New bounds and relations for some basic geometric problems

I will give an overview on recent conditional lower bounds for some basic (geometric) problems in P, then proceed with our recent upper bounds for these problems. Particularly, 3SUM, Fréchet Distance, Dynamic Time Warping, Geometric Edit Distance, Dominance Products, and high-dimensional Closest Pair problems. I will review the results and touch some of the techniques used to obtain them.

I will finish with our ongoing work towards breaking the quadratic barrier of some archetypal geometric problems, such as testing whether there are three collinear points in a set of n points in the plane.
(All works are joint with Micha Sharir.)