Higher-dimensional incidence theorems and applications in geometric combinatorics

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.