# Title and Abstract

**May 23**^{rd}, Monday

^{rd}, Monday

**9:30-10:30 am Nageswari Shanmugalingam, University of Cincinnati**

Title: Besov function spaces on compact doubling metric measure spaces as traces of Sobolev functions.

Abstract: Besov classes arise naturally in the context of quasisymmetric mappings between compact doubling metric measure spaces as well as in the context of (non-local) fractional Laplacian operators. In this talk we will describe the construction of hyperbolic filling of a compact doubling metric measure space and a lift of the measure to the hyperbolic filling so that Besov spaces on the compact space arise naturally as traces of Sobolev functions on the uniformization of the hyperbolic filling. The talk is based on work done in collaboration with Anders Bjorn and Jana Bjorn.

**11:00-12:00 Hiroaki Aikawa, Chubu University**

Title: Intrinsic Ultracontractivity for domains in negatively curved manifolds

Abstract: Let *M *be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. We give a sufficient condition for an open set *D *⊂ *M *to be intrinsically ultracontractive. That condition is formulated in terms of capacitary width. The crucial observation is that the bottom of the spectrum of the Dirichlet Laplacian acting in *L*^{2}(*D*) is estimated by the capacitary width for *D*, in case it is sufficiently small. The technical key ingredients are the volume doubling property, the Poincar´e inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale. Joint work with Michiel van den Berg and Jun Masamune.

**2:30-3:30 pm Shouhei Honda, Tohoku University**

Title: Topological stability theorem from nonsmooth to smooth spaces with Ricci curvature bounded below

Abstract: In this talk, inspired by a recent work of Bing Wang and Xinrui Zhao, we prove that for a fixed n-dimensional closed Riemannian manifold (Mn,g), if an RCD(K,n) space (X,d,m) is Gromov-Hausdorff close to Mn, then there exists a homeomorphism F from X to Mn such that F is Lipschitz continuous and F−1 is H¨older continuous, where the Lipschitz constant of F, the H¨older exponent and the H¨older constant of F−1 can be chosen arbitrary close to 1. Moreover if X is smooth, then such a map F can be chosen as a diffeomorphism. This is a joint work with Yuanlin Peng (Tohoku University).

**4:00-5:00 pm Luigi Ambrosio****, SNS Pisa**

Title: A Gamma-convergence result and an application to the derivation of the MongeAmpere gravitational model

Abstract: I will present a general Gamma-convergence result for action functionals involving a kinetic term plus a term depending on the gradient of a possibly nonsmooth convex function. The motivation for this problem came from the formal derivation of the Monge-Ampere gravitational model as a limit of particle system. This is joint work with A.Baradat and Y.Brenier. In the end I will also illustrate a generalization to metric spaces, proved in a joint work with C.Brena.

**5:10-6:10 pm Andrea Mondino****, University of Oxford**

Title: Minimal boundaries in non-smooth spaces with Ricci Curvature bounded below

Abstract: The goal of the talk is to report on recent joint work with Daniele Semola. Motivated by a question of Gromov to establish a “synthetic regularity theory” for minimal surfaces in non-smooth ambient spaces, we address the question in the setting of nonsmooth spaces satisfying Ricci curvature lower bounds in a synthetic sense via optimal transport.

**May 24**^{th}, Tuesday

^{th}, Tuesday

**9:30-10:30 am Piotr Hajlasz, University of Pittsburgh**

Title: Approximation of mappings with derivatives of low rank

Abstract: My talk is based on two recent joint papers with Pawe l Goldstein. While the problem that I will discuss is purely Euclidean and analytic, the proofs of theorems and constructions of counterexamples are based on methods of analysis on metric spaces.

Jacek Galeski in 2017, in the context of his research in geometric measure theory, formulated the following conjecture:

**Conjecture. **Let 1 ≤ *m < n *be integers and let Ω ⊂R* ^{n }*be open. If

*f*∈

*C*

^{1}(Ω

*,*R

*) satisfies rank*

^{n}*Df*≤

*m*everywhere in Ω, then

*f*can be uniformly approximated by smooth mappings

*g*∈

*C*

^{∞}(Ω

*,*R

*) such that rank*

^{n}*Dg*≤

*m*everywhere in Ω.

One can also modify the conjecture and ask about a local approximation: smooth approximation in a neighborhood of any point. These are very natural problems with possible applications to PDEs and Calculus of Variations. However, the problems are difficult, because standard approximation techniques like the one based on convolution do not preserve the rank of the derivative. It is a highly nonlinear constraint, difficult to deal with.

In 2018 Goldstein and Haj\l{}asz obtained infinitely many counterexamples to this conjecture. Here is one:

**Example. **There is *f *∈ *C*^{1}(R^{5}*,*R^{5}) with rank*Df *≤ 3 that cannot be locally and uniformly approximated by mappings *g *∈ *C*^{2}(R^{5}*,*R^{5}) satisfying rank*Dg *≤ 3.

This example is a special case of a much more general result and the construction heavily depends on algebraic topology including the homotopy groups of spheres and the Freudenthal suspension theorem.

More recently Goldstein and Hajlasz proved the conjecture in the positive in the case when m = 1. The proof is based this time on methods of analysis on metric spaces and in particular on factorization of a mapping through metric trees.

The method of factorization through metric trees used in the proof of the conjecture when m = 1 is very different and completely unrelated to the methods of algebraic topology used in the construction of counterexamples. However, quite surprisingly, both techniques have originally been used by Wenger and Young as tools for study of Lipschitz homotopy groups of the Heisenberg group, a problem that seems completely unrelated to problems discussed in this talk.

**11:00-12:00 Panu Lahti****, Chinese Academy of Sciences**

Title: Quasiconformal, Sobolev, and BV mappings in metric measure spaces

Abstract: Consider Ahlfors regular spaces X,Y and a homeomorphism f between them. In the theory of quasiconformal mappings, it is known that if the distortion number *h _{f }*is finite outside a set of sigma-finite

*Q*− 1-dimensional Hausdorff measure, and has suitable integrability, then

*f*is a Sobolev mapping. An analogous result holds for the local Lipschitz number lip f. I will introduce generalized versions of

*h*and lip f, and show how these can be used to give new sufficient conditions for f to be a Sobolev or BV mapping.

_{f }**2:30-3:30 pm Riikka Korte****, Aalto University**

Title: Fractional Laplacian on metric spaces and transforming unbounded domains into bounded domains.

Abstract: In [1] we used spectral theory to construct the fractional Laplacian corresponding to a Cheeger differential structure on complete doubling metric measure spaces supporting a Poincar´e inequality. The methods depended strongly on the availability of the Poincar´e inequality and it is not adaptable to fractional powers of the nonlinear pLaplacian operator. There the metric space Z is naturally seen as the boundary of the unbounded domain *X *× (0*,*∞). In contrast, in [2], we consider fractional Laplacians on the boundary of a bounded domain that supports a Poincar´e inequality. In order to show that these two approaches lead to the same notion of nonlocal Laplacian, we show that we can conformally transform *Z *×(0*,*∞) into a bounded doubling metric measure space supporting a Poincar´e inequality so that harmonic functions in *Z *× (0*,*∞) are also harmonic in this modified space and that *Z *is isometric to the boundary of this modified space. We will also discuss transforming a more general unbounded set with compact boundary into a bounded space.

This talk is related to three projects:

[1] Eriksson-Bique, Giovannardi, Korte, Shanmugalingam, Speight: Regularity of solutions to the fractional Cheeger-Laplacian on Domains in metric spaces of bounded geometry, JDE ’22. arXiv:2012.09450.

[2] Capogna, Kline, Korte, Shanmugalingam, Snipes: Neumann problems for p-harmonic functions, and induced nonlocal operators in metric measure spaces. arXiv:2204.00751.

[3] Ongoing project with Gibara and Shanmugalingam.

**4:00-5:00 pm Pekka Koskela, University of Jyväskylä **

Title: Existence and uniqueness of limits at infinity

Abstract: We consider an unbounded doubling metric space and an associated homogeneous Newtonian Sobolev space. Even when the space supports a Poincare inequality, it may happen that there exists a Sobolev function for which there does not exist even a single curve that “tends to infinity” in such a way that our Sobolev function would have a limit along this curve. We give criteria for the positive outcome, discuss the uniqueness of the potential limits and give initial results on the sets of “directions” along which one obtains limits. The results are based on collaboration with Khanh Nguyen.

**5:10-6:10 pm Estibalitz Durand-Cartagena, Universidad Nacional de Educación a Distancia**

Title: The least doubling constant on graphs

Abstract: In this expository talk, we study the least doubling constant among all possible doubling measures defined on a graph. We present some results that establish connections between spectral graph theory and doubling constants of certain measures on graphs.

The talk is based on a joint work with Pedro Tradacete (ICMAT) and Javier Soria (UCM).

**May 25**^{th}, Wednesday

^{th}, Wednesday

**9:30-10:30 am Jun Kigami, Kyoto University**

Title: Conductive homogeneity of compact metric spaces and construction of *p*-energy

Abstract: In the ordinary theory of Sobolev spaces on domains of R* ^{n}*, the

*p*-energy is defined as the integral of |∇

*f*|

*. In this talk, we try to construct*

^{p}*p*-energy on compact metric spaces as a scaling limit of discrete

*p*-energies on a series of graphs approximating the original space. In conclusion, we propose a notion called conductive homogeneity under which one can construct a reasonable

*p*-energy if

*p*is greater than the Ahlfors regular conformal dimension of the space. In particular, if

*p*= 2, then we construct a local regular Dirichlet form and show that the heat kernel associated with the Dirichlet form satisfies upper and lower sub-Gaussian type heat kernel estimates. As examples of conductively homogeneous spaces, we present a new class of square-based self-similar sets and rationally ramified Sierpinski crosses, where no diffusion was constructed before.

**11:00-12:00 Mario Bonk, UCLA**

Title: The visual sphere of an expanding Thurston map

Abstract: A Thurston map is a branched covering map on a topological 2-sphere for which the forward orbit of each critical point under iteration is finite. Each such map gives rise to a fractal geometry on its underlying 2-sphere. It is an open problem to determine the conformal dimension of this sphere if the Thurston map is obstructed and not realized as a rational map. In my talk I will report on some recent progress.

**2:30-3:30 pm Shin-ichi Ohta****, Osaka University**

Title: Some quantitative stability estimates for the Bakry-Ledoux isoperimetric inequality

Abstract: We discuss some quantitative stability estimates for the Bakry-Ledoux isoperimetric inequality of weighted Riemannian manifolds. Our methods and statements are both based on the needle decomposition (a.k.a. localization). Assuming that equality nearly holds in the Bakry-Ledoux isoperimetric inequality, we establish an *L*^{1}-estimate exhibiting that the push-forward of the reference measure by the guiding function is close to the Gaussian measure. This is joint work with Cong Hung MAI (Osaka University).

**4:00-5:00 pm Antoni Kijowski, OIST**

Title: H-convex hull and h-quasiconvex envelope in the Heisenberg group

Abstract: In my presentation I discuss h-convex sets and h-quasiconvex functions in the Heisenberg group H. A set *E *⊂ H is called h-convex (or horizontally convex, weakly h-convex), if the horizontal segment connecting any two points in *E *lies in *E*. A function *u *in a h-convex domain *E *is called h-quasiconvex, if all sublevel sets of *u *are h-convex. The main motivations are: (1) construction of the h-convex hull of a given set *E *⊂ H (the smallest h-convex set containing *E*) and (2) construction of the h-quasiconvex envelope *Q*(*u*) of a given function *u *(the largest h-quasiconvex function bounded above by *u*). I show how to reach (2) by employing in the Heisenberg group a scheme developed by Barron–Goebel–Jensen in the Euclidean space. The main tool is based on a viscosity characterization of h-quasiconvexity. This enables to construct *Q*(*u*) as a limit of nonincerasing iterative sequence *u _{n }*of viscosity solutions to nonlocal Hamilton–Jacobi equations with

*u*

_{0 }=

*u*. Next, I use the fact that every sublevel set of

*Q*(

*u*) is h-convex hull of the related sublevel set of

*u*to solve problem (1). I illustrate complexity of solving problems (1) and (2) with examples.

The talk is based on a joint work with Xiaodan Zhou (Analysis on Metric Spaces Unit, OIST) and Qing Liu (Geometric Partial Differential Equations Unit, OIST).

**May 26**^{th}, Thursday

^{th}, Thursday

**9:30-10:30 am Luca Capogna, Smith College**

Title: The Neumann problem and the fractional laplacian in measure metric spaces

Abstract: In this talk we will report on some recent joint work with Josh Kline, Riikka Korte, Marie Snipes and Nages Shanmugalingam, concerning the Neumann problem in PI spaces, and a new definition of fractional p-Laplacians in arbitrary doubling measure metric space. We extend some earlier work by Lukas Maly and Nages Shanmugalingam, proving well posedness in appropriate Lebesgue classes for the Neumann problem for pLaplacians, and then leverage these results to prove existence, stability and regularity for the corresponding non homogeneous non-local PDE.

**11:00-12:00 Yoshito Ishiki, RIKEN**

Title: Branching and confluent geodesics in the Gromov–Hausdorff space

Abstract: In talk, we construct branching and confluent geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, satisfying that (1) they pass through or avoid sets of all compact spaces satisfying some of the doubling property, the uniform disconnectedness, and the uniform perfectness, or (2) they pass through the sets of all infinite-dimensional spaces, or (3) they pass through set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov–Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.

**2:30-3:30 pm. Sylvester Eriksson-Bique, University of Oulu**

Title: New Approximations of Sobolev and Lipschitz functions

Abstract: I will present recent work on a few new approximation methods for Sobolev functions. These methods, or their variations, show the following: a) Lipschitz functions are dense in energy even for p=1 b) there is a sharp lower bound for duality c) there is a differential structure associated to p=1 d) continuous functions are dense in the Sobolev space for any locally complete and separable space. In the talk, I will discuss such approximation methods. It is intriguing that they arise from fusing classical methods, relating to moduli and capacities, with discrete approximations. The results include joint work with Pietro Poggi-Corradini and Elefterios Soultanis.

**4:00-5:00 pm Karl-Theodor Sturm, University of Bonn**

Title: Distribution-valued Ricci Bounds for Metric Measure Spaces

Abstract: We will study metric measure spaces (*X,d,m*) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE_{1}(*κ,*∞) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary *ψ *∈ Lip* _{b}*(

*X*), and which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets

*Y*⊂

*X*.

In the latter case, the distribution-valued Ricci bound will be given by the signed measure *κ *= *km _{Y }*+

*ℓσ*where

_{∂Y }*k*denotes a variable synthetic lower bound for the Ricci curvature of

*X*and

*ℓ*denotes a lower bound for the “curvature of the boundary” of

*Y*, defined in purely metric terms.

**5:10-6:10 pm ** **Nicola Gigli, SISSA**

Title: Lipschitz continuity of harmonic maps from RCD to CAT(0) spaces

Abstract: In ‘classical’ geometric analysis a celebrated result by Eells-Sampson grants Lipschitz continuity of harmonic maps from manifolds with Ricci curvature bounded from below to simply connected manifolds with non-negative sectional curvature. All these concepts, namely lower Ricci bounds, upper sectional bounds and harmonicity, make sense in the setting of metric-measure geometry and is therefore natural to ask whether a regularity result like the one of Eells-Sampson hold in this more general setting. In this talk I will survey a series of recent papers that ultimately answer affirmatively to this question.

**May 27**^{th}, Friday

^{th}, Friday

**9:30-10:00 am Ryosuke Shimizu, Kyoto University**

Title: Construction of a canonical *p*-energy on the Sierpinski carpet

Abstract: It is known that the Brownian motion on the Sierpinski carpet has sub-Gaussian behavior, and so we can not develop the upper gradient based analysis on the Sierpinski carpet. In this talk, I will give a construction of a “canonical” *p*-energy on the Sierpinski carpet as a scaling limit of discrete *p*-energies on approximating graphs when *p *is strictly greater than the Ahlfors regular conformal dimension. This *p*-energy reflects the “geometry” of the carpet (symmetry and self-similarity) and satisfies inequalities of *L ^{p}*, e.g. Clarkson’s inequality. I also plan to describe associated energy measures and

*p*-energy form on the Sierpinski carpet.

**10:10-10:40 am Zhangkai Huang, Tohoku University**

Title: Isometric immersions of RCD(*K,N*) spaces via heat kernels

Abstract: For an RCD(*K,N*) space (X*,*d*,*m), one can use its heat kernel *ρ *to embed it into *L*^{2}(X*,*m) by a locally Lipschitz map Φ* _{t}*(

*x*) :=

*ρ*(

*x,*·

*,t*). In particular, an RCD(

*K,N*) space is said to be an isometric heat kernel immersion space, if its associated Φ

*is an isometric immersion multiplied by a constant depending on*

_{t }*t*for any

*t >*0. We prove that any compact isometric heat kernel immersion RCD(

*K,N*) space is isomorphic to an unweighted closed smooth Riemannian manifold. Besides, it is proved that M(

*K,n,D*) has finitely many members up to diffeomorphism, where M(

*K,n,D*) is the isometric class of

*n*-dimensional closed Riemnannian manifolds (M

*g), with a lower Ricci curvature bound*

^{n},*K*∈ R, an upper diameter bound

*D >*0, and an eigenmap realizing an isometric immersion into some Euclidean space. This improves Honda’s topological finiteness result for M(

*K,n,D*).

**11:00-11:30 am Kohei Sasaya, Kyoto University**

Title: Some inequalities between spectral dimensions and the Ahlfors regular conformal dimension of resistance metrics

Abstract: In this talk, we will consider the relation between the spectral dimension *d _{S }*of the heat kernel determined from a resistance form on a metric space (

*X,d*) and the Ahlfors regular conformal dimension dim

*of (X,d). Kigami(2020) showed that for the standard Dirichlet forms on the Sierpin´ski gasket or generalized Sierpin´ski carpets with Euclidean distance, either dim*

_{AR }*(*

_{AR}*X,d*) ≤

*d*2 or dim

_{S }<*(*

_{AR}*X,d*) ≥

*d*≥ 2 holds. We will show the followings: (a) There are examples of metric spaces and Dirichlet forms where

_{S }*d*dim

_{S }<*(*

_{AR}*X,d*)

*<*2, even when certain homogeneity and reflection symmetry are satisfied. (b) Let (

*X,d*) be a complete metric space with dim

*(*

_{AR}*X,d*)

*<*∞

*, µ*be a nontrivial locally finite Borel measure satisfying the volume doubling condition, and (E

*,*F) be a resistance form on

*X*with the associated resistance metric is quasisymmteric to

*d*. Then for the upper spectral dimension

*d*

^{¯}

*which is defined by modifying the way of the limit of the spectral dimension, the inequality dim*

_{S},*(*

_{AR}*X,d*) ≤

*d*

^{¯}

*2 holds for the associated heat kernel.*

_{S }<**11:40-12:10 Hiroshi Tsuji, Osaka University**

Title: Improved log-Sobolev and transportation-cost inequalities under log-concavity and log-convexity

Abstract: In this talk, we discuss an improvement of the Gaussian logarithmic Sobolev inequality and transportation-cost inequality under log-concavity and log-convexity. The similar results have been already shown by Eldan-Lehec-Shenfeld for the log-Sobolev inequality and Mikulincer for the transportation-cost inequality in terms of covariance matrices of probability measures. Moreover, we discuss the same improvements for the log-Sobolev inequality under log-subharmonicity. This talk is based on a joint work with Neal Bez (Saitama University) and Shohei Nakamura (Osaka University).