Arrangements of Hyperplanes and Combinatorial Constructions in Topology

我的研究涉及组合数学和代数，拓扑和几何领域的技术之间的相互作用。 组合数学是数学的一个分支，是计算机科学和运筹学的基础。 它涉及离散结构以及计数问题的研究。 在数学的其他领域中，许多有趣的对象本质上是组合的，因此组合学的方法自然地出现在理解“测试”对象的家庭中，如复曲面簇和超平面排列。 这些可以作为可访问的例子，以帮助识别和理解一般的数学现象。 在另一个方向上，代数和几何的成熟技术有时可以回答组合问题，并为离散数学中观察到的或确定的对象行为提供深层原因。 例如，如果一个计数问题的答案是一个本身没有额外结构的数，那么人们可能会发现这个数实际上是一个多项式的求值。 也许多项式的系数本身可以解释为某些多面体的体积，或者它们计算了可以从原始问题构建的几何对象中的高维“孔”的数量。 通过这种方式，人们可以更好地理解组合对象，将其视为具有更多结构的事物的影子，并且人们希望，更好地了解或至少更容易分析的属性。

My research involves interactions between combinatorics and techniques from areas of algebra, topology, and geometry. Combinatorics is a branch of mathematics which sits at the foundation of computer science and operations research. It involves the study of discrete structures as well as of counting problems. Many interesting objects in other areas of mathematics are fundamentally combinatorial in nature, so methods of combinatorics appear naturally in understanding families of "test" objects such as toric varieties and hyperplane arrangements. These can be used as accessible examples to help identify and understand general mathematical phenomena. In a different direction, the well-developed techniques of algebra and geometry can sometimes answer combinatorial questions and offer deep reasons for observed or conjectured behaviour of objects in discrete mathematics. For example, if the answer to a counting problem is a number which in itself shows no extra structure, one might find that this number is, in fact, some evaluation of a polynomial. Perhaps the polynomial has coefficients which themselves may be interpreted as volumes of certain polyhedra, or they count the number of high-dimensional "holes" in a geometric object which could be built from the original problem. In this way, one may obtain a better understanding of a combinatorial object, by viewing it as a shadow of something with much more structure and, one hopes, properties which are better known or at least more easily analyzed.