List of Talks
Giovanni Alberti
Title: The geometric Vanishing Mass Conjecture
Abstract:
G. Bouchitté formulated the “Vanishing Mass Conjecture” about twenty years ago [1], motivated by an optimization problem for elastic light
structures. Since then, the only progress has been obtained by J.F. Babadjian, F. Iurlano and F. Rindler in [2]. In this talk I will illustrate the meaning of this conjecture, placing the emphasis on its geometric nature (which is rather independent of its mechanical interpretation), and describe some partial results obtained in collaboration with Andrea Marchese (Trento) and Andrea Merlo (Bilbao).
References
[1] Guy Bouchitté: Optimization of light structures: The vanishing mass conjecture, in "Homogenization, 2001 (Naples)". Gakuto Internat. Ser. Math. Sci. Appl. 18, Gakkotosho, Tokyo, 2003, pp. 131-145.
[2] Jean-Francois Babadjian, Flaviana Iurlano, Filip Rindler: Shape optimization of light structures and the vanishing mass conjecture. Duke Math. J., Vol. 172, (2003), pp. 43-103.
Adolfo Arroyo-Rabasa
Title: The Geometric Structure and Fine Properties of PDE-Constrained Measures
Abstract: The classical theory of linear partial differential operators with constant coefficients, pioneered by L. Hörmander and many others, is a cornerstone of modern analysis. A central theme is the connection between the algebraic properties of the principal symbol polynomial and the behavior of solutions. For example, a one-to-one symbol guarantees the transfer of regularity properties. Recent developments extend this concept to measure-valued solutions, exploring their functional, geometric, and variational properties solely through symbol analysis. This talk delves into the history of these developments in the context of geometric measure theory. Our journey begins with a geometric rigidity theorem for A-free measures, generalizing the structure rigidity of De Philippis and Rindler. I will then explain how this result shares deep connections with symbolic conditions (ellipticity and cancellation) that govern the existence of traces, Korn-type inequalities, slicing properties, and other fundamental fine properties of BV^A functions. During the talk, I will also share with you a few conjectures and challenging open problems.
Paolo Bonicatto
Title: Representation of the total variation as a Gamma-limit of BMO-type seminorms
Abstract: In recent years there has been a significant interest in the relations between the gradient seminorm $|Df|$ of a function of bounded variation $f: \mathbb R^n \to \mathbb R$ and certain BMO-type seminorms defined in terms of the oscillation of the function $f$ over a collection of disjoint cubes in $\mathbb R^n$. We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the Gamma-limit, with respect to the L^1_loc topology, of such BMO-type seminorms is given by 1/4 times the total variation seminorm. Our method also yields an alternative proof of previously known lower bounds for the pointwise limit and conveys a compactness result in L^1_loc in terms of the boundedness of the BMO-type seminorms.
Based on joint works with A. Arroyo-Rabasa (Bonn) and G. Del Nin (MPI Leipzig).
Giacomo Canevari
Title: The Ginzburg-Landau functional on complex line bundles: Gamma-convergence and asymptotics for critical points.
Abstract: The Ginzburg-Landau functional was originally proposed as a model for superconductivity in Euclidean domains. However, invariance with respect to gauge transformations - which is one of the most prominent features of the model - suggests that the functional can be naturally defined in the setting of complex line bundles, where it can be regarded as an Abelian Yang-Mills-Higgs theory. In this talk, we shall consider the Ginzburg-Landau functional on a Hermitian line bundle over a closed Riemannian manifold, in the scaling inherited from superconductivity theory. We shall discuss the asymptotic behaviour, in the so-called "London limit", of minimisers and critical points whose energy grows at most logarithmically in the coupling parameter. The talk is based on a joint work with Federico Dipasquale (Università Federico II, Napoli) and Giandomenico Orlandi (Verona).
Camillo De Lellis
Title: Besicovitch's 1/2 problem and linear programming.
Abstract: In 1928 Besicovitch formulated the following conjecture. Let E be a Borel subset of the plane with finite H^1 measure. If the lower 1-dimensional density of E is strictly bigger than 1/2 at H^1-a.e. x in E, then E rectifiable. 1/2 cannot be lowered, while Besicovitch himself showed that the conclusion does hold if 1/2 is replaced by 3/4. His bound was improved by Preiss and Tiser in the nineties to an (algebraic) number which is approximately 0.735. In this talk I will report on further progress stemming from a joint work with Federico Glaudo, Annalisa Massaccesi, and Davide Vittone. Besides improving the bound of Preiss and Tiser to a substantially lower number, our work proposes a family of variational problems to progress further. While we can improve the bound of Preiss and Tiser with a pen-and-paper proof, in order to reach the tightest result we need the assistance of a computer, which is used to examine a very large, but finite, number of cases. The practical feasibility of these computations depend however on some theoretical understanding of the variational problems and one important ingredient is that many subroutines can be formulated as linear programming tasks.
Thierry De Pauw
Title: Radon-Nikodymification.
Abstract: Motivated by the question "What is SBV dual?" I will explain why, for a general measure space the canonical map from $L^\infty$ to the dual of $L^1$ may be neither injective nor surjective. I will review the situation for Hausdorff measures including undecidable statements related to surjectivity. Finally, I will explain how to associate with any measure space a smallest "bundle" of it for which the above map is an isometric isomorphism and I will give an explicit description of this "Radon-Nikodymification" in case of integral geometric measure. The latter applies to describing SBV dual.
Michael Goldman
Title: From local energy bounds to dimensional estimates in a reduced model for type-I superconductors.
Abstract: In the limit of vanishing but moderate external magnetic field, we derived a few years ago together with S. Conti, F. Otto and S. Serfaty a branched transport problem from the full Ginzburg-Landau model. In this regime, the irrigated measure is the Lebesgue measure and, at least in a simplified 2d setting, it is possible to prove that the minimizer is a self-similar branching tree. In the regime of even smaller magnetic fields, a similar limit problem is expected but this time the irrigation of the Lebesgue measure is not imposed as a hard constraint but rather as a penalization. While an explicit computation of the minimizers seems here out of reach, S. Conti, F. Otto and S. Serfaty recently conjectured that the irrigated measure should be of dimension 8/5. I will present some recent progress in this direction.
Toni Ikonen
Title: Metric Sobolev spaces: equivalence of definitions
Abstract: Abstract: In this talk, we overview four definitions of Sobolev spaces on metric measures spaces for p greater or equal to one: based on approximation by Lipschitz functions (Cheeger), integration by parts relative to derivations (Di Marino), and two generalizations of the ACL property (Shanmugalingam and Ambrosio-Gigli-Savaré, respectively). The approaches are known to be equivalent when p > 1 (Ambrosio-Gigli-Savaré) and we recently extended the equivalence to the case p = 1.
The approach when p = 1 requires new ideas since the gradient flow methods yield the BV theory instead (Ambrosio-Di Marino) and the 1-modulus is not a Choquet capacity (V. H. Exnerová, O. F.K. Kalenda, J. Malý, O. Martio).
Based on joint work with L. Ambrosio (Scuola Normale Superiore), D. Lučić and E. Pasqualetto (University of Jyväskylä).
Cole Jeznach
Title: Boundary singular set estimates for solutions to degenerate elliptic equations.
Abstract: Recent advances in quantitative unique continuation properties for solutions to uniformly elliptic, divergence form equations (with Lipschitz coefficients) has led to a good understanding of the vanishing order and size of singular and zero set of solutions. Such estimates also hold at the boundary, provided that the domain is sufficiently regular. In this talk, I will discuss joint work with M. Engelstein and Y. Sire where we investigate the boundary behavior of solutions to a class of elliptic equations in the higher co-dimension setting, whose coefficients are neither uniformly elliptic, nor uniformly Lipschitz. Despite these challenges, we are still able to show analogous estimates on the singular set of such solutions near the boundary.
Linhan Li
Title: Uniform rectifiability, smooth distance, and Green function.
Abstract: Uniformly rectifiable sets are often viewed as good sets for singular integrals, potential theory, and geometric measure theory. Roughly speaking, those sets have big overlaps with Lipschitz images at every scale. In this talk, I will survey several equivalent characterizations of uniform rectifiability of an Ahlfors regular set and highlight a characterization in terms of oscillation of the Green function for a class of elliptic operators on the complement of the set. In contrast to some earlier results in analysis and PDE that are only valid for sets of co-dimension 1, this Green function characterization holds also in the higher co-dimensional setting.
This is based on joint works with Joseph Feneuil and Svitlana Mayboroda (DOI: 10.1007/s00208-023-02715-6, DOI: 10.1016/j.aim.2023.109220).
Francesco Maggi
Title: Plateau’s laws for soap films, the Allen–Cahn equation, and a hierarchy of Plateau-type problems.
Abstract: The incompatibility between Plateau’s laws and stable solutions to the Allen–Cahn equation is resolved by the formulation and analysis of a new model for soap films as small volume regions with diffused interfaces. As a result, Plateau-type singularities are approximated by stable solutions to free boundary problems for modified Allen–Cahn equations. Underlying our approach is the study of a hierarchy of Plateau problems that showcases the newly introduced diffused interface model at the top, a soap film capillarity model with sharp interfaces and bulk spanning at the intermediate level, and the classical Plateau model at the bottom. Central to our analysis is a measure-theoretic revision of the topological notion of homotopic spanning that has been behind much recent progress on the classical Plateau problem. This is joint work with Michael Novack (CMU Pittsburgh) and Daniel Restrepo (Johns Hopkins University).
Mihalis Mourgoglou
Title: Varopoulos extensions and Applications to Boundary Value Problems in rough domains.
Abstract: In this talk I will talk about some recent advances in solvability of Boundary Value Problems for second order elliptic operators in rough domains. The main ingredient is the construction of Varopoulos' type extensions. Namely, those are extensions $u$ of boundary data $f \in L^p$ and $w$ of boundary data $g$ in the Haj\l asz-Sobolev space $W^{1,p}$ so that the Carleson functional of $\nabla u$ and $Lw$ are in $L^p$ and the non-tangential maximal function of $u$ and $\nabla w$ are in $L^p$ with norms bounded by the $L^p$ norm of $f$ and the $ W^{1,p}$ norm of $g$ respectively. My plan is to show how those extensions appear in a natural way and sketch their construction.
Giuseppe Savaré
Title: The superposition principle for measure-valued P_1-solutions to the continuity equation.
Abstract: We present an overview of results concerning time-dependent measure-valued solutions to the continuity equation in the case when the vector momentum measure has finite total variation but is not absolutely continuous with respect to the evolving measures. Under a suitable minimality condition, it is possible to represent the solutions by a superposition of BV curves characterised by a probability measure.
We will also show the connection with solutions of an augmented continuity equation and with a reparametrization technique, which extends a typical tool of rate-independent evolutions to the mean-field setting. (In collaboration with Stefano Almi and Riccarda Rossi).
Anna Skorobogatova
Title: Regularity for minimizers of an Allen-Cahn approximation for the Plateau problem with homotopic spanning.
Abstract: I will discuss the regularity of minimizers and their associated free boundary, for a variational problem arising from the diffuse interface/Allen-Cahn approximation of the set-theoretic Plateau problem recently introduced by Maggi-Novack-Restrepo. Here, a homotopic spanning constraint, first considered by Harrison-Pugh, forces the surfaces (and also the corresponding interface for the diffuse approximation), to remain attached to the given wires frame. The presence of the spanning condition allows for minimizers to exhibit codimension 1 singularities such as triple junctions and tetrahedral singularities, in stark contrast to the work of Tonegawa-Wickramasekera which shows that any stable minimal hypersurface arising as a limit of interfaces for stable critical points of classical Allen-Cahn. This is a joint work with Mike Novack and Daniel Restrepo.
Luca Spolaor
Title: Dimension of the singular set of 2-valued stationary graphs.
Abstract: As a consequence of the celebrated Allard’s epsilon regularity theorem, it is well known that the singular set of an integral stationary n-varifold is meager. However, all known examples suggests that the Hausdorff dimension of such singular set should be (n-1). In this talk I will present a recent result, joint with J. Hirsch (Leipzig), where we show that if the stationary varifold is a 2-valued Lipschitz n-graph, then indeed its singular set is of dimension (n-1). I will spend ample time introducing the problem and explaining what are the main difficulties.
Salvatore Stuvard
Title: On the notion of dynamical (in)stability for minimal surfaces
Abstract: In this talk, I will prove that certain types of singularities of minimal surfaces (intended as stationary integral varifolds) are “dynamically unstable”, in the sense that they can be perturbed away with a non-trivial, area reducing mean curvature flow (in the sense of Brakke). I will compare this notion of dynamical stability with the classical notion of stability defined in terms of the spectrum of the second variation operator, and I will argue that the mean curvature flow may be used as a selection principle for “well-behaved” solutions to Plateau’s problem even possibly in the case of very wild boundaries. This is joint work with Yoshihiro Tonegawa (Tokyo Institute of Technology).
Emanuele Tasso
Title: Rectifiability of a class of integralgeometric measures and applications.
Abstract: In his textbook "Geometric Measure Theory" Federer proposed the following problem: is the restriction of the m-dimensional Integralgeometric measure to a finite set a m-rectifiable measure?
After a brief introduction to the problem, I will introduce a novel class of measures based upon the idea of slicing and having integralgeometric type of structure. The central result of this talk will follow, which is a sufficient condition for rectifiability in the previously introduced class. I will then focus on the solution to Federer's problem and its application to a part of Vitushkin's conjecture still not completely understood. Eventually, I will present a novel rectifiability criterion for Radon measures via slicing, reminiscent of White's rectifiable slices theorem for flat chains.
Xavier Tolsa
Title: Carleson's $\epsilon^2$-conjecture in higher dimensions, Faber-Krahn inequalities, and the Alt-Caffarelli-Friedman monotonicity formula.
Abstract: In a joint work with Ian Fleschler and Michele Villa, we have extended the $\epsilon^2$-conjecture of Carleson about the characterization of tangent points for Jordan domains in the plane to the higher dimensional setting. The solution of this problem has led us to explore the connections with the Alt-Caffarelli-Friedman monotonicity formula and with the so-called Faber-Krahn inequality. This inequality asserts that among the domains $\Omega\subset \mathbb R^n$ with a fixed volume, the ball minimizes the first Dirichlet eigenvalue of the Laplacian.
Besides reporting on the extension of the $\epsilon^2$-conjecture to higher dimensions and the above connections, I will explain a new quantification of the Faber-Krahn inequality in terms of capacitary functions and Hausdorff contents which turns out to be an important ingredient for our extension of the $\epsilon^2$-conjecture.
Yoshihiro Tonegawa
Title: Existence of curvature flow with forcing in a critical Sobolev space.
Abstract: Given a curve with finite length and a time-dependent vector field in a parabolically critical Sobolev space, we prove a global-in-time existence of evolving curves whose velocity is given by the sum of the curvature and the given vector field in a suitable weak sense of measure. This is a joint-work with Yuning Liu (NYU Shanghai).
Bozhidar Velichkov
Title: Regular and singular one-phase free boundaries
Abstract: The focus of this talk is on the local structure of the one-phase Bernoulli free boundaries, in particular, on the ones obtained as local minimizers of the Alt-Caffarelli functional with weight Q.
We will give a brief overview of the classical regularity theory for minimizers in the case of H\"older continuous weights Q. Finally, we will discuss the case of discontinuous Q and we will present some recent results obtained in collaboration with Lorenzo Ferreri (Scuola Normale Superiore).
Benedikt Wirth
Title: Patterns emerging in shape optimization of elastic structures.
Abstract: So-called compliance minimization seeks the optimal geometry of an elastic material to withstand a fixed given mechanical load and, at the same time, to consume as little material as possible. In general this problem is ill-posed: Minimizers may not exist, and microstructure may form along minimizing sequences. However, for some externally applied loads and some loading geometries minimizers actually do exist, sometimes even without microstructure. Indeed, the problem features an interesting dichotomy: If the load has eigenstresses of opposite sign, minimizers do not exist, and the only optimal microstructure consists of a laminate. If both eigenstresses are of the same sign, there is suddenly a plethora of optimal microstructures, and even minimizers of fractal geometry are known. But is microstructure really necessary, and if yes, how does the microstructure of minimizers look like? In the talk we will discuss some answers to these problems. The work is joint with Peter Bella and Jonathan Fabiszisky.
Qinglan Xia
Title: Partial Plateau’s problems related to optimal transportation and H-mass.
Abstract: In this talk, we’ll describe the motivations, set-up, and results of some partial Plateau’s problems with various masses.
Robert Young
Title: Surfaces in Heisenberg groups and quantitative rectifiability.
Abstract: The $2n+1$--dimensional Heisenberg groups $H_n$ are some of the simplest examples of subriemannian manifolds, and the subriemannian structure has distinctive effects on surfaces in $H_n$. In this talk, we explore the geometry of surfaces in the Heisenberg group. We will describe some techniques for visualizing, constructing, and analyzing surfaces, and use these to explain how the geometry and analysis of surfaces in $H_n$ depends on $n$. Parts of this talk are joint with Naor and Chousionis-Li.
Zihui Zhao
Title: Boundary unique continuation of harmonic functions.
Abstract: Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular or critical set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.