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（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。用方程描述抛物线是加拿大人在高中时可能还记得的一个例子。类似地，我们可以将代数与组合学联系起来，允许人们从多个角度研究问题。尽管自笛卡儿时代以来，新工具不断发展，但从许多不同角度研究问题的主题始终如一。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。特别地，我们希望将交换代数的工具应用于图论中的开放问题，反之亦然。我建议的研究，具体如下，深入研究与此主题相关的一些领域。我提出的研究的一个方面集中在称为与图相关的环面理想的代数对象上。图是一组节点，其中一些节点用线连接在一起，有时用于模拟真实的单词场景。例如，用节点表示机场，如果在相应的机场之间有航班，则连接两个节点。从图中我们可以得到一个环理想，一个代数对象，可以用几何、代数或组合的方法来研究（环理想也出现在与生物学等领域相关的应用中）。我想确定图形结构是如何被编码成一个称为最小自由分辨率的代数对象的。例如，已知图中的闭偶行走对应于环面理想的生成。在我们的机场示例中，封闭偶数步行将对应于在返回原始机场之前前往偶数个机场。我的目标是确定可以从图中确定关于环面理想的其他信息。图也可以与称为边理想的代数对象相关联。我的建议的另一方面是研究边缘理想的各种幂，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次幂（与代数信息有关），也可以构造它的符号r次幂（与几何信息有关）。比较这些幂是交换代数中被称为包容问题的一般主题的一部分。通过使用边缘理想，我想利用图论信息来获得对包含问题的新见解。这个建议，这是我正在进行的研究的一部分，从组合的角度来理解代数不变量，将为未来的应用（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.