Analysis and Geometry of Fractals and Metric Spaces:
Recent Developments and Future Prospects
5-9 March 2023
Okinawa, Bankoku Shinryokan, Ocean Hall
Organisers: Naotaka Kajino (Kyoto University, Chair), David Croydon (Kyoto University),
Daisuke Shiraishi (Kyoto University)

Sunday 5th
March

Monday 6th
March
09:30–10:20

Tuesday 7th
March
09:30–10:20

Kleiner
10:00–14:00

10:40–11:30

Registration
12:00–14:00

10:40–11:30

Bonk
!

Lunch
14:00–14:50

11:40–12:30

15:10–16:00

Murugan
15:10–16:00

Meyer
16:20–17:10
Shanmugalingam

11:40–12:30

16:20–17:10

Aikawa09:10–10:00

09:20–10:10

MackaySasaya10:50–11:40

Hambly14:00–14:50

TanakaChen
10:20–10:45

Shimizu
11:10–12:00

Saksman
13:20–14:10

Saloff-Coste
15:10–16:00

Grigor’yan*Thursday 9th
March

10:10–10:35

Lapidus*

Tsukamoto
14:00–14:50

Barlow

Ohta

Wednesday 8th
March

Miermont
14:20–15:10

Tyson
16:20–17:10

Teplyaev

Kumagai
15:15–15:25

Closing∗ = online presentation.

Martin Barlow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forty years of analysis on fractals
Daniel Meyer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Quasisymmetric uniformization and quasivisual approximations
Nageswari Shanmugalingam . . . . . . . . . . . . . . . . . . . . . . Constructing boundaries of domains in metric spaces using prime ends
Bruce Kleiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigidity and flexibility of mappings between Carnot groups
Mario Bonk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Green functions in metric measure spaces
Masaki Tsukamoto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to mean dimension
Mathav Murugan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasisymmetry and the elliptic Harnack inequality
Alexander Grigor’yan* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Off-diagonal estimates of heat kernels for jump processes
Hiroaki Aikawa . . . . . Elliptic and parabolic boundary Harnack principles via Harnack inequality admitting exceptional set
Shin-ichi Ohta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On weighted Ricci curvature with negative dimension parameter
Michel Lapidus* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An introduction to the theory of complex dimensions
Ben Hambly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral asymptotics for some domains with random fractal boundaries
Laurent Saloff-Coste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limit theorems for stable-like random walks on nilpotent groups
Jeremy Tyson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On H-type and polarizable Carnot groups
Alexander Teplyaev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BV and harmonic functions on fractals
John Mackay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poincaré inequalities and hyperbolic groups
Kôhei Sasaya . . . . . . . . . . A system of dyadic cubes of a complete, doubling, uniformly perfect metric space without detours
Ryokichi Tanaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-noise sensitivity for random walks on word hyperbolic groups
Zhen-Qing Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence of effective resistances on generalized Sierpinski carpets
Ryosuke Shimizu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linear potential theory on the Sierpiński carpet
Eero Saksman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GFF, chaos and Riemann zetaGrégory Miermont . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact Brownian surfaces
Takashi Kumagai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral dimension of simple random walk on random media
https://www.kurims.kyoto-u.ac.jp/∼croydon/agfms.html

1Abstracts
Martin Barlow (University of British Columbia)
Forty years of analysis on fractals
It is now about 40 years since the first papers by mathematical physicists which studied statistical physics models on fractals, and introduced mathematicians to this topic. This talk willsurvey some parts of the history of the field of analysis and diffusions on fractals. It will also
recall some unsolved problems and (largely) unexplored directions.
Daniel Meyer (University of Liverpool)
Quasisymmetric uniformization and quasivisual approximations
Quasisymmetric maps are homeomorphisms where ratios of distances are controlled. They are
generalizations of conformal maps. Originating in complex analysis, they appear in a variety ofcontexts. In particular they appear in geometric group theory (at the boundary at infinity ofGromov hyperbolic groups). The purpose of this talk is to give an overview of quasisymmetricuniformization and to present a recent framework for this. Namely, “quasivisual approximations” provide a sequence of discrete approximations of spaces as well as necessary and sufficientconditions for quasisymmetric equivalence. This is closely related to recent work by Jun Kigami.
The talk is based on joint work with Mario Bonk.
Nageswari Shanmugalingam (University of Cincinnati)
Constructing boundaries of domains in metric spaces using prime ends
To obtain continuous extensions of conformal mappings between planar domains to their boundaries, Carathéodory constructed a notion of prime ends for simply connected planar domains,and demonstrated that conformal maps between simply connected planar domains do extend as
homeomorphisms to their respective prime end boundaries. For non-simply connected domains
Carathéodory’s construction need not work. In this talk we will discuss an alternative notionof prime ends and discuss its properties, with application to potential theory and branched
quasisymmetric maps.
Bruce Kleiner (New York University)
Rigidity and flexibility of mappings between Carnot groups
The lecture will discuss some recent results concerning the regularity, rigidity, and flexibilityof bilipschitz, quasiconformal, and more generally Sobolev mappings between Carnot groups.
This is based on joint work with Stefan Muller, Laszlo Szekelyhidi, and Xiangdong Xie.

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Mario Bonk (University of California, Los Angeles)
Green functions in metric measure spaces
Let X be an unbounded Ahlfors Q-regular Q-Loewner space with Q ≥ 2. Then X admits ameasurable differential structure in the sense of Cheeger and one can define a notion of “Cheeger
harmonic” functions on X. In my talk I will discuss the statement that in this setting for every
point x0 ∈ X there exists a Green function Gx0 with pole at x0 . It is harmonic in X \ {x0 } andhas logrithmic blow-up near x0 and near ∞ (with different signs). With suitable normalization,
this Green function Gx0 is unique. This is joint work with Luca Capogna and Xiaodan Zhou.
Masaki Tsukamoto (Kyoto University)
Introduction to mean dimension
Mean dimension is a topological invariant of dynamical systems introduced by Gromov in 1999.
It evaluates the number of parameters per unit time for describing a given dynamical system.
Gromov introduced this notion for the purpose of exploring a new direction of geometric analysis.
Independently of this original motivation, Elon Lindenstrauss and Benjamin Weiss found deep
applications of mean dimension in topological dynamics. I plan to survey some highlights ofthe mean dimension theory.
Mathav Murugan (University of British Columbia)
Quasisymmetry and the elliptic Harnack inequality
Jun Kigami made an elementary but useful observation that the elliptic Harnack inequality ispreserved under a quasisymmetric change of metric. In this talk, I will survey three works that
rely on this observation: a characterization of quasisymmetry of circle packing metric using
heat kernel estimates and Harnack inequalities, the stability of the elliptic Harnack inequality,
and the notion of conformal walk dimension. This talk is based on joint works with Martin
Barlow, Zhen-Qing Chen, and Naotaka Kajino.
Alexander Grigor’yan* (Universität Bielefeld)
Off-diagonal estimates of heat kernels for jump processes
We prove upper bounds of the heat kernel for a jump type Dirichlet form on a doubling metricmeasure space, where the off-diagonal term depends on the Lq tail estimate of the jump kernel.
An important ingredient of the proof is a parabolic mean value inequality. This is a joint workwith Jiaxin Hu and Eryan Hu.

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Hiroaki Aikawa (Chubu University)
Elliptic and parabolic boundary Harnack principles via Harnack inequality admitting
exceptional set
The Harnack inequality is one of the most fundamental inequalities for positive harmonic
functions. Various generalizations of Harnack inequality have been developed for elliptic andparabolic equations. We observe a different type extension for classical harmonic functions,
i.e., even if there is a small exceptional set in the Harnack chain, the same Harnack inequality
still holds. This extended Harnack inequality has applications to elliptic boundary Harnack
principle and parabolic boundary Harnack principle (intrinsically ultracontractivity) for a very
nasty domain.
Shin-ichi Ohta (Osaka University)
On weighted Ricci curvature with negative dimension parameter
The weighted Ricci curvature is a modification of the Ricci curvature for a Riemannian manifold
equipped with a measure which can be different from the volume measure. Geometric andanalytic studies using the weighted Ricci curvature go back to Lichnerowicz’ splitting theorem
and Bakry-Emery’s curvature-dimension condition, in the latter the dimension corresponds toa real parameter appearing in the weighted Ricci curvature. In this talk, we review the recent
development of comparison geometry and geometric analysis of weighted Ricci curvature withnegative dimension parameter; in this case we have a weaker curvature bound and can cover a
wider class of spaces.
Michel Lapidus* (University of California, Riverside)
An introduction to the theory of complex dimensions
We will give some sample results from the new higher-dimensional theory of complex fractal dimensions developed jointly with Goran Radunovic and Darko Zubrinic in the researchmonograph [2]. We will also explain its connections with the earlier one-dimensional theory ofcomplex dimensions developed, in particular, in the research monograph [1].
In particular, to an arbitrary compact subset A of the N -dimensional Euclidean space(or, more generally, to any relative fractal drum), we will associate new distance and tube zetafunctions, as well as discuss their basic properties, including their holomorphic and meromorphic
extensions, and the nature and distribution of their poles (or ‘complex dimensions’). We will
also show that the abscissa of convergence of each of these fractal zeta functions coincides
with the upper box (or Minkowski) dimension of the underlying compact set A, and that theassociated residues are intimately related to the (possibly suitably averaged) Minkowski content
of A. Examples of classical fractals and their complex dimensions will be provided.
Finally, if time permits, we will discuss and extend to any dimension the general definitionof fractality proposed by the author (and Machiel van Frankenhuijsen) in their earlier work [1],as the presence of nonreal complex dimensions. We will also provide examples of “hyperfractals”, for which the ‘critical line’ {Re(s) = D}, where D is the Minkowski dimension, is not
only a natural boundary for the associated fractal zeta functions, but also consist entirely of
singularities of those zeta functions.

4Fractal tube formulas are obtained which enable us to express the intrinsic oscillations offractal objects in terms of the underlying complex dimensions and the residues of the associatedfractal zeta functions. Intuitively, the real parts of the complex dimensions correspond to theamplitudes of the associated “geometric waves”, while their imaginary parts correspond to thefrequencies of those waves. This is analogous to Riemann’s explicit formula in analytic number
theory, expressing the counting function of the primes in terms of the underlying zeros of thecelebrated Riemann zeta function. These results are used, in particular, to show the sharpness ofan estimate obtained for the abscissa of meromorphic convergence of the spectral zeta functionsof fractal drums. Furthermore, we will also briefly discuss recent joint results in which we obtaingeneral fractal tube formulas in this context (that is, for compact subsets of Euclidean spaceor for relative fractal drums), expressed in terms of the underlying complex dimensions. We
may close with a brief discussion of a few of the many open problems stated at the end of theaforementioned book, [1].
In a series of joint papers with Claire David [3-6], this general theory of complex dimensions
has recently been significantly extended and applied to the Weierstrass nowhere differentiable
function (and a large class of other fractal curves, including the Koch snowflake curve) in orderto obtain a corresponding fractal tube formula expressed in terms of the underlying complex
dimensions, as well as the associated fractal cohomology. However, in this talk, we will not havethe time to expound about the latter work.
[1] Michel L. Lapidus and Machiel van Frankenhuijsen. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. 2013. Springer
Monographs in Mathematics, Springer-Verlag, New York; 2nd rev. and enl. edn. of the2006 edn.
[2] Michel L. Lapidus, Goran Radunovic and Darko Zubrinic. Fractal Zeta Functions andFractal Drums: Higher Dimensional Theory of Complex Dimensions. 2017. Springer
Monographs in Mathematics, Springer International Publishing, Switzerland.
[3] Claire David and Michel L. Lapidus. Weierstrass fractal drums – I – A glimpse ofcomplex dimensions. 2022. hal-03642326, Sorbonne Université, CNRS, Paris, France.
[4] Claire David and Michel L. Lapidus. Weierstrass fractal drums – II – Towards a fractal
cohomology. 2022. hal-03758820v3, Sorbonne Université, CNRS, Paris, France.
[5] Claire David and Michel L. Lapidus. Fractal cohomology and complex dimensions of theWeierstrass curve. 2022. hal-03797595v2, Sorbonne Université, CNRS, Paris, France.
[6] Claire David and Michel L. Lapidus. Iterated fractal drums – some new perspectives:
Polyhedral measures, atomic decompositions and Morse theory. 2023. hal-039446104,
Sorbonne Université, CNRS, Paris, France.
Ben Hambly (University of Oxford)
Spectral asymptotics for some domains with random fractal boundaries
We consider the eigenvalue counting function on fractals and domains with fractal boundary.
In the classical case this function is known to have an asymptotic expansion where the leading
term is proportional to the volume of the domain and, if the boundary is sufficiently smooth, asecond order term proportional to the volume of the boundary. We will focus on the situationof random fractals and random boundaries arising from random recursive constructions. Using
limit theorems for the general branching process we will show what happens to this asymptotic
expansion in the random case and give examples where the fluctuations due to the randomness
are visible in the spectral asymptotics.
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Laurent Saloff-Coste (Cornell University)
Limit theorems for stable-like random walks on nilpotent groups
This talk reports on joint work with Z-Q. Chen, T. Kumagai, J. Wang and T. Zheng. I will
explain operator limit theorems for random walks on finitely generated nilpotent groups thattake stable-like long jumps in various directions. The limit processes are Lévy processes oncertain nilpotent Lie groups. An important aspect is that the Lie group which carries the limit
process depends both on the original discrete group carrying the random walk and on the stableparameters of the random walk.
Jeremy Tyson (University of Illinois Urbana-Champaign)
On H-type and polarizable Carnot groups
This talk concerns nonlinear potential theory in stratified Lie groups (Carnot groups) andanalytic characterizations for a class of well-behaved Carnot groups of step two. According toa classical result of Folland, the fundamental solution for the Laplacian in the nth Heisenberg
group is a multiple of N 2−Q , where N denotes Korányi’s homogeneous norm and Q = 2n + 2 isthe homogeneous dimension. Capogna, Danielli and Garofalo later observed that an analogous
statement is true for the p-Laplacian for every 1 < p < ∞ and in every Kaplan H-type group,
to wit, the fundamental solution of the p-Laplacian is a multiple of N (p−Q)/(p−1) (or log(1/N )if p = Q). Balogh and Tyson introduced the class of polarizable groups as the largest class of
Carnot groups equipped with a similar one-parameter family of fundamental solutions for thep-Laplacians, all defined in terms of a fixed homogeneous norm. These groups also support apolar coordinate decomposition using horizontal radial curves; such a decomposition allows forthe explicit computation of moduli of ring domains and systems of measures, with applications
to the Hölder regularity of quasiconformal mappings. We conjecture that the only polarizablegroups are Kaplan H-type groups. Towards this end, we seek to understand the class of H-typegroups geometrically within the class of all step two Carnot groups (equipped with a coherent
Riemannian metric). We introduce and study a numeric measurement which quantifies thedegree to which a given step two Carnot group fails to be of H-type. As an application, we
provide several characterizations for H-type groups in terms of analytic identities satisfied by
Folland’s norm.
Alexander Teplyaev (University of Connecticut)
BV and harmonic functions on fractals
The talk will present the first steps aimed to understand bounded variation functions on selfsimilar fractals and related estimates of harmonic functions and flows. This is part of a long-term
joint project with Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, and Nageswari
Shanmugalingam. We build on the classical pioneering work of Barlow, Bass, Fukushima,
Kigami, Kusuoka, and Perkins on Dirichlet and resistance forms, and heat kernel estimates on
fractals. In particular, we relate Besov critical exponents and Hölder continuity of harmonic
functions with the recently introduced topological-Hausdorff dimension of fractals.

6

John Mackay (University of Bristol)Poincaré inequalities and hyperbolic groups
One way to view a Poincaré inequality on (say) a graph is as a tool to measure how wellconnected the graph is. I will discuss a family of invariants based on these inequalities which
generalise the “separation profile” of Benjamini-Schramm-Timar. In the case of Gromov hyperbolic groups, these invariants relate to analytic properties of the boundary at infinity, such as
Poincaré inequalities in the sense of Heinonen-Koskela, and Pansu’s conformal dimension, andhave applications to embedding problems. This is joint work with David Hume and Romain
Tessera.
Kôhei Sasaya (Kyoto University)
A system of dyadic cubes of a complete, doubling, uniformly perfect metric space without
detours
In this talk, we consider an extension of the notion of the standard dyadic cubes in the Euclidean
space to which in a general metric space in Christ’s way. We will construct a system of “dyadic
cubes” of a complete, doubling, and uniformly perfect metric space, such that any two points
in the space are connected by a chain of up to three cubes whose diameters are comparable tothe distance of the points.
Ryokichi Tanaka (Kyoto University)
Non-noise sensitivity for random walks on word hyperbolic groups
The noise sensitivity question for random walks on groups asks the following: Given a sample
of a random walk with independent increments, produce another by refreshing each incrementwith an independent copy with a small probability (noise) or retain it otherwise; are these
two walks asymptotically independent? We discuss the ℓ1 -noise sensitivity which uses the total
variation norm to measure the distance against the distributions of a pair of independent copies,
introduced by Benjamini and Brieussel ’19. We show that a broad class of random walks on
non-elementary word hyperbolic groups are not ℓ1 -noise sensitive in a strong sense, explaining
how Poisson boundaries and the dimension of harmonic measures are related to the problem.
Zhen-Qing Chen (University of Washington)
Convergence of effective resistances on generalized Sierpinski carpets
Let F be a generalized Sierpinski carpet inside a d-dimensional unit hypercube with d ≥ 2 andFn be its n-stage construction. Denote by dw and L ≥ 3 the walk dimension and the length scaleof the carpet F . Let X n be the reflected Brownian motion on Fn running at speed L(dw −2)n .
In this talk, we show that X n converges weakly to a Brownian motion on F . We further showthat the effective resistance between two opposite faces of Fn with respect to X n converges toa positive constant as n tends to infinity. This gives a positive answer to an open problem ofBarlow and Bass (1990). Based on a joint work with Shiping Cao.

7

Ryosuke Shimizu (Kyoto University)
Non-linear potential theory on the Sierpiński carpet
I will provide a review of constructions of (1, p)-Sobolev space, self-similar p-energy and p-energy
measures on the Sierpiński carpet. For p = 2, our self-similar 2-energy and (1, 2)-Sobolev spacecorrespond to the canonical self-similar Dirichlet form on the Sierpiński carpet given by Barlow–
Bass (1989)/Kusuoka–Zhou (1992). I will also explain the connection between p-energy and thenotion of Ahlfors regular conformal dimension.
Eero Saksman (University of Helsinki)
GFF, chaos and Riemann zeta
We survey some recent results (mostly due to others) on probabilistic properties of the Riemann zeta function on the critical line, including connections to Gaussian free field (GFF) and
Gaussian multiplicative chaos (GMC).

Grégory Miermont (École Normale Supérieure de Lyon)
Compact Brownian surfaces
We describe the compact scaling limits of uniformly random quadrangulations with boundaries
on a surface of arbitrary fixed genus. These limits, called Brownian surfaces, are homeomorphic
to the surface of the given genus with or without boundaries depending on the scaling regime of
the boundary perimeters of the quadrangulation. They are constructed by appropriate gluingsof pieces derived from Brownian geometrical objects (the Brownian plane and half-plane). In
this talk, I will review their definition and discuss possible aleternative constructions. This isbased on joint work with Jérémie Bettinelli.
Takashi Kumagai (Waseda University)
Spectral dimension of simple random walk on random media
We give a general method that implies on-diagonal heat kernel bounds, in particular, the lower
bounds on stationary random media that may have long-range correlations. As an application,
we determine the spectral dimension of simple random walk on the long-range percolation modelin Zd . This is joint work with Van Hao Can (Hanoi) and David A. Croydon (Kyoto).

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