Applications of commutative algebra

交换代数的应用

• 批准号: RGPIN-2014-03898
• 负责人: VanTuyl, Adam
• 金额: $ 1.31万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=610711
• 关键词: Applications; commutative; algebra

My research lies in the intersection of three areas of mathematics: algebra, geometry, and combinatorics. The overlap of algebra and geometry is a rich area of mathematics that dates back to at least the first half of the 17th century. The mathematician/philosopher Rene Descartes was among the first to demonstrate the power of associating geometric objects with algebraic equations and vice versa. Describing a line or a parabola with an equation is an example of this technique that many Canadians will remember from their mathematics courses in high school. In a similar way, there are ways to connect algebra with problems in combinatorics (counting problems and discrete structures), allowing one to study the same problem through multiple lenses. While many new tools and techniques have been developed since Descartes’ time, the theme of studying problems from many different angles has remained a constant. The research of the proposal is interested in exploring and uncovering new connections between algebra, geometry, and combinatorics, and in particular, applying the tools and techniques of commutative algebra to questions in other areas of mathematics. A brief summary of three of my proposed projects are given below. One facet of my proposed research focuses on the algebraic structures attached to a graph, in order to build a bridge between graph theory and algebra. A graph is a collection of nodes with lines joining some of the nodes together. When we colour a graph, we want to assign a colour to each node subject to the rule that nodes that are joined together by a line must receive a different colour. One would then like to know what is the least number of colours you need to colour the graph. Colouring problems can be related to questions about scheduling and even solving Sudoku problems. Over the last couple of years, I have been interested in studying how this colouring information can be encoded algebraically. My current research proposal is interested in finding graphs with interesting colouring properties in order to build unique and interesting algebraic structures. Another project of my proposal also hopes to strengthen this bridge between graph theory and algebra. I am interested in objects called toric ideals which can be constructed from a graph. Toric ideals have rich structure, and can be studied geometrical, algebraically, or even combinatorially. Toric ideals are of great interest because they have applications to areas such as biology and statistics. I am interested in determining how the graph theory information gets encoded into an algebraic object called a minimal free resolution. A third project of my research proposal is to use geometry and algebra to study problems about matrices. Matrices, which are rectangular arrays of numbers, are commonly introduced as a tool to study systems of equations. I am currently interested in problems of the following type: what properties about the matrix can we deduce if we are only given some partial information about the matrix, e.g. the location of the non-zero entries? The origin of this type of question can be traced back to work of P. Samuelson, the Nobel Prize winner in Economics. More recently, problems of this type have appeared in the context of modelling diseases. My proposed research plans to study problems of this type through the lenses of algebraic geometry and commutative algebra, thus introducing new tools and techniques to the area. These projects, which are part of my ongoing research programme to understand homological and algebraic invariants, will hopefully provide a theoretical basis for future applications.

我的研究领域是数学三个领域的交叉：代数、几何和组合学。代数和几何的重叠是数学的一个丰富领域，至少可以追溯到17世纪的上半叶。数学家兼哲学家勒内·笛卡尔（Rene Descartes）是最早证明将几何对象与代数方程联系起来的力量之一，反之亦然。用方程描述一条直线或一条抛物线是这种技巧的一个例子，许多加拿大人在高中的数学课程中都会记得。同样，也有一些方法将代数与组合数学中的问题（计数问题和离散结构）联系起来，允许人们通过多个角度研究同一问题。虽然自笛卡尔时代以来已经开发了许多新的工具和技术，但从许多不同角度研究问题的主题仍然是一个常数。 该提案的研究兴趣在于探索和揭示代数，几何和组合学之间的新联系，特别是将交换代数的工具和技术应用于其他数学领域的问题。下面是我提出的三个项目的简要摘要。 我提出的研究的一个方面集中在图的代数结构，以建立图论和代数之间的桥梁。一个图是一个节点的集合，其中一些节点用线连接在一起。当我们给一个图着色时，我们想给每个节点分配一种颜色，但要遵守这样的规则，即由一条线连接在一起的节点必须接收不同的颜色。然后，人们想知道你需要给图着色的最少颜色是多少。着色问题可以与调度问题甚至解决数独问题有关。在过去的几年里，我一直有兴趣研究如何用代数编码这种颜色信息。我目前的研究计划是寻找具有有趣的着色特性的图，以构建独特而有趣的代数结构。 我提议的另一个项目也希望加强图论和代数之间的这座桥梁。我感兴趣的对象称为环面理想，可以从一个图构造。复曲面理想具有丰富的结构，可以用几何、代数甚至组合的方法来研究。复曲面理想是非常有趣的，因为它们可以应用于生物学和统计学等领域。我感兴趣的是确定图论信息如何被编码到一个称为最小自由分辨率的代数对象中。 我的研究计划的第三个项目是利用几何和代数来研究有关矩阵的问题。矩阵是数字的矩形阵列，通常被引入作为研究方程组的工具。我目前对以下类型的问题感兴趣：如果我们只给出矩阵的部分信息，例如非零元素的位置，我们可以推导出矩阵的哪些性质？这类问题的起源可以追溯到诺贝尔经济学奖赢家萨缪尔森的工作。最近，这种类型的问题出现在建模疾病的背景下。我提出的研究计划通过代数几何和交换代数的镜头来研究这类问题，从而为该领域引入新的工具和技术。 这些项目，这是我正在进行的研究计划，以了解同调和代数不变量的一部分，将有望为未来的应用提供理论基础。