We are pleased to announce that Dr Damian Dąbrowski has received a grant from the European Research Council (ERC) for the implementation of the project: QPROJECT – Quantitative projection problems in geometric measure theory, the abstract of which is attached below. We warmly congratulate Dr Dąbrowski.
dr Damian Dąbrowski
The project will run from the 1st of October 2025 for five years until the 30th of September 2030. Four researchers are planned to be recruited to the research team implementing the project – three postdocs and
one doctoral student.
This is already the third project funded by the ERC to be carried out at the Institute of Mathematics of the Polish Academy of Sciences (the previous ones were: INDEX by Prof. P. Nowak and KAPIBARA by Prof. P. Achingera). It is worth noting that these three grants are the only ERC projects awarded in Poland within the mathematics panel (PE1).
Link to the competition: link 1 / link 2
Link to the results: https://erc.europa.eu/news-events/news/erc-2025-starting-grants-results
Link to the project website: damiandabrowski.eu/qproject
In the QPROJECT project, we will seek answers to questions from geometric measure theory (GTM), 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.