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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587779
• 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.

这个提议的目的是探索被称为“单项理想”的代数对象与 几何上称为“单纯复形”。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。一个著名的单纯复形的例子 是一个图表。 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 已经开发了组合工具来捕获这种理想的行为。 由于强大的 工具，如“Groebner基地”，研究单项式理想代数提供了深入了解代数 任何理想的性质。由于这些原因，单项理想是例子的滋生地 以及代数中的反例，在那里它们可以作为一个衡量标准， 为了一个普遍的理想而发生。 组合数学领域，开发计数工具，一直存在于数学世界， 如果不总是突出。一些最深刻的数学论证归结为组合数学。所以作为 数学发展到新的前沿，组合学调整和更新其结构和工具， 沿着。新技术的进步和发明似乎证实了这一点， 加强旧的，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，但最近十年 我们看到该领域重新活跃起来，有许多新工具和数百篇新论文。本提案的一部分 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，其存在于代数或组合结构中， 确保“工作”，即使不是完美的。一旦一个物体是科恩-麦考利的，它的行为就很漂亮 复杂的计算变得简单。此外，一旦你理解了是什么让一个对象成为科恩-麦考利， 你对那个物体的结构有内部了解Cohen-Macaulay天体的分类 代数、几何或组合语言是流行的、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”。 代数的分解 对象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， 它本身，它的分辨率用更简单的对象来描述它。 对决议的研究可以追溯到 到希尔伯特著名的世纪的朔望定理。 Cohen-Macaulayness概念 以上所述的问题可以用决议来描述，并对决议有很大的影响。关于决议的文献 特别是一般和组合的分辨率是巨大的。我最近的一些工作和我近期的研究 计划涉及新的想法，找到不变量，只画一个图或单纯复形。 我提出的研究旨在产生方法来“计数”单项式理想的代数不变量，或检查是否 它们是科恩-麦考利方程，不需要进行复杂的代数计算。这样的结果是最 在数学中，这是一个很好的例子，因为它们简化了原本应该复杂的东西。因此，我期望高 我的研究成果的影响和许多应用。