The project "Approximate incidence geometry" aims to make progress in a number of open problems in geometric measure theory, for example the Furstenberg set conjecture, the Distance set conjecture, and the Visibility conjecture. Here is what the visibility conjecture is about. Picture yourself standing at some distance from a thick forest, staring into the trees. You cannot see deep into the woods, because the view to the forest is blocked by the trees. How large is, in fact, the part of the forest that you can see? In precise mathematical terms: what is the dimension of the visible part of the forest? The conjecture predicts that the visible part is at most 1-dimensional, at least from almost all vantage points. The problem is trickier to solve than to state: at the moment, the best existing result tells us that the visible part cannot have dimension higher than 1.9. What is even more surprising, the proof of this fact is not elementary, but is based on Fourier-analysis.

2023

2027