On February 9, 2021, Doowon Koh (고두원) from Chungbuk National University gave a talk at the Discrete Math Seminar on the restriction conjecture for the cone in four dimensions and its applications to the upper bound on the point-sphere incidences in the field field setting. The title of his talk was “On the cone restriction conjecture in four dimensions and applications in incidence geometry“.
Main purpose of this talk is to introduce a connection between restriction estimates for cones and point-sphere incidence theorems in the finite field setting. First, we review the finite field restriction problem for cones and address new results on the conical restriction problems. In particular, we establish the restriction conjecture for the cone in four dimensions. Second, we study how to apply the conical restriction results to the point-sphere incidence bounds. As a consequence, we obtain sharp point-sphere incidence bounds when sphere sets are not too big.