The Geometry of Singular Sets

From August 2–13, 2027, the second two-week program in our inaugural IAS / South America season, The Geometry of Singular Sets: School on Geometric Measure Theory, Harmonic Analysis and PDE, will take place exclusively at the Universidad de Buenos Aires.

All of the courses in this program approach a single question from different angles: how much structure must a set or a solution have, and where can that structure break down? The answers draw on dimension, rectifiability, and the geometry of singular sets. Harmonic measure and free-boundary problems read the analysis of an elliptic equation off the geometry of a boundary; the singular sets of minimal surfaces and other variational problems are governed by rectifiability; regularity theory for nonlocal and degenerate equations asks what survives when classical geometry breaks down; and the dimension theory of fractal sets, together with projection, Fourier-analytic and multifractal methods supply the measuring tools common to all of them.

Format at a Glance

The program will last for two weeks, from Monday to Friday, with both Wednesday and Friday afternoons kept unscheduled to allow for research and informal discussion. The program also includes two 90-minute distinguished plenary lectures (one each week) and a series of plenary lectures.

• Monday-Friday Teaching Blocks: 10:00–11:30, 12:00–13:30, 15:00–16:30
• Coffee Breaks: 11:30-12:00 (11:00-11:30 on Wednesdays)
• Lunch: 13:30–15:00
• Evening Sessions: 17:00-19:00 (Mondays, Tuesdays, and Thursdays)

Week 1: August 2–6, 2027

Minicourses 

• Eugenia Malinnikova — Stanford University
  Critical Sets of Harmonic Functions: The critical set of a nonconstant harmonic function on a domain in n-dimensional space has Hausdorff dimension at most (n–2). In this minicourse, we will study local bounds for its (n–2)-dimensional Hausdorff measure in terms of the growth of the function, measured by its frequency or doubling index. We will first present quantitative Cauchy uniqueness for harmonic functions and then discuss the estimate of Naber and Valtorta for the critical set. Then we will prove a sharper bound in terms of the frequency. The proof combines new inequalities for harmonic polynomials with geometric arguments. The minicourse is based on recent work with Ben Foster, Josef Greihuber, and Fedor Nazarov.
• Camillo De Lellis — Institute for Advanced Study, Princeton, New Jersey
  Sets and the Plateau Problem: The Plateau problem concerns, famously, surfaces which minimize area among all those spanning a given contour, and the Belgian physicist Joseph Plateau formulated it to model the geometry of soap films. Assume the contour is a classical smooth embedded loop Γ in ³. A natural attempt to formulate the problem mathematically is to look for ℝ the embedded oriented 2-dimensional smooth surface which minimizes the area among all those whose boundary is, in the sense of smooth manifolds, Γ. One of the crowning achievements of geometric measure theory is that such a surface always exists and it is as smooth as one could possibly hope (real analytic away from Γ and, modulo minor technical adjustments, as smooth as Γ allows at boundary points). However, any classroom experiment with suitably complicated loops shows that in fact actual soap films form singularities. For this reason, there are several alternative formulations of the Plateau problem that instead propose minimizing area in much more general classes of subsets of Euclidean space. In these lectures, we will review several such concepts, and give a glimpse of the underlying rich existence and regularity theory. Given the vastness of the material, rather than insisting on detailed arguments, the course will focus on providing a bird's-eye view of the subject.
• Luis Silvestre — University of Chicago
  Regularity for Elliptic PDE: Elliptic equations describe states in equilibrium. The Laplace equation is the prime example. Solutions to elliptic equations are typically quite smooth, and this is the property that we will investigate. Studying equations with irregular coefficients is useful to understanding nonlinear equations. An inequality, based on an old idea of Aleksandrov, plays a central role in the regularity theory for elliptic equations in nondivergence form. We will discuss old and new ideas to prove this inequality in various contexts, and its consequences. 

Distinguished Plenary Lecture

Carlos Kenig — University of Chicago
Interior and Boundary Unique Continuation: In this expository lecture we will describe classical interior unique continuation results and boundary unique continuation results, for elliptic equations. We will also describe connections with geometric problems dealing with nodal sets, singular sets and critical sets, both in the interior and at the boundary, on domains with low regularity assumptions on the normal direction.

Plenary Lectures

• Stéphane Jaffard — LAMA, Université Paris-Est Créteil; CNRS
  Wavelet Methods in Multifractal Analysis: Multifractal analysis quantifies the fluctuations of regularities in functions through the estimation of their “multifractal spectrum” which encapsulates the fractional dimensions of the singularity sets of various “strength”. Wavelet techniques supply robust tools in order to perform multifractal analysis which became a standard signal/image processing method for classification or model selection. We will describe recent advances in several topics, such as the joint multifractal analysis of several functions, the challenges posed by the lack of regularity of the wavelets used, and the multifractal properties of stochastic processes generated using wavelet series with random coefficients.
• Malabika Pramanik — University of British Columbia
  Analytic and Geometric Perspectives on Sparse Sets: Sparse sets arise naturally in harmonic analysis and geometric measure theory, where their size alone often does not determine their 
  analytic behavior. This lecture will explore how geometric structure and distribution across scales influence problems involving Fourier decay, maximal operators, and geometric configurations. Through examples and selected results, we will discuss complementary ways of understanding 
  sparsity and the connections between them.
• Zihui Zhao — Johns Hopkins University
  Singularity Models for the One-Phase Free Boundary Problems: Free boundary problems are problems of solving an elliptic or parabolic PDE, but on a domain that is determined by the solution; they arise naturally in the study of water waves, the propagation of flames, etc. In this talk, we will focus on the regularity of the free boundaries for the one-phase free boundary problem, firstly studied by Alt and Caffarelli. In a joint work with Firester and Tsiamis, we study a rich family of singularity models for this problem using isoparametric hypersurfaces of the sphere. Together with Engelstein and Restrepo, we study the behavior of solutions near some well-known singularity models; and in some special cases the solutions are fully classified. Our results bear analogy with Allard–Almgren and Simon–Solomon in minimal surface theory.
• Svitlana Mayboroda — ETH Zürich; University of Minnesota
  Building Turbulence and Understanding Geometry of Flames: This course will examine the behavior of tracers (solutions to a parabolic PDE with a turbulent drift) and related geometric problems. 

Week 2: August 9–13, 2027

Minicourses 

• Pablo Shmerkin — University of British Columbia
  Projection Theory and Additive Combinatorics: This mini-course will survey some recent developments on projections of fractal-type sets and their discretized variants, emphasizing the role of additive combinatorics in some of the proofs. No previous background on additive combinatorics or fractal geometry will be required.
• Xavier Tolsa — ICREA; Universitat Autònoma de Barcelona
  Harmonic Measure and Rectifiability: Harmonic measure plays a fundamental role in the solution of the Dirichlet problem for the Laplace equation. Understanding its metric and geometric properties has been the subject of intensive research for many years. In this minicourse we will review some recent developments in this area, focusing on the connections between harmonic measure, quantitative rectifiability, and harmonic analysis, particularly through the use of Riesz transforms.
• Francesco Maggi — University of Texas at Austin
  Sets of Finite Perimeter and Capillarity Theory: This course will examine how the theory of sets of finite perimeter can be used in the study of capillarity theory and illustrate these ideas by looking at three types of physical system: bubbles, foams, and soap films.

Distinguished Plenary Lecture

François Ledrappier — CNRS / Sorbonne Université; University of Notre Dame
A Survey of Dimension, Entropy and Lyapunov Exponents: We consider stationary evolutions that are described by a smooth transformation and an invariant probability measure. Probabilistic global complexity is reflected in the Kolmogorov entropy. Local instability of the trajectories is recorded by the Lyapunov exponents. Geometric complexity is described by the Hausdorff dimension properties of the measure. It turns out that these three quantities are closely related. We shall explain this relation in the basic examples, present the form it takes in general and indicate some open problems. An application of these ideas is the calculation (by Weixiao Shen) of the Hausdorff dimension of the graph of some Weierstrass-like functions.
 

Plenary Lectures

• Joshua Zahl — Chern Institute of Mathematics, Nankai University
  A Survey of the Kakeya and Restriction Conjectures (provisional): A Besicovitch set is a compact subset of ⁿ that contains a unit line segment pointing in every direction. The Kakeya ℝset conjecture asserts that every Besicovitch set in ⁿ must have Minkowski and Hausdorff dimension n. Stein's ℝFourier restriction conjecture is a question about what happens when the Fourier transform of a function f: ⁿ → ℝℂis restricted to a hypersurface S. This course examines these two questions, the connections between them, and applications of these problems to other areas of mathematics.
• Pablo Lessa — Universidad de la República, Montevideo
  Dimension of Stationary Measures and Limit Sets on Boundaries of Non-Compact Type Symmetric Spaces: This course examines results and questions arising from joint work with François Ledrappier on the stationary measures associated to random walks on certain Lie groups.
• Chun-Kit Lai — San Francisco State University
  Cantor–Moran Sets, Deterministic Salem Sets, and Spectral Measures: This course will study Fourier analysis of the generalization of the middle-third Cantor sets known as the Cantor–Moran sets, whose contraction ratios are reciprocals of different integers. Such sets are natural and have a wide range of Fourier analytic properties. In this talk, we will use this to construct a new class of deterministic Salem sets, and a large class of spectral measures admitting Fourier orthonormal bases. The talk is based on a joint work with Ruxi Shi and Yu-Hao Xie.