- A reduction from the distinct distances problem to a problem about line intersections in . This part is referred to as the Elekes-Sharir reduction or as the Elekes-Sharir framework.
- The introduction of polynomial partitioning.
- Applying 19th century analytic geometry tools that are related to ruled surfaces, such as flecnode polynomials.
- The polynomial technique presented in Guth and Katz’s previous paper. (That is, relying on the existence of a polynomial of a small degree that vanishes on a given point set to reduce incidence problems between points and lines to the interactions between the lines and an algebraic variety.)
“Dip the apple in the brew / Let the Sleeping Death seep through.”
Eventually, Turing reenacted the following scene, committing suicide by taking a bite out of a poisoned apple (injected with cyanide).
Structure of the optimal solution
“My most striking contribution to geometry is, no doubt, my problem on the number of distinct distances. This can be found in many of my papers on combinatorial and geometric problems.”
Names that appear quite often in this set of posts: Paul Erdős, Larry Guth, Nets Katz, and Endre Szemerédi.
Constrained sets of points
|Variant||Lower Bound||Upper Bound|
I’m Adam and you can find various details about me in this link.
In this blog I plan to write about combinatorial geometry in general, and mainly about the algebraic techniques that are recently being used in it (the most famous example is probably the novel proof of Guth and Katz for the distinct distances problem). I will start by trying to clearly explain the basics of these methods, and then move to more advanced stuff, recent developments, etc.
Regarding other topics, I’m not yet sure what these will be yet, but I will perhaps discuss connections between incidence problems and other branches of mathematics (such as this famous connection to additive number theory), numbers of planar configurations that can be embedded on a point set, and various personal musings.
I hope that you will find something interesting to read here.