Westlake University, Hanghzou, February 1 - 6, 2026

The study of rectifiability and pure unrectifiability started in the first half of the twentieth century with the work of Besicovitch and his collaborators concerning the geometry of Hausdorff measures in the Euclidean plane. These have been later generalized to higher dimensions and applied to the calculus of variations. For instance, Besicovitch’s projection theorem – a geometric discrepancy between rectifiable and fractal sets – was generalized by Federer and used in his original proof with Fleming of the compactness of integral currents.

The projection theorem of Besicovitch-Federer has received renewed attention in recent years, for instance with the work of Bate in metric spaces and the work of Dabrowski giving a quantitative version of the projection theorem in the plane. Damian Dabrowski’s mini-course will present the history of the problem and recent tools that provide bounds for the size of projection of sets that are close to being purely unrectifiable. It is a first step toward answering the remaining open part of the Vitushkin’s conjecture asserting that a set E is removable for bounded complex analytic functions if and only if its Favard length is null, i.e. the average length of its orthogonal projections on lines vanishes. If E has zero Favard length and has finite 1-dimensional Hausdorff measure then it is removable for bounded complex analytic functions, as has been proved by G. David building on work of Christ, Jones, Mattila, Melnikov, Verdera among others, using tools from uniform rectifiability and harmonic analysis. For sets of  σ-finite length this follows from additional work of Tolsa. This implication fails in general, though, as shown by Mattila, Jones and Murai. The converse implication for sets of finite length follows from the work of Calderón, but is open in general.

Uniform rectifiability is a quantitative notion of rectifiability that was introduced in the early 1990’s, first in connection with boundedness of singular integral operators on sets. Guy David’s lectures will offer a quiet introduction to the notion, culminating with at least one example of corona construction, a tool that has proved to be very effective in this context. In recent years, the connection between the Riesz transform and rectifiability has been essential for the study of removable singularities for Lipschitz harmonic functions and also for the solution of some free boundary problems involving harmonic measure, such as the so called one-phase and two-phase problems for harmonic measure. Xavier Tolsa’s mini-course will survey some joint results which characterize the measures μ with L²(μ) bounded Riesz transform in terms of the β₂ coefficients of the measure.

The regularity theory of harmonic maps has long been a central topic in geometric analysis and non-linear PDE. It has been known that weakly harmonic maps can be everywhere discontinuous (Rivière), stationary harmonic maps are smooth away from a set of codimension 2 (Bethuel, Evans), while minimizing harmonic maps are smooth away from a set of codimension 3 (Schoen-Uhlenbeck). However, refined characterization of the singular set, of even minimizing harmonic maps, had long remained elusive until the seminal work by Naber-Valtorta. Among many other important results, they proved that the top-dimensional stratum of the singular set of a minimizing harmonic map has uniformly locally finite Hausdorff measure. Changyou Wang’s lectures will first introduce the partial regularity theory of (minimizing and) stationary harmonic maps by Hélein, Evans, Bethuel, then discuss the rectifiability property of the singular sets (and the associated defect measures) of stationary harmonic maps following the theory developed by Lin, Naber-Valtorta, and finally discuss some general properties of singular set of Chen-Struwe’s heat flow of harmonic maps. 

Following Naber-Valtorta’s original work, various new ideas and techniques have been developed and exploited, including quantitative stratification (Cheeger-Naber), new (W^1,p, rectifiable, and discrete) Reifenberg theorems, and novel energy covering/neck analysis methods. Daniele Valtorta’s mini-course will study rectifiable versions of Reifenberg-type conditions involving Dini summability of Jone’s β-numbers and calibrations.

Rademacher’s theorem asserts that the set of points where a real-valued Lipschitz function defined on a Euclidean space fails to be differentiable is Lebesgue-null. The converse holds for functions of one variable, up to a necessary descriptive-set-theoretical condition, i.e. every Lebesgue-null set in the real line is contained in the set of points where some Lipschitz function fails to be differentiable. For functions of two or more variables the situation is drastically different as illustrated by a theorem of Preiss: There exist Lebesgue-null sets in the plane that contain a point of differentiability of each Lipschitz function of two variables. Moreover, there is a wider collection of Lebesgue-null sets which contain points of differentiability of a typical Lipschitz function, and for all sets outside of this class Lipschitz mappings are non-differentiable in an extremal way. Olga Maleva’s mini-course will tie together differentiability of typical Lipschitz functions and property of being covered by countably many closed purely unrectifiable sets.