Quantitative projection problems in geometric measure theory

Duration: October 2025 – September 2030

Host Institution: Institute of Mathematics of the Polish Academy of Sciences (IMPAN)

Grant number: 101219218 QPROJECT

About QPROJECT

This project is in the field of geometric measure theory (GMT), an area of analysis seeking to solve geometric problems using the tools of measure theory. A classical line of research in GMT concerns estimating the size of orthogonal projections of planar sets, and the most important results in this topic are the projection theorems of Besicovitch and Marstrand.

In the last few decades, it became increasingly clear that obtaining stronger, more quantitative projection results is connected to open questions at the intersection of GMT, complex analysis, harmonic analysis, and additive combinatorics. The main goal of this project is proving quantitative projection results, with special focus on three concrete questions.

The first is Vitushkin’s conjecture from 1967. One of the key objectives of QPROJECT is completing the solution to this conjecture by showing that removable sets for bounded analytic functions have Lebesgue-null orthogonal projections in almost every direction. This will be achieved by proving a quantitative Besicovitch projection theorem. The second question this project aims to answer is an old conjecture of Besicovitch about the radial projections of purely unrectifiable sets.

The plan is to solve these two problems using the tools of quantitative rectifiability, and it is the right time to tackle them due to the PI’s recent solution to a closely related conjecture of David and Semmes. The new techniques introduced in that work, such as the directional stopping time arguments, are likely to lead to breakthroughs on the two old questions.

The third problem is the visibility conjecture from fractal geometry, which is closely related to quantifying Marstrand’s classical slicing theorem. Building on the PI’s earlier work on this conjecture, the key to the full solution will be proving lower bounds on incidences in multiscale incidence geometry.

Team

PI: Damian Dąbrowski

Hiring plan:

• 3 postdocs, starting from 2025, 2026, and 2028,
• 1 PhD student, starting from 2027.

If you are interested in applying for one of these positions, send me an email!

Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or European Research Council. Neither the European Union nor the granting authority can be held responsible for them.