The main topics of the semester:

1.  Geometric function and mapping theory in Euclidean spaces, Riemannian manifolds, Heisenberg and Carnot-Carathéodory groups and in more general metric measure spaces.

2.  Function spaces on metric measure spaces.

Senior leaders

Piotr Hajlasz (University of Pittsburgh, USA)

Stanislav Hencl (Charles University, Prague, Czechia)

Giona Veronelli (University of Bicocca, Milano, Italy).

Junior leaders

Daniel Campbell (Charles University, Prague, Czechia)

Sylvester Eriksson-Bique (University of Jyväskylä, Finland)

Aleksis Koski (Aalto University, Finland)

Damaris Meier (ETH, Switzerland)

David Tewodrose (Vrije Universiteit Brussel)

Scott Zimmerman (Ohio State University, OH, USA)

Key events of the semester

 8-12.VI.2026, IM PAN Warsaw: Workshop in geometric analysis organized as a series of lectures. 

The lectures will be aimed largely at young researchers. Moreover, the session of contributed talks will be held. 

Preliminary list of speakers: 

    Daniel Campbell (Charles University, Prague, Czechia)

    Sylvester Eriksson-Bique (University of Jyväskylä, Finland)

    Piotr Hajłasz (University of Pittsburgh, USA), series of lectures

    Stanislav Hencl (Charles University, Prague, Czechia), series of lectures

    Aleksis Koski (Aalto University, Finland)

    Damaris Meier (ETH, Switzerland)

    David Tewodrose (Vrije Universiteit Brussel)

    Daniele Valtorta (University of Bicocca, Milano, Italy), series of lectures

    Giona Veronelli (University of Bicocca, Milano, Italy), series of lectures

    Scott Zimmerman (Ohio State University, OH, USA).

21-27.VI.2026, Będlewo Conference Center: 

A conference in geometric analysis to honor Piotr Hajłasz’ 60th birthday 

See the conference webpage for more details:

www.hajlasz60.com 

List of plenary speakers:

Luigi Ambrosio, Giovanni Alberti, Zoltan Balogh, Jana Bjorn, Stanislav Hencl, Paweł Goldstein, Zofia Grochulska, Agnieszka Kałamajska, Pekka Koskela, Enrico Le Donne, Juan Manfredi, Jani Onninen, Pekka Pankka, Armin Schikorra, Nageswari Shanmugalingam, Paweł Strzelecki, Xiaodan Zhou.

Simons Seminar in Geometric Analysis

Apart from the workshop and the conference, the semester plans to invite several guests who could cooperate scientifically with the semester participants, as well as deliver a talk at the Simons semester seminar.  

The tentative list of the semester guests (the exact dates of their visits during the semester will be announced later):

• Riku Anttila,

• Fabrice Baudoin,

• Miguel Garcia Bravo,

• Milica Caković, 

• Baptiste Devyver,

• Manuel Dias,

• Manisha Garg,

• Chris Gartland,

• Asma Hassannezhad

• Sahojar Khan,

• Fedya Manin,

• Pekka Pankka

• Andrea Pinamonti,

• Carmelo Puliatti 

• Kai Rajala,

• Leah Schaetzler,

• Nageswari Shanmugalingam,

• Elefterios Soultanis,

• Janne Taipalus,

• Daniele Valtorta,

• Noa Vikman,

• Veikko Vuolasto,

• Robert Young,

• Zheng Zhu. 

Seminar schedule (tentative): 

                         Mondays, 10-14, room 403

                         Tuesdays and Thursdays 14-17, room 321

                         Reading group:  3, 5, 17, 19 of June  (Wednesdays and Fridays)  14-16, room 1

                         Additional meetings:

                            Friday 5.VI, 14-17, room 321

                            Friday 3.VII, 10-12.30, room 321

Week 1-5.VI

1. Fabrice Baudoin

2. Baptiste Devyver (Université Grenoble Alpes)

Title: Best constants in Hardy-Poincaré-Sobolev inequalities for some weighted  hyperbolic spaces
Abstract: In this talk, we will consider optimal Sobolev inequalities with their best constant on some weighted Riemannian manifolds with positive bottom of the spectrum of the Laplacian. The question we will investigate is whether it is possible to control the energy, shifted by the first eigenvalue, by an Lp norm, keeping a ''Euclidean'' optimal constant. The work of Benguria-Frank-Loss and Mancini-Sandeep (2007) shows that in the case of usual hyperbolic space of dimension 3, such an optimal Sobolev inequality indeed holds. On the other hand, such a phenomenon do not occur for dimension at least 4. Results on this problem will be presented in the context of some ''weighted'' hyperbolic spaces which appear naturally in the study of Caffarelli-Kohn-Nirenberg inequalities. It is a joint work with Pierre-Damien Thizy and Louis Dupaigne from Université Lyon 1.

3. Manuel Dias (Vrije Universiteit Brussel)

Title: Symmetrized AMV Laplacians on unions of stitched Riemannian manifolds.

Abstract: The symmetrized Asymptotic Mean Value Laplacian \tilde{\Delta} is obtained as limit of approximating operators  \tilde{\Delta}_r, and is an extension of the Laplace Beltrami operator on maniolds. In the work we introduce the notion of stitched Riemannian manifolds, and study the spectral convergence of of these operators, obtaining limit eigenfunctions that have a Kirchhoff condition in the intersection.

4. Andrea Pinamonti (University of Trento)

Title: A discrete-time overdetermined problem for the heat equation

Abstract: In this talk, I consider a parabolic counterpart of Serrin's overdetermined problem, in which the overdetermined condition (constant flux condition) is imposed only on a discrete infinite set of time values. I show that, under suitable regularity assumptions on the domain, such a discrete-time overdetermined problem admits a solution if and only if the domain is a ball.  I study both the case in which the constant flux condition is imposed on the boundary and the case in which the constant flux condition is imposed on an interior surface. The results are based on a joint paper with Lorenzo Cavallina.

5. Carmelo Puliatti (Universitat Autònoma de Barcelona) 

Title:  Free boundary problems for elliptic measure and rectifiability

Abstract: Absolute continuity of harmonic measure with respect to surface measure provides significant geometric information about the boundary of a domain. It was shown by Azzam, Hofmann, Martell, Mayboroda, Mourgoglou, Tolsa, and Volberg (GAFA, 2016) that this condition suffices to imply rectifiability.

A two-phase analogue of this result, due to Azzam, Mourgoglou, Tolsa, and Volberg, asserts that, given two disjoint domains in R^{n+1}, mutual absolute continuity of the corresponding harmonic measures on a common portion of their boundaries implies absolute continuity with respect to surface measure and rectifiability on that subset.

In this talk, I will discuss how these results generalize to the elliptic measure associated with the PDE

  div (A(\cdot)\nabla u) = 0,

where A(\cdot) is an (n+1)\times(n+1) uniformly elliptic matrix whose L^1 mean oscillation satisfies suitable Dini-type conditions.

The proof uses techniques from harmonic analysis and, in particular, quantitative rectifiability criteria formulated in terms of the gradient of the single layer potential. This is joint work with A. Merlo and M. Mourgoglou.

Colloquium talk, Wednesday 3.VI: Stanislav Hencl

Title: Models of Nonlinear Elasticity: Questions and progress

Abstract: In this talks we study mappings that could serve as defomations in models on Nonlinear Elasticity. We have a body Omega in R^n and we  have a  mapping f: Omega \to R^n that describes the deformation of this body.

 Natural questions considered are when the mapping is continuous (the material does not break during the deformation), when is it invertible (no "interpenetration of matter" happens) or when does it map sets of zero volume to sets of zero volume (no material is created or lost during our deformation). In particular we will study when the deformation does not change orientation (you cannot map your right hand to your left   hand) or when can we approximate injective mappings by piecewise linear (or smooth) injective mappings.  

             Series of two lectures by Scott Zimmerman and Aleksis Koski

Scott Zimmerman

Title: Extending curves in metric spaces

Abstract: Both of my talks will address the following problem. Suppose $K$ is a subset of the real line and $X$ is a metric space, and fix $f:K \to X$ such that some property $P(f)$ is true (e.g. $f$ is Lipschitz, bi-Lipschitz, metric differentiable). What conditions on $X$ and $f$ are necessary and/or sufficient to guarantee the existence of a curve $F:\mathbb{R} \to X$ such that $F|_K = f$ and $P(F)$ is true? Moreover, can we relate $P(F)$ and $P(f)$ quantitatively?

In my first talk, I will discuss a project that has developed over the past decade with Gareth Speight. Suppose $X$ is $\mathbb{R}^n$ with a sub-Riemannian structure. In other words, assume that there is a distribution of vector fields on $\mathbb{R}^n$ with dimension less than $n$ (i.e., a subbundle of the tangent bundle $T\mathbb{R}^n$). Intuitively, this distribution controls the directions through which a curve may travel in space. Indeed, a curve in $X$ is horizontal if it lies tangent to this distribution almost everywhere. We may define a metric on $X$ by infimizing over lengths of horizontal curves just as one infimizes over lengths of curves in a Riemannian manifold to make it a metric space. The property $P$ that we will consider here is ``f is smooth''. For real valued maps, this extension problem was solved by Whitney in 1934, and I will discuss my work with Speight on this problem over the past decade for maps into Carnot groups. I will also present applications of this result and related rectifiability properties of sets in these metric spaces.

My second talk will discuss joint work with Vyron Vellis and Jacob Honeycutt. In this talk, $X$ will be a metric space with a particular geometric structure, and the property $P$ will be ``f is bi-Lipschitz''. This problem is challenging even when $X$ is Euclidean. I will show that, if $X$ is $Q$-Ahlfors regular and supports a $p$-Poincare inequality (for certain ranges of $Q$ and $p$), then bi-Lipschitz extensions of curves must always exist. Moreover, I will show that the assumptions on $X$ and the ranges of $p$ and $Q$ are (almost) strict.

Aleksis Koski 

Title: Sobolev homeomorphic extensions 

 Week 15-19.VI

• Miguel Garcia Bravo 

Title: The Morse-Sard theorem and its generalizations

Abstract:  In 1935, H. Whitney constructed  a function $f: \mathbb R^2 \to \mathbb R$ of class $C^1$ whose set of critical points has positive measure. At that time, the conditions under which such examples could exist, where the image of the critical set has positive measure, were poorly understood. In the next years, a more complete vision of the problem was obtained by Morse and Sard.

The classical Morse--Sard theorem, from 1942, states that any function $f:\mathbb{R}^n \to \mathbb{R}^m$ of class $C^k$, with $k \geq \max\{1, n - m + 1\}$, satisfies that its set of critical values has Lebesgue measure zero in $\mathbb R^n$. Whitney’s example shows that this regularity requirement is sharp with respect to the scale of $C^j$ spaces. Since then, many mathematicians have explored different generalizations of the Morse-Sard theorem, with deep applications  in various areas of mathematics, including Analysis, Differential Topology, and Partial Differential Equations.

In this talk, we will make an overview of several generalizations of the Morse-Sard theorem to classes of functions (in what follows $k= n-m+1$ and $n\geq m$):

1. The space $C^{k-1,1}(\mathbb R^n;\mathbb R^m)$: Bates (1993).

2. Sobolev spaces $W^{k,p}(\mathbb R^n;\mathbb R^m)$: De Pascale (2001).

3. Functions of bounded variation $BV_n(\mathbb R^n;\mathbb R)$: Bourgain, Korobkov and Kristensen (2015).

4. Sobolev-Lorenz spaces $W^{k,n/k}_{l}(\mathbb R^n,\mathbb R^m)$:  Korobkov, Kristensen and Haj{\l}asz (2017).

5. Real-valued functions with Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$:  Azagra, Ferrera, G\'omez-Gil  (2017).

6. Approximate differentiable functions of order $k$: Azagra and Garc\'ia-Bravo (2020).

• Chris Gartland  (UNC Charlotte) 

Title: Wasserstein Metrics over Spaces with Property A

Abstract: The Wasserstein metric over a metric space X is an optimal-transport based distance on the set of probability measures on X. Metric spaces for which the optimal transport problem is "easiest" to solve are trees,  in the sense that the Wasserstein metric on trees isometrically embeds into L1. Property A is a coarse invariant of metric spaces introduced by Yu as an approach to solving the coarse Baum-Connes conjecture. We prove a new characterization of bounded degree graphs X with Property A as precisely those that are coarsely equivalent to another space Y whose Wasserstein metric admits a biLipschitz embedding into L1. Applications to group actions on Banach spaces will be discussed. Based on joint work with Tianyi Zheng and Ignacio Vergara.

• Fedya Manin 

   TBA

• Kai Rajala

Title: Rigid circle domains with non-removable boundaries 

Abstract: We give a negative answer to the rigidity conjecture of He and Schramm by constructing a rigid circle domain on the Riemann sphere with a conformally non-removable boundary. Rigidity here means that every conformal map onto another circle domain is a Möbius transformation, while non-removability means the existence of a global homeomorphism that is conformal off the boundary but not globally. The proof relies on Wu’s theorem on the non-removability of products of Cantor sets and on Ntalampekos’ metric characterization of quasiconformal maps, building on foundational work of Gehring, Heinonen, Kallunki, Koskela, and others. 

• Elefterios Soultanis

Title: Differentiability spaces which are subsets of Banach spaces and rectifiability

Abstract: There is a conceptually simple relationship between embeddability, differentiability, and rectifiability: If X bi-Lipschitz embeds into a Banach space V, and all Lipschitz maps from X to V are differentiable, then X itself must be rectifiable. This principle is present already in the seminal work of Cheeger, and is the basis for several non-embedding results in the literature. In this talk I discuss obtaining rectifiability of spaces embeddable in certain Banach spaces B without knowing the differentiability of all B-valued Lipschitz maps. Joint work with Ivan Caamano and Sylvester Eriksson-Bique.

• Robert Young (Courant Institute of Mathematical Sciences,New York University)

Title: Areas of curves in R^n and the coarea formula in the Heisenberg group

Abstract: In this talk, we will explore the problem of extending the coarea formula to the subriemannian Heisenberg group $H$.

The coarea formula relates the Jacobian of a Lipschitz map $f$ to the measure of the fibers $f^{-1}(w)$. This is a classical result for maps from $\mathbb{R}^m\to \mathbb{R}^n$, where fibers can be approximated by regular surfaces. This is impossible, however, for fibers of maps from $H$ to $\mathbb{R}^2$, which have topological dimension 1 but can have Hausdorff dimension 2 or more. Furthermore, the measure of a fiber depends on the area of the projection of the fiber to $\mathbb{R}^{2}$, and since the projection can have Hausdorff dimension 2, this area can be infinite or undefined.

We will present recent work with Gioacchino Antonelli that solves this problem by giving a geometric condition for the area of a Hölder curve to be well-defined and showing that while the individual fibers of a map can behave wildly, fibers satisfy strong geometric bounds on average.

Colloquium talk, Wednesday 17.VI: Piotr Hajłasz 

Title: Geometry and Topology of Mappings with Derivatives of Small Rank

Abstract: The main theme of this talk is the study of mappings—primarily continuously differentiable and Lipschitz—that are critical everywhere, in the sense that the rank of their derivative is small at every point. Such mappings arise naturally in a variety of contexts across analysis, geometry, and topology. I will show how ideas from different areas combine to address fundamental questions about these mappings, with emphasis on problems related to approximation, homotopy, contact structures, Heisenberg groups, and analysis on metric spaces.

Series of two lectures by David Tewodrose and  Sylvester Eriksson-Bique

 David Tewodrose

Title: Spectral properties of symmetrized AMV Laplacians

Abstract: The symmetrized Asymptotic Mean Value (AMV) Laplacians extend the classical Laplace operator from Rn to metric measure spaces through suitable averaging integrals. On complete Riemannian manifolds, they provide an alternative approximation of the Laplace—Beltrami operator. In this series of two talks, I will first introduce these operators and illustrate their behavior through several key examples. I will then present recent results obtained with Manuel Dias (VUB) on the spectral properties of these operators on compact doubling metric measure spaces, including manifolds with boundary and unions of intersecting Riemannian manifolds. If time permits, I will also briefly discuss ongoing work on progress on sub-Riemannian and Finsler manifolds.

Sylvester Eriksson-Bique

Title: Decomposability bundles via modulus

Abstract: Alberti and Marchese introduced the powerful notion of decomposability bundles - a type of tangent space to measures in Euclidean space. This notion plays a key role in the resolution of Cheeger's conjecture by Marchese, de Philippis and Rindler, and the study of closability of differential operators by Marchese, Bate and Alberti. It is also the main tool in resolving the Energy Image Density conjecture in my joint work with Murugan. This concept is a bit tedious to work with, partly due to the definition involving Alberti representations. In this talk, I will try to give you a more quick introduction to this notion by avoiding all talk about Alberti representations. This gives a new approach to this concept and some slick proofs that illustrate key ideas.

In the first lecture, we will define infinity Modulus and use it to define decomposability bundles. We show that the decomposability bundle exists, and iimmediately relate the decomposability bundle to upper gradients of C^1 functions. Time permitting, we mention also pointwise tangent fields and relate decomposability bundles to them.

There are two main concepts the decomposability bundle is useful for: studying differentiability and finding approximations. In the second lecture, we study differentiability by giving a Rademacher's theorem for Lipschitz functions in the direction of the tangent space. The proof here is quite slick when employing the graph of a Lipschitz function. Then, depending on time, we study approximations by describing how an upper gradient can be turned into a sequence of approximating Lipschitz functions via a compactness argument.

The arguments here are partially unpublished, and I will provide lecture notes to describe the proofs. These arose through joint work  and discussions with Elefterios Soultanis, Iván Caamaño Aldemunde, Guy C. David, Raanan Schul, David Bate and Mathav Murugan.

          Week 29.VI-3.VII

• Riku Anttila

• Ivan Caamano Aldemunde (3. VII at 11.15 am)

• Asma Hassannezhad

• Dmitrios Ntalampekos (online talk on 3. VII at 10 am )

• Leah Schaetzler

• Nageswari Shanmugalingam

• Pekka Pankka

• Noa Vikman

Colloquium talk, Wednesday 1.VII: Giona Veronelli

Title: Isoperimetric inequalities via the ABP method on manifolds

Abstract: The ABP method is a technique introduced around 1960 to prove a celebrated maximum principle. In this talk, we will survey the application of the ABP method to prove geometric results, with a special focus on (sharp) isoperimetric inequalities—first in Euclidean space, and subsequently on minimal hypersurfaces and non-negatively curved Riemannian manifolds. In particular, we will outline the remarkable contributions in this direction by X. Cabré and S. Brendle, as well as, time permitting, more recent results concerning Sobolev inequalities on manifolds.

Series of two lectures by Damaris Meier and Daniel Campbell

Damaris Meier

Title: Uniformization of metric surfaces

Abstract: The classical uniformization theorem states that every simply connected Riemann surface is conformally equivalent to the unit disc, the complex plane, or the Riemann sphere. Over the past few decades, there has been an increasing interest in extending this fundamental result to non-smooth metric surfaces. In this series of two lectures talk we will introduce the theory of metric surfaces, study different generalizations of conformal mappings, and survey key developments toward the uniformization of metric surfaces with locally finite Hausdorff 2-measure. 

Information for the semester participants.

 If any scientific publications result from participation in the Simons Semester or from collaborations initiated during the semester, the Acknowledgments section should include the text below (the wording has been approved by the Ministry – please do not introduce any changes other than inserting the correct name of the work or event). Thank you!

“This [work/event] was partially supported by the Simons Foundation grant (award no. SFI-MPS-T-Institutes-00010825) and from State Treasury funds as part of a task commissioned by the Minister of Science and Higher Education under the project “Organization of the Simons Semesters at the Banach Center - New Energies in 2026-2028” (agreement no. MNiSW/2025/DAP/491).”