- 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.)

# Month: May 2013

# Was Disney trying to kill mathematicians during the 1930’s?

* A more optimistic Disney-mathematics connection.*

“

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).

# Distinct Distances: Open Problems and Current Bounds (part 2)

*Elekes-Sharir framework*. Once again, thanks to Micha Sharir for helping in the preparation of this post.

## Structure of the optimal solution

*optimal*if . Since even the asymptotic value of is still unknown, we also consider sets of points such that , and refer to such sets as

*near-optimal*. All the point sets in this section are planar.

**Theorem 1. (Landau-Ramanujan)**

*The number of positive integers smaller than that are the sum of two squares is .*

# Distinct Distances: Open Problems and Current Bounds (part 1)

“

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.”

**Problem 1.**

*Find the exact asymptotic value of*.

*Names that appear quite often in this set of posts: Paul Erdős, Larry Guth, Nets Katz, and Endre Szemerédi.*

**Problem 2.**

*Find the exact asymptotic value of*.

## Constrained sets of points

Variant | Lower Bound | Upper Bound |
---|---|---|

# First post

Hi,

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.