Continuous Incidence Theory and its Applications to Number Theory and Geometry

Friday, June 1, 2012 - 4:30pm - 5:00pm
Keller 3-180
It is a classical problem to study incidences between a nite number
of points and a nite number of geometric objects. In this presentation, we see that continuous incidence theory can be used to derive a number of results in geometry, geometric measure theory, and analytic number theory. The applications to geometry include a fractal variant of the regular value theorem. The applications to geometric measure theory include a generalization of Falconer distance problem in which we prove that a compact subset of Rd of sufficiently large Hausdorff dimension determines a positive proportion of all (k+1)-configurations
described by certain restrictions. More specifically we consider the set

{(x, x1,...,xk)∈Ek+1:x - xiB = ti;1≤ik},

where B is any convex centrally symmetric body with a smooth bound-
ary and non-vanishing curvature.
The applications to Number Theory include counting integer lat-
tice points in the neighborhood of variable coecient families of sur-
MSC Code: