Higher-dimensional incidence theorems and applications in geometric combinatorics

在Guth和Katz的2010年论文取得突破后，几何组合学取得了很大进展，这是由于新一代关联结果在更高维度上的发展。这些结果可以分为两大类。其中一个适用于一般域，例如鲁德涅夫的点平面定理。另一个是具体的实数，另外使用多项式分割方法。Sam曼斯菲尔德论文的主题是发展高维关联理论，讨论了Sharir和所罗门最近的两篇论文，以及M.沃尔什在初始阶段，这将旨在加强具体的关联定理的代数方面，并使用其更强的版本来约束三维圆锥曲线相交的数量。萨姆在关联理论方面有很强的背景，对他的论文的进一步研究将涉及其最新进展和设计应用程序。

After a breakthrough 2010 paper by Guth and Katz geometric combinatorics has seen a lot of progress, owing to the development of a new generation of incidence results in higher dimension. These results can be put into two main categories. One holds for general fields, such as Rudnev's point-plane theorem. The other is specific for the reals, using in addition the polynomial partitioning methods. The main theme of Sam Mansfield thesis is expected to be development of higher-dimensional incidence theory, taking up on two recent papers by Sharir and Solomon, as well as one by M. Walsch. At the initial stage this will aim at strengthening the algebraic aspect of specific incidence theorems and using their stronger versions to bound the number of conic curve intersections in three dimensions.Sam has a strong background in incidence theory, and further research towards his thesis will involve its most recent advances and designing applications for them.