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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=707957
• 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.

这项提议的目标是探索被称为“单项理想”的代数对象和 几何复形被称为“单纯复形”。单项式是变量和单项式理想的乘积 是一组单项式的组合的集合。一个众所周知的单纯复形的例子 是一张图表。 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，许多人 已经开发出组合工具来捕捉这些理想的行为。多亏了强大的 像“Groebner基”这样的工具，研究单项式理想的代数提供了对代数的洞察 任何理想的性质。出于这些原因，单项理想是例子的温床。 以及代数中的反例，在代数中，它们充当一个人可以和不能期望的东西的标尺 为了一个普遍的理想而发生。 开发计算工具的组合学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，作为 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移动 和它一起。非常神奇的是，新技术的进步和发明似乎验证了 加固老建筑，就好像这些建筑只是躺在那里等着被发现。 使用组合学来理解理想的想法可以追溯到几十年前，但在过去的十年里 看到该地区新的活动，有许多新的工具和数百份新的论文。此提案的一部分 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 加强我们最新的工具。 我研究的一个方向是研究组合对象，其相关理想是“Cohen-Macaulay”。 Cohen-Macaulay性质是一种微妙性质，它存在于代数或组合结构中 即使不是十全十美，也要确保“一切正常”。一旦一个物体是Cohen-Macaulay，它就会表现得很漂亮 复杂的计算变得容易了。此外，一旦你理解了是什么让一个物体成为科恩-麦考利， 你对那个物体的结构有内幕知识。Cohen-Macaulay对象的分类 代数、几何或组合语言很流行、很重要，也很难。 我感兴趣的一个相关概念是单项理想的“解决”。代数的归结 对象是一种使用一组不变量来描述它的方式，例如“射影维度”、“Betti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 它本身，它的分辨率用更简单的对象来描述它。对解决方案的研究可以追溯到 希尔伯特19世纪著名的协和定理。Cohen-Macaulayness的概念 上面描述的可以用决议来描述，并且对决议有很大的影响。中有关决议的文献 特别是一般和组合分辨率是巨大的。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来计算单项式理想的代数不变量，或者检查 它们是Cohen-Macaulay，不需要进行复杂的代数计算。这样的结果是最多的 在数学中很受欢迎，因为它们简化了本应复杂的东西。因此，我期望很高 对我的研究成果产生了很大的影响并得到了很多应用。