。 **我提出的研究旨在找到“计算”单项式理想的代数不变量的方法，或者检查它们是否是科恩-麦考利的，而无需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事情。因此，我期望我的研究成果具有高影响力和广泛的应用。