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。

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.