Multi-scale incidence geometry (MUSING)
Päärahoittaja
Rahoittajan antama koodi/diaarinumero: 101087499
Päärahoittajan myöntämä tuki (€)
- 1 362 842,50
Rahoitusohjelma
Hankkeen aikataulu
Hankkeen aloituspäivämäärä: 01.08.2023
Hankkeen päättymispäivämäärä: 31.07.2028
Tiivistelmä
The ERC CoG project MUSING aims to make progress in several old problems in geometric measure theory (GMT), including Vitushkin's conjecture, the Furstenberg set conjecture}, and related projection problems, after Besicovitch and Marstrand. Recent work on these questions combines techniques from GMT, additive combinatorics, harmonic analysis, and incidence geometry. Many of them have direct links to other key problems in the area, such as \emph{Falconer's distance set problem}, and the Erdös-Szemer\'edi sum-product problem.
MUSING will tackle its problems with techniques from multi-scale analysis, and via the special case of Ahlfors-regular sets. Studying this case separately is justified for several independent reasons. The first one is methodological: Ahlfors-regular sets are uniform at different scales and locations, so they are amenable to multi-scale methods, such as the entropy method, induction on scales, and bootstrapping schemes. In many questions in this proposal, world records for Ahlfors-regular sets are far better than for general sets.
Second, Ahlfors-regular sets have great independent interest: they arise naturally from dynamical systems, and are, for example, more general than self-similar sets with separation. Recent works of Hochman, Shmerkin, Wu, and others have marked tremendous progress in incidence-geometric problems for dynamically defined sets. 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?
Third, in all the key problems in project MUSING, mechanisms exist for transferring results from (almost) Ahlfors-regular special cases to more general sets. For Vitushkin's problem, this mechanism is known as \emph{corona decomposition}, pioneered by David and Semmes in the 90s. For Furstenberg sets and projection problems, the relevant mechanism was devised by Keleti and Shmerkin in 2017 to make progress on Falconer's distance set problem.
MUSING will tackle its problems with techniques from multi-scale analysis, and via the special case of Ahlfors-regular sets. Studying this case separately is justified for several independent reasons. The first one is methodological: Ahlfors-regular sets are uniform at different scales and locations, so they are amenable to multi-scale methods, such as the entropy method, induction on scales, and bootstrapping schemes. In many questions in this proposal, world records for Ahlfors-regular sets are far better than for general sets.
Second, Ahlfors-regular sets have great independent interest: they arise naturally from dynamical systems, and are, for example, more general than self-similar sets with separation. Recent works of Hochman, Shmerkin, Wu, and others have marked tremendous progress in incidence-geometric problems for dynamically defined sets. 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?
Third, in all the key problems in project MUSING, mechanisms exist for transferring results from (almost) Ahlfors-regular special cases to more general sets. For Vitushkin's problem, this mechanism is known as \emph{corona decomposition}, pioneered by David and Semmes in the 90s. For Furstenberg sets and projection problems, the relevant mechanism was devised by Keleti and Shmerkin in 2017 to make progress on Falconer's distance set problem.
