Program

       The series of lectures: 

Piotr Hajłasz (University of Pittsburgh) 

Advanced Calculus in Metric Spaces

Advanced calculus as we usually know it is formulated in Euclidean spaces and, more generally, on smooth manifolds. In the lectures, I will discuss a perhaps surprising fact: advanced calculus can be developed in arbitrary metric spaces.

More precisely, I will explain how notions such as differentiability, the implicit function theorem, the change of variables formula, and the area and coarea formulas extend to Lipschitz mappings with values in general metric spaces. This theory was initiated by Bernd Kirchheim in 1994 and has since become a cornerstone of contemporary geometric measure theory.

The lectures will be elementary and accessible to anyone with a basic background in advanced calculus, metric spaces, and measure theory.


 Stanislav Hencl (Charles University, Prague) 

Homeomorphisms in the Sobolev space 

In Nonlinear Elasticity we model our deformations as mappings in Sobolev spaces from a subset of Rn into Rn that need to be continuous as the material does not break during the deformation and invertible because of the "Noninterpenetration of the matter". Thus we often work with the class of homeomorphism in the Sobolev space. 

The main aim of this series of talks is to study some natural properties of homeomorphisms in the Sobolev space: When can we get differentiability a.e. of our mappings? When do we know that zero measure sets are mapped to zero measure sets and what are the possible counterexamples? Under which conditions do we know that the mapping does not change orientation, i.e. the Jacobian does not change sign? Is it possible to approximate Sobolev homeomorphisms by diffeomorphisms in the Sobolev norm?


 Daniele Valtorta (University of Bicocca, Milano) 


  Quantitative Stratification: Introduction And Recent Developments

In this course, we will study the basic ideas of quantiative stratification, applied to the study of singularities of different objects in GMT. Introduced by Cheeger and Naber, this technique is based on quite simple ideas and is flexible enough to be applied to different contexts. In the course, we will focus on rectifiability and quantitative estimates for singularities of harmonic maps, and use this as an example to model other situations. We will also mention the energy identity in different contexts.

See the pdf file below for the prerequisities and the literature.


Giona Veronelli (University of Bicocca, Milano) 


L^p Liouville theorems on incomplete spaces

In this series of three talks, we will discuss Liouville-type theorems for both harmonic and positive subharmonic L^p-functions. One of the motivations is their connection to the essential self-adjointness of the Laplacian and Schrödinger operators, although our treatment will mostly remain at a PDE level. The general setting we work in consists of not necessarily complete Riemannian manifolds; a special focus will be placed on subsets of Euclidean space, where most of these phenomena already appear. 


    Lectures:

Daniel Campbell (Charles University, Prague)


Everything interesting happens in a Cantor set


We explore the utility of Cantor sets in constructing counterexamples to various conjectures regarding mapping behavior. We focus on how these fractal sets serve as foundational structures in analysis, allowing for the construction of maps that show the sharpness of hypothesis in claims and the existence of maps with interesting pathalogical behaviour.



    Sylvester Eriksson-Bique (University of Jyväskylä)


Energy Image Density property

 On many fractals, such as the Sierpinski carpet, one can define a family of energies and Sobolev spaces that are biLipsxhitz or quasisymmetric invariants. These are not built by using upper gradients, but instead by using rescaled energies. These invariants are inexplicit and difficult to work with in general. In this talk, I describe how these energies come about, how they relate to conformal dimension, and how one can get a handle on them via a ``change of variables''. The key tool of decomposability bundles of Alberti and Marchese are mentioned. This final step answers an old problem on Dirichlet forms posed by Bouleau and Hirsch. This is joint work with Mathav Murugan and Riku Anttila.


       Aleksis Koski (Aalto University)

Establishing energy minimizers via free lagrangians


Proving that a map from an appropriate class of functions is the minimizer of a given energy functional is simply a matter of proving one inequality, but the intricate methods of establishing such inequalities are an artform by itself. When minimizing a functional defined among a class of homeomorphisms between two given domains without any fixed boundary values, the general strategy is to take advantage of integral expressions called free lagrangians. I will explain this method and discuss some recent new ways of applying it to solve a wider array of minimization problems for mappings between Euclidean annuli. This talk is based on joint work with my student Akseli Jussinmäki.


    Damaris Meier (ETH)


Monotone Sobolev extensions in metric surfaces

Every rectifiable Jordan curve in the plane admits a monotone Lipschitz extension over the disc. This fails in general for metric surface targets due to the possibility of 2‑unrectifiable regions. In the Sobolev setting, the situation changes. We show that any monotone W^{1,2} map from S^{1} to the boundary of a Jordan domain in a metric surface with locally finite Hausdorff 2-measure admits a monotone W^{1,2} extension to the disc. Our proof combines energy minimization methods with a collar construction. This is based on joint work with Noa Vikman and Stefan Wenger. 


    David Tewodrose (Vrije Universiteit Brussel)


A precompactness theorem for manifolds with boundary


I will present joint work with Marie Bormann (University of Bonn), in which we prove a precompactness theorem for Riemannian manifolds with boundary via a suitable time-change transformation. In contrast with previous results, we do not assume any pointwise bound on the second fundamental form of the boundary, but instead a Dynkin-type bound on a measure involving both the Ricci curvature in the interior and the second fundamental form of the boundary. I will introduce this Dynkin bound, explain how it enables the required time-change transformation, and discuss some perspectives.


    Scott Zimmerman (Ohio State University, OH)


Carnot groups as metric spaces

 It is easier to navigate through a room while walking as opposed to riding a bicycle. This is because the directions you can move while riding a bike are more limited than while walking. How does one model this scenario mathematically? In this talk, I will define and explore a class of metric spaces called Carnot groups. These spaces are studied by those working in these kinds of controlled dynamics, but Carnot groups are of independent interest as mathematical objects in their own right. While Carnot groups are heavily studied by algebraic geometers, no prior algebraic or differential geometric knowledge will be necessary beyond calculus on Euclidean spaces. My talk will approach Carnot groups as Euclidean spaces on which a new metric is defined.



  Contributed talks: 


Ivan Caamano Aldemunde (IM PAN)


Derivative Harnack inequalities for quasiregular maps


For quasiconformal mappings, the classical Koebe distortion theorem holds true allowing to easily infer a Harnack inequality for the derivative of the map. However, Koebe's result fails for quasiregular mappings, where we lack injectivity, and thus a different approach is needed for developing analogous Harnack estimates. We will discuss the role of the multiplicity and how a weak quantitative Harnack estimate holds under mild assumptions.



Milica Caković (University of Jyväskylä) 


On the pmG-convergence of pointed metric measure spaces 


In this talk, we recall the notion of pointed measured Gromov (pmG) convergence of pointed metric measure spaces, introduced by Gigli--Mondino--Savaré, which is suitable for non-compact settings. Using this notion of convergence, we study some properties of tangents of pointed metric measure spaces, in particular Preiss's phenomenon -- namely, that a tangent of a tangent is itself a tangent. Furthermore, we present an Arzelà--Ascoli-type compactness theorem for pmG-convergence of metric measure spaces.

This talk is based on joint work with Elefterios Soultanis.


Manisha Garg (University of Illinois, Urbana-Champaing)


Geodesic trees and bi-Lipschitz embeddings

 A natural question in the bi-Lipschitz geometry of trees is whether a large class of geodesic trees admits a single universal element, that is, a fixed tree into which every member of the class embeds in a bi-Lipschitz manner. Furthermore, Kinnenberg asked if every quasiconformal tree of Assouad dimension < 2 admits a bi-Lipschitz embedding into R^2. Chrontsios-Garitsis, Ioannidis, and Vellis prove that the class of quasiconformal trees with uniform separation can quasisymmetrically embed into a universal element T, and moreover, this T admits a biLipschitz embedding into R^2. In this talk, we will show that such a universal element cannot be found if one replaces quasisymmetric maps with bi-Lipschitz maps, not even in the case of the geodesic trees. This answers a question of the aforementioned authors.

 This is joint work with Sylvester Eriksson-Bique.


 Zofia Grochulska  (University of Jyväskylä)

Challenges of interpolating Sobolev spaces on domains


A normed space A is an interpolation space between X and Y if, roughly, it is contained in between these spaces and linear operators continuous on X and Y are also continuous on A. I will explain why it is more difficult to interpolate Sobolev spaces on arbitrary domains than Sobolev spaces on the whole Euclidean space. I will discuss some new (yet unpublished) results on this topic, obtained with Pekka Koskela and Riddhi Mishra (both from University of Jyväskylä).



Alexandru Pirvuceanu 


Hypercontractivity of the Heat Flow: Sharpness and Rigidities on RCD(0, N) Spaces


In this talk, we establish sharp hypercontractivity bounds for the heat flow $({\sf H}_t)_{t\geq 0}$ on RCD(0, N) metric measure spaces. The best constant in these estimates involves the asymptotic volume ratio, and its optimality is obtained via the sharp L^2-logarithmic Sobolev inequality on RCD(0, N) spaces and a blow-down rescaling argument. The equality case in these estimates is completely characterized, and applications of our results include an extension of Li's rigidity result, almost rigidities, as well as topological rigidities of non-collapsed RCD(0, N) spaces. This is joint work with Shouhei Honda and Alexandru Kristaly.


Marcin Walicki (Warsaw Polytechnic)


Harmonic mappings on Grushin planes


Harmonic functions (or more generally, maps) on the Grushin spaces are usually studied as minimisers of the Dirichlet energy given by the squared L^2 norm of the horizontal gradient w.r.t. the Lebesgue measure. We study analogous energy, with the difference being the underlying measure is a natural, weighted Lebesgue measure, which is p-Muckenhoupt for p>1. The talk will focus on defining appropriate Sobolev space, exploring the relevant minimisation problem, as well as, stating new theorems including the H^{2,2} regularity, Bochner formula and the equivalence between the Dirichlet and the Korevaar-Schoen energy.


The talk is based on joint work with Tomasz Adamowicz and Ben Warhurst: "Harmonic mappings on Grushin planes".