Project description
Solving the puzzles hidden in projections
Geometry often hides its deepest secrets in shadows and projections (how shapes appear when viewed from different angles). For nearly a century, mathematicians have sought to understand these patterns, but some of the hardest questions in geometric measure theory remain unresolved. With this in mind, the ERC-funded QPROJECT aims to tackle three of them, including long-standing conjectures by Vitushkin and Besicovitch about how complicated sets behave when projected along lines or radii. By developing new quantitative tools that blend geometry, analysis, and combinatorics, the project hopes to bring unprecedented precision to classical projection theorems. In doing so, QPROJECT could illuminate fundamental connections between geometry and analysis — and solve puzzles that have challenged mathematicians for decades.
Objective
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.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
This project's classification has been human-validated.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
This project's classification has been human-validated.
Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2025-STG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
00-656 Warszawa
Poland
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