Multi-scale incidence geometry

The ERC CoG project MUSING aims to make progress in several old problems in geometric measure theory (GMT), including Vitushkin's conjecture from the 60s, and the Furstenberg set conjecture proposed by Wolff in the 90s.

Both problems are related to sharpening and generalising some cornerstone results in GMT, such as the projection theorems of Besicovitch and Marstrand.

Recent work on these questions combines techniques from GMT, additive combinatorics, harmonic analysis, and incidence geometry.

Vitushkin's conjecture is motivated by the Painlev problem on finding a geometric characterisation for the removable singularities of bounded analytic functions.

The Furstenberg conjecture has direct links to other key open problems in continuum incidence geometry, such as Falconer's distance set problem, and the Erds-Szemerdi sum-product problem.

MUSING will tackle its problems with techniques from multi-scale analysis, and via the special case of Ahlfors-regular sets. These sets are uniform at different scales and locations, so they are particularly amenable to multi-scale methods.

On the other hand, progress in the Ahlfors-regular special cases can often be extended to more general sets via mechanisms such as the corona decompositions of David and Semmes, and the scale block decomposition technique, devised by Keleti and Shmerkin in their work on Falconer's distance set problem.

Apart from being a stepping stone on the way to general sets, Ahlfors-regular sets also have great independent interest.

Evidence is accumulating that incidence geometric problems may admit far stronger solutions for Ahlfors-regular sets than for general sets.

Conclusive results of this type already exist for classes of dynamically generated sets, notably self-similar sets, due to the works of Hochman, Shmerkin, Wu, and others.

To what extent can these results be extended to Ahlfors-regular sets, which share the spatial uniformity of self-similar sets, but lack an underlying dynamical system?