Talks (titles and abstracts)
Guido De Philippis: On the converse of Rademacher Theorem and the rigidity of measures in Lipschitz differentiability spaces
Rademacher Theorem asserts that every Lipschitz function is differentiable almost everywhere with respect to the Lebesgue measure. Results in this spirit have been established by Pansu in Carnot groups and by Cheeger in abstract metric measure spaces. A natural question is then the rigidity of those measures for which every Lipschitz function is differentiable almost everywhere. Aim of the talk is to discuss this issue in increasing generality. In particular I will present a proof of the fact that Rademacher Theorem can hold for a measure if and only if it is absolutely continuous with respect to the Lebesgue measure. I will also present some consequences of this fact concerning the structure of measures in Lipschitz differentiability spaces. Finally I will present some ongoing work concerning the converse of Pansu Theorem.
Alessio Figalli: The De Giorgi conjecture for the half-Laplacian in dimension 4
The famous the Giorgi conjecture for the Allen-Cahn equation states that global monotone solutions are 1D if the dimension is less than 9. This conjecture is motivated by classical results about the structure of global minimal surfaces. The analogue of this conjecture in half-spaces can be reduced to study the problem in the whole space for the Allen-Cahn equation with the half-Laplacian. In this talk I will present a recent result with Joaquim Serra where we prove the validity of the De Giorgi conjecture for stable solutions in dimension 3.
As a corollary, we obtain the corresponding result for monotone solutions in dimension 4.
Peter Haïssinsky: Quasi-isometric rigidity of fundamental groups of compact 3–manifolds
The talk will focus on the following results: a finitely generated group quasi-isometric to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group contains a finite index subgroup isomorphic to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group.
Tuomas Hytönen: New thoughts around bounded mean oscillation
I will discuss two topics related to the John-Nirenberg BMO space of functions of bounded mean oscillation:
(1) A space \(JN_p\) that is related to \(L^p\) in a similar way as BMO is related to \(L^\infty\). This space was already introduced by John and Nirenberg in the same paper as BMO, but it has been considerably less studied. Until recently, it remained open whether this space is just another name for \(L^p\), and answering this turned out surprisingly tricky. (joint with Dafni, Korte and Yue)
(2) Two-weight inequalities for commutators of BMO functions and singular integrals. A classical results of Coifman, Rochberg and Weiss characterises BMO as the space of functions whose commutators with singular integrals give bounded operators on \(L^p\). A version of this result even holds for boundedness between two weighted \(L^p\) spaces with different weights. While new techniques are needed, it turns out that the case of higher commutators can still be deduced from the first order case by an adaptation of an old Cauchy path integral trick of Coifman et al.
Enrico Le Donne: Conformal maps between boundaries of pseudoconvex domains
We will discuss a new point of view for a proof of a theorem by Fefferman.
Inspired by the proof of Mostow's rigidity, we shall show that any biholomorphism between strongly pseudoconvex domains induces a smooth map between the boundaries.
The result is now obtained in two steps.
1) From Balogh and Bonk we know that such domains equipped with the Bergman metrics are Gromov hyperbolic and hence the biholomorphism extends on the boundary as a quasi-conformal map. After refining Balogh-Bonk estimates, we show that the boundary extensions are 1-quasi-conformal with respect to the subRiemannian metric induced by the Levi form.
It will be crucial to use Bonk-Schramm hyperbolic fillings and prove the conformal invariance of various visual distances.
2) We will study 1-quasi-conformal maps between subRiemannian manifolds. We prove a morphism property for harmonic functions and show that such maps are smooth in the case of contact manifolds.
This is a joint work with L.Capogna, G.Citti, A.Ottazzi.
Vladimir Markovic: Caratheodory's Metrics on Teichmuller Spaces
One of the central results in classical Teichmuller theory is the theorem of Royden which says that the Teichmuller and Kobayashi metrics agree on any Teichmuller space. On the other hand, it has recently been shown that the Teichmuller and Caratheodory's metrics disagree on Teichmuller spaces of closed surfaces. I shall discuss this result, its implications in Teichmuller theory, and the role of Teichmuller dynamics in this study.
Giuseppe Mingione: Nonlinear potential and Calderón-Zygmund theories
The classical, linear potential and Calderón-Zygmund theories deal with integral and pointwise estimates for solutions to linear elliptic and parabolic equations. On the other hand, in the last years, we have witnessed a number of results and methods, that cast a rather satisfying nonlinear analog of the linear theories. In my talk I'll give a number of results in this direction. Most of the results are in collaboration with Tuomo Kuusi.
Pekka Pankka: Deformation of cubical Alexander maps
In his 1985 paper on the sharpness of the Picard theorem for quasiregular mappings, Rickman introduced new piece-wise linear methods to construct quasiregular mappings in the \(3\)-space. One of the methods is deformation of \(2\)-dimensional Alexander maps, that is, a branched cover extension method for piece-wise linear branched covers from a (planar) surface to the \(2\)-sphere having three critical values.
In this talk I will discuss Rickman's deformation theory for higher dimensional cubical Alexander maps and its applications: a Hopf theorem for shellable Alexander maps between spheres, a Berstein--Edmonds type extension theorem, and an extension of Rickman's large local index theorem for quasiregular maps to dimensions \(n>3\).
This is joint work with Jang-Mei Wu.
Pierre Pansu: Large scale conformal maps
Benjamini and Schramm's work on incidence graphs of sphere packings suggests a notion of conformal map between metric spaces which is natural under coarse embeddings. We show that such maps cannot exist between nilpotent or hyperbolic groups unless certain numerical inequalities hold.
Kai Rajala: Uniformization of metric surfaces
Non-smooth versions of the classical uniformization theorem have recently appeared, motivated by several types of questions. We discuss the uniformization of metric spaces homeomorphic to the standard plane and with locally finite Hausdorff area, and give some remarks on the case of fractal spaces.
Tatiana Toro: Almost minimizers with free boundary
In recent work with G. David and M. Engelstein, we study almost minimizer for functionals which yield a free boundary, as in the work of Alt-Caffarelli and Alt-Caffarelli-Friedman. The almost minimizing property can be understood as the defining characteristic of a minimizer in a problem which explicitly takes noise into account. In this talk we will discuss regularity results for these almost minimizers and as well as the structure of the corresponding free boundary. Almost monotonicity formulas and the properties of harmonic functions on non-tangentially accessible domains play a central role in this work.
Anna Wienhard: tba
David Bate: Rectifiability via arbitrarily small perturbations
The Besicovitch-Federer projection theorem is a fundamental result that characterises n-rectifiable subsets of Euclidean space with finite n-dimensional Hausdorff measure. However, it was recently shown that it significantly fails in any infinite dimensional Banach space and so it is very natural to ask of other characterisations in this setting. Very recently, Pugh gave a "localised" version of the projection theorem for Ahlfors n-regular subsets of Euclidean space. It involves reducing the Hausdorff measure of a purely n-unrectifiable set to an arbitrarily small value under a Lipschitz mapping arbitrarily close (in the uniform metric) to the identity.
This talk will present significant generalisations of this result to various classes of Banach spaces. Unlike the classical projection theorem, these are not dependent on the linear structure and so, via suitable embeddings, can be formulated purely in the language of metric spaces. As a corollary, we obtain some simple consequences of the classical projection theorem in the metric setting.
Maria Colombo: The structure of transport equations and the Vlasov-Poisson system
The transport equation describes the evolution of a distribution of particles moving along the flow of a prescribed smooth vector field. An accurate description of its solutions, even when the smoothness assumption is dropped, is motivated by several applications, among which the study of kinetic equations such as the Vlasov-Poisson system.
Given a vector field in \(R^d\), the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth; this, in turn, translates in existence and uniqueness results for the transport equation. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. Their theory relies on a growth assumption which prevents the trajectories from blowing up in finite time. In this seminar we give an overview of the topic and we introduce a new notion of maximal flow for non-smooth vector fields which allows for finite-time blow up of the trajectories. We show structure results for the transport equation under only local assumptions on the vector field and we apply them to the Vlasov-Poisson system, where we describe the solutions as transported by a suitable flow.
Estibalitz Durand Cartagena: Self-contracted curves: recent developments and applications
Self-contracted curves were introduced to provide a unified framework for the study of convex and quasiconvex gradient dynamical systems in the euclidean setting. However, the definition of self-contractedness is purely metric and does not require any regularity neither on the space, nor on the curve. In this expository talk we will recall motivations, main ideas and results concerning this notion, presenting the current state of the art for future developments.
Daniel Faraco: Mixing Solutions for the Muskat Problem
The Muskat Problem describes the evolution of the interphase between two fluids evolving through a porous media. The theory is very different depending on whether the heavier or lighter fluid are in top of each other. The case of the heavier fluid on top is ill posed in Sobolev Spaces. In spite of that there is a number of results and numerical experiments showing the presence of a so called mixing zone, related to the phenomena of fingering. In this talk I will describe how a combination of convex integration and contour dynamics yields the existence of weak solutions for an arbitrary initial interphase. This is a joint work with Angel Castro y Diego Cordoba.
Philip Isett: A Proof of Onsager’s Conjecture for the Incompressible Euler Equations
In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration using the “Mikado flows” introduced by Daneri-Székelyhidi with a new “gluing approximation” technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.
Alexandru Kristaly: Intrinsic Jacobian determinant inequalities on corank 1 Carnot groups
We establish a weighted pointwise Jacobian determinant inequality on corank 1 Carnot groups related to optimal mass transportation akin to the work of Cordero-Erausquin, McCann and Schmuckenschläger. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space including the presence of abnormal geodesics. Our inequality interpolates in some sense between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established. The talk is based on a joint work with Z. M. Balogh and K. Sipos.
Tuomo Kuusi: The additive structure of elliptic homogenization
One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this talk, I will address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the first-order correctors which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument: using the regularity theory recently developed for stochastic homogenization, we accelerate the weak convergence of the energy density, flux and gradient of the solutions as we pass to larger and larger length scales, until it saturates at the CLT scaling. This is a joint work with S. Armstrong and J.-C. Mourrat.
Kazumasa Kuwada: Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces
The W-entropy is introduced by G. Perelman in his seminal work on Ricci flow, and this notion is brought to (time-homogeneous) Riemannian manifolds by L. Ni. He proves that, on Riemannian manifolds with nonnegative Ricci curvature, the W-entropy is monotone in time along the heat flow. Moreover, this monotonicity enjoys a rigidity in the sense that vanishing time derivative can happens only in a very special case. Indeed, the space must be Euclidean.
In this talk, we show the corresponding results on “Riemannian” metric measure spaces with nonnegative Ricci curvature and upper dimension bound (more precisely, RCD (0,N) spaces). Because of the lack of usual differentiable structure, we have to develop new approaches. As a by-product of our new approach, some parts of our result are new even on Riemannian manifolds. In addition, we can find more spaces in the class of RCD (0,N) spaces in the rigidity.
This talk is based on a joint work with X.-D. Li (Chinese Academy of Science).
Cyril Lecuire: Deformations of hyperbolic 3-manifolds, convergence and local topology
The deformation space AH(M) can be viewed as the analogous of Teichmüller space for a compact hyperbolic 3-manifold M. But, in contrast to Teichmüller space, the topology of AH(M) is quite complicated. I will explain some of the topological features of AH(M) and explain how they are related to the behaviour of the ends invariants of M. This a joint work with R. Canary, J. Brock, K. Bromberg and Y. Minsky.
Andrea Mondino: Some smooth applications of non-smooth Ricci curvature lower bounds
The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the 80s and was pushed by Cheeger and Colding in the 90s who investigated the fine structure of possibly non-smooth limit spaces. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago. Via such an approach one can give a precise definition of what it means for a non-smooth space to have Ricci curvature bounded below. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seem to be new even for smooth Riemannian manifolds.
Roberto Monti: Excess and tangents of sub-Riemannian geodesics
We present some recent results on the regularity problem of sub-Riemannian length minimizing curves. This is a joint work with A. Pigati and D. Vittone. After introducing the notion of excess for a horizontal curve, we show that at any point of a length minimizing curve excess is infinitesimal at some sequence of scales. This implies the existence of a linear tangent. We also discuss other results related to excess.
Shin-ichi Ohta: Spectral gap and rigidity under positive Ricci curvature
We consider a rigidity problem for the spectral gap of Laplacian on an RCD\((K,\infty)\)-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive \(K\). For a weighted Riemannian manifold, Cheng-Zhou (2013) showed that the sharp spectral gap is achieved only when a 1-dimensional Gaussian space is split off. Generalizing this to RCD-spaces is not straightforward due to the lack of smooth structure as well as the absence of upper dimension bound. In order to overcome these difficulties, we lift an eigenfunction to the Wasserstein space and employ the regular Lagrangian flow recently developed by Ambrosio-Trevisan. This is a joint work with N. Gigli (SISSA), C. Ketterer (Freiburg) and K. Kuwada (Tohoku).
Mircea Petrache: Sharp large-N asymptotics for an N-marginal optimal transport problem inspired from physics
Density Functional Theory is a simplification of the Schroedinger equations introduced by Hohenberg and Kohn, in order to tackle the curse of dimensionality in quantum mechanical molecular computations. This minimization problem is intimately related to a symmetric Optimal Transport problem with N marginals representing the densities of N electrons, and interacting by Coulomb power-law pair interactions. I will show how this Optimal Transport problem fits in a larger class in which sharp asymptotics for the minimizers can be obtained. Several intriguing open questions arise naturally from this discussion.
Istvan Prause: Parametrization of frozen boundaries
Random surfaces arising from the dimer model exhibit limit shape formation. There is a sharp spatial separation between frozen facets and disordered liquid regions. The arctic circle of domino tilings of the Aztec diamond is a celebrated example of this. We study the geometry and parametrization of such frozen boundaries using Kenyon-Okounkov theory and the intrinsic complex structure on the liquid region. This complex structure is described by a quasilinear Beltrami equation which degenerates at the frozen boundary. Thus the equation can be used to detect the boundary and its properties.
The talk is based on ongoing joint work with K. Astala, E. Duse and X. Zhong.
Sévérine Rigot: Besicovitch covering property on graded groups and applications to measure differentiation
In this talk we will give a characterization of graded groups admitting homogeneous distances for which the Besicovitch covering property (BCP) holds. In particular it follows from this characterization that a stratified group admits homogeneous distances for which BCP holds if and only if it has step 1 or 2. We will illustrate this result with explicit examples of homogeneous distances satisfying BCP on the Heisenberg group. We will also discuss applications to measure differentiation which is one of the motivations for considering such covering properties.
Andrea Schioppa: Calculus on Metric Spaces: Beyond the Poincaré Inequality
We discuss a framework introduced by J. Cheeger (1999) to differentiate Lipschitz maps defined on metric measure spaces which admit Poincaré inequalities, and discuss (the first) examples on which it is still possible to differentiate despite the infinitesimal geometry being incompatible with the Poincaré inequality.
Gareth Speight: Maximal directional derivatives and universal differentiability sets in Carnot groups
Rademacher's theorem asserts that Lipschitz functions from \(R^n\) to \(R^m\) are differentiable almost everywhere. Such a theorem may not be sharp: if \(n>1\) then there exists a Lebesgue null set N in \(R^n\) containing a point of differentiability for every Lipschitz mapping \(f:R^n->R\). Such sets are called universal differentiability sets and their construction relies on the fact that existence of an (almost) maximal directional derivative implies differentiability. We will see that maximality of directional derivatives implies differentiability in all Carnot groups where the Carnot-Caratheodory distance is suitably differentiable, which include all step 2 Carnot groups (in particular the Heisenberg group). Further, one may construct a measure zero universal differentiability set in any step 2 Carnot group. Finally, we will observe that in the Engel group, a Carnot group of step 3, things can go badly wrong... Based on joint work with Andrea Pinamonti and Enrico Le Donne.
Daniele Valtorta: Structure of the singular sets of Q-valued harmonic functions
In this talk we present the new regularity results proved for the singular sets of Q-valued harmonic functions in collaboration with Camillo de Lellis, Andrea Marchese and Emanuele Spadaro (see arXiv: 1612.01813).
We prove that the singular set of Q-valued Dirichlet minimizing harmonic functions is m-2 rectifiable with uniform volume bounds on the set of Q-points. This estimate is based on a Reifenberg-type theorem obtained in collaboration with Aaron Naber (see arXiv:1504.02043), and the technique is very versatile in nature and can be used to tackle many different problems in GMT.
Jean Van Schaftingen: Uniform boundedness principle for Sobolev maps between manifolds
The classical uniform boundedness principle of Banach and Steinhaus for linear operators between Banach spaces is a theorem that transforms a bound depending on the point in the domain into a global uniform bound. The nonlinear character of Sobolev spaces between manifolds makes it unapplicable in these spaces. By relying on the structure of the domain of Sobolev maps, we have obtained a quite general uniform boundedness principle for energies of Sobolev maps, which allows us to recover known estimates and counterexamples for the problems of weak-bounded approximation, of extension of traces, of lifting and of superposition. The result covers fractional and first order Sobolev spaces.