You may have noticed that I have a bit of an obsession with geometric incidences. I do believe that incidences are a natural mathematical object that is connected to many different parts of math. This belief seems at least partially justified by the developments of the recent years. Less than two years ago I posted a list of some applications of incidences outside of Discrete Geometry. The purpose of that list was to show how incidences are becoming useful in a variety of fields, such as Harmonic Analysis, Theoretical Computer Science, and Number Theory. It seems that this process did not slow down in the past two years – incidences have continued to demonstrate their usefulness in the aforementioned fields, and there is even a new interest in the subject by model theorists.
This post is part 2 of the list of incidence uses outside of Discrete Geometry. It is just a list of references, and does not include many details. In future posts I might focus on specific applications and provide actual explanations. Hopefully new applications will continue to appear and I’ll have to keep adding more and more parts to this list!
The Kakeya conjecture. Katz and Zahl derived an improved bound for the Kakeya conjecture. Specifically, they improved Wolff’s longstanding bound for the Hausdorff dimension problem in . This is the latest in a sequence of Harmonic Analysis works that have strong connections to incidences (see the first part of this list and also below).
Model theory. In Logic, a group of model theorists generalized incidence results to Distal structures. Another similar work extended incidence results to o-minimal structures. In general, there seems to be some interest in generalizing incidence-related problems to various models.
Number theory. A very recent number theoretic result is relying on incidence bounds. Admittedly, I still do not understand what this paper is about, and am hoping to learn that soon.
Algorithms. Moving to Theoretical Computer Science, a recent work analyzes point covering algorithms using incidence results.
Quantum Information. A few years ago, incidences in spaces over finite fields were used to study a problem in Quantum Information. This result is not from the past two years, but I was not aware of it before (thanks Ben Lund).
More Harmonic Analysis. An older survey of Łaba contains a nice review of previous appearances of incidences in Harmonic Analysis. I write “older” although not even a decade have passed. I just mean that this survey appeared before the new era of polynomial methods in Discrete Geometry.
There are obviously more results that are still missing from this list. If you noticed anything that I missed, I would be happy to hear about it.