Arrangements of Hyperplanes and Combinatorial Constructions in Topology

我的研究涉及组合数学与代数、拓扑学和几何学领域的技术之间的相互作用。 组合学是数学的一个分支，它是计算机科学和运筹学的基础。它既涉及离散结构的研究，也涉及计数问题的研究。在数学的其他领域中，许多有趣的对象本质上都是组合性的，所以组合学的方法在理解诸如环簇和超平面排列等“测试”对象的族时自然出现。这些可以作为可访问的例子来帮助识别和理解一般的数学现象。 在不同的方向上，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推测的对象行为提供深刻的理由。例如，如果计数问题的答案是一个本身没有额外结构的数字，人们可能会发现这个数字实际上是对多项式的某种求值。也许多项式的系数本身可以被解释为某些多面体的体积，或者他们计算可以从原始问题建立的几何对象中的高维“洞”的数量。通过这种方式，人们可以通过将组合对象视为具有更多结构的东西的阴影，并希望将其视为更为人所知或至少更容易分析的性质，从而更好地理解组合对象。

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.