Applications of commutative algebra

我的研究涉及数学的三个领域：代数、几何和组合学。代数和几何的重叠是一个丰富的数学领域，至少可以追溯到17世纪上半叶。数学家兼哲学家勒内·笛卡尔（Rene Descartes）是第一批证明几何物体与代数方程相关联的力量的人之一，反之亦然。用方程描述直线或抛物线是这种技巧的一个例子，许多加拿大人在高中数学课程中都会记得。类似地，也有一些方法可以将代数与组合学中的问题（计数问题和离散结构）联系起来，从而允许人们从多个角度研究同一个问题。自笛卡儿时代以来，虽然出现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不变的。该提案的研究兴趣在于探索和发现代数、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总结。我提出的研究的一个方面集中在图的代数结构上，以便在图论和代数之间建立一座桥梁。图是节点的集合，其中一些节点用线连接在一起。当我们给一个图上色时，我们想给每个节点分配一种颜色，但要遵守由一条线连接在一起的节点必须接收不同颜色的规则。然后我们想知道给图形上色最少需要多少种颜色。着色问题可能与日程安排甚至数独问题有关。在过去的几年里，我一直对研究如何用代数方法编码这些着色信息很感兴趣。我目前的研究计划是寻找具有有趣着色性质的图，以建立独特而有趣的代数结构。我提案的另一个项目也希望加强图论和代数之间的桥梁。我感兴趣的是可以从图中构造的被称为环面理想的对象。环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。环面理想之所以引起人们的极大兴趣，是因为它们在生物学和统计学等领域都有应用。我感兴趣的是如何将图论信息编码成一个称为最小自由分辨率的代数对象。我的研究计划的第三个项目是使用几何和代数来研究矩阵问题。矩阵是数字的矩形数组，通常作为研究方程组的工具而引入。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，例如非零项的位置，我们可以推断出矩阵的哪些性质？这类问题的起源可以追溯到诺贝尔经济学奖得主萨缪尔森的研究。最近，这类问题出现在疾病建模的背景下。我提出的研究计划是通过代数几何和交换代数的透镜来研究这类问题，从而为该领域引入新的工具和技术。这些项目是我正在进行的研究计划的一部分，以理解同调和代数不变量，将有望为未来的应用提供理论基础。

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.