Algebraic and Probabilistic examples in combinatorial geometry

DMS 0432237网卡茨印第安纳大学组合几何中的代数和几何实例摘要我们将研究一系列来自数学不同部分的类似问题。这些问题都是组合性质的，涉及到对各种几何对象之间重叠的估计。我们将尝试用代数和概率方法给出这些问题的例子。大量的数学分析涉及到处理无穷大。然而，在这里，我们从事一个有限的深入研究。准确地说，我们研究了几何对象的有限集合，如矩形、网格中的点的模式和高维球面上的向量的排列方式，以获得某些不寻常的效果。我们将试图发现新的和更好的这种排列方式。这样做的一种方法是使用已知的代数结构来建议一种构造。另一种可能性是在一定的约束条件下随机选择例子。

DMS 0432237Nets KatzIndiana UniversityAlgebraic and Geometric Examples in Combinatorial GeometryABSTRACTWe will study a collection of problems of similar flavor comingfrom different parts of mathematics. The problems are all combinatorialin nature and involve estimates on the overlap between various geometricobjects. We will attempt to produce examples in these problems byboth algebraic and probabilistic methods.A great deal of mathematical analysis involveshandling the infinite. Here however we engage ina deep study of the finite. To be precise, we study the waysin which finite collections of geometric objects like rectangles,patterns of points in a grid, and vectors on a high dimensionalsphere can be arranged to achieve certain unusual effects.We will attempt to discover new and better such arrangements.One way of doing this is by using a known algebraic structureto suggest a construction. Another possibility is to choose the example randomlywithin certain constraints.