UBC Math Department Colloquium: Hong Wang

Given a set T of distinct \delta-tubes and a set P of disjoint \delta-balls in R^n, the set of incidences between T and P is defined as I(P,T)={(p,l)∈P×T:p∩l≠∅}. The well-known Szemeredi-Trotter theorem in combinatorics studies a discrete analogue, the number of incidences between points and lines in the plane. On the other hand, the incidence for tube problems are natural generalizations of projection theorems in geometric measure theory.

In this talk, we will survey recent progress on incidence bounds for tubes including the Furstenberg sets estimate in the plane, restricted projections, Kakeya problem and discuss their connections and future problems. A lot of progress is made by members of the analysis group at UBC.