Higher-dimensional incidence theorems and applications in geometric combinatorics

自2010年Guth和Katz的论文取得突破性进展以来，由于新一代关联结果在更高维度上的发展，几何组合学已经取得了很大的进展。这些结果可以归结为两大类。一种适用于一般领域，如鲁德涅夫的点面定理。另一种是针对实数的，另外还使用了多项式划分方法。预计萨姆·曼斯菲尔德论文的主要主题将是高维关联理论的发展，包括沙里尔和所罗门最近发表的两篇论文以及M.Walsch的一篇论文。在初始阶段，这将旨在加强特定关联定理的代数方面，并使用它们的更强版本来限制三维圆锥曲线交点的个数。Sam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。

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.