Problems in combinatorial commutative algebra

我的研究涉及三个领域：代数、几何和组合学(离散结构)。数学家勒内·笛卡尔是最早证明将几何对象与代数方程联系在一起的力量之一。用方程式描述抛物线是加拿大人可能从高中就记得的这种技术的一个例子。同样，我们可以将代数与组合学联系起来，允许人们通过多个镜头研究一个问题。虽然自笛卡尔时代以来不断开发新的工具，但从许多不同角度研究问题的主题一直是不变的。*我研究的长期目标是在代数和组合学(如果可能的话，还有几何)之间建立新的联系。特别是，我们想要应用交换代数的工具来解决图论中的公开问题，反之亦然。我提出的研究，以及下面的一些具体内容，深入探讨了与这个主题相关的一些领域。*我提出的研究的一个方面集中在与图相关的称为环面理想的代数对象上。图形是一组节点，其中一些节点用线连接在一起，有时用于模拟真实世界的场景。例如，用节点表示机场，如果相应的机场之间有航班，则连接两个节点。从一个图中我们可以得到一个圆理想，一个代数对象，可以用几何、代数或组合的方法来研究(圆理想也出现在与生物等领域相关的应用中)。我想确定如何将图结构编码到一个称为最小自由分解的代数对象中。例如，已知图中的闭偶游动对应于圆理想的生成元。在我们的机场示例中，封闭的偶数步行对应于在返回原始机场之前到偶数个机场的旅行。我的目标是确定关于环面理想的其他哪些信息可以从图中确定。*图也可以与称为边理想的代数对象相关联。我提议的另一个方面是研究边缘理想的各种力量，并对它们进行比较。对于任何理想(代数对象)，可以构造它的正则r次方(与代数信息有关)，也可以构造它的r次符号次方(与几何信息有关)。比较这些幂是交换代数中称为包容问题的一般主题的一部分。通过使用边理想，我想利用图论信息来获得关于包容问题的新见解。*这个建议是我正在进行的从组合角度理解代数不变量的研究的一部分，将为未来的应用(例如生物学、统计学、线性规划)提供理论基础，并将揭示代数和组合学之间的新联系。

My research lies in the intersection of three areas: algebra, geometry, and combinatorics (discrete structures). The mathematician Rene Descartes was among the first to demonstrate the power of associating geometric objects with algebraic equations and vice versa. Describing a parabola with an equation is an example of this technique that Canadians may remember from high school. Similarly, we can link algebra with combinatorics, allowing one to study a problem through multiple lenses. While new tools have been developed since Descartes' time, the theme of studying problems from many different angles has remained a constant.******The long term goal of my research is to buid new connections between algebra and combinatorics (and if possible, geometry). In particular, we want to apply the tools of commutative algebra to open problems in graph theory, and vice-versa. My proposed research, with some specifics below, delves into some areas related to this theme. ******One facet of my proposed research focuses on algebraic objects called toric ideals associated with a graph. A graph is a set of nodes with lines joining some of the nodes together, sometimes used to model real word scenarios. E.g., denote airports by nodes, and join two nodes if there is a flight between the corresponding airports. From a graph we can make a toric ideal, an algebraic object, that can be studied geometrically, algebraically, or combinatorially (toric ideals also appear in applications related to areas such as biology). I want to determine how the graph structure gets encoded into an algebraic object called a minimal free resolution. For example, it is known that closed even walks in a graph correspond to the generators of the toric ideal. In our airport example, a closed even walk would correspond to a trip to an even number of airports before returning to the original airport. My goal is to determine what other information about the toric ideal can be determined from the graph.******Graphs can also be associated with algebraic objects called edge ideals. Another facet of my proposal is to study various powers of edge ideals and compare them. For any ideal (an algebraic object), one can construct its regular r-th power (related to algebraic information), or one can construct its r-th symbolic power (related to geometric information). Comparing these powers is part of general theme in commutative algebra called the containment problem. By using edge ideals, I want to exploit the graph theory information to gain new insights on the containment problem.******This proposal, which is part of my ongoing research to understand algebraic invariants from a combinatorial point-of-view, will provide a theoretical basis for future applications (e.g. biology, statistics, linear programming) and will uncover new connections between algebra and combinatorics.