Program

Discrete Geometric Analysis and its Applications

Date : 7 January, 2022 (Fri.) ̶ 9 January, 2022 (Sun.)
Venue: Online (Zoom)

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Program
07 Jan. (Fri.)
14:00 - 14:10 Motoko Kotani (Chair of the organizing committee, Tohoku Univ.)
Opening
14:10 - 15:10 Wayne Rossman (Kobe Univ.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Broadening discrete surface classes
15:30 - 16:30 Wai Yeung Lam (Tsinghua Univ.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Deformation space of circle patterns
17:00 - 17:30 Kosuke Naokawa (Hiroshima Inst. Tech.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Discrete developable surfaces and their singularities
17:40 - 18:10 Masashi Yasumoto (Kyushu Univ.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Discrete timelike minimal surfaces
18:30 - 19:30 Alexander Bobenko (TU Berlin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
On a discretization of confocal quadrics: Geometric parametrizations a integrable
systems and incircular nets
08 Jan. (Sat.)
11:00 - 12:00 Jonah Gaster (Univ. of Wisconsin-Milwaukee) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Convergence for discrete harmonic maps between nonpositively curved surfaces(break)13:30 - 14:00 Tatsuya Mikami (Kyoto Univ.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
Percolation on crystal lattices and covering monotonicity of percolation clusters
14:10 - 14:40 Shu Kanazawa (Kyoto Univ.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
On the large deviation principle for persistence diagrams of random cubical ﬁltration
14:50 - 15:20 Hiroki Kodama (Tohoku Univ.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Tension tensors of harmonic nets
15:50 - 16:50 Toru Kajigaya (Tokyo Sci Univ.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
The energy minimizing discrete harmonic maps and its application to closed surface
17:10 - 18:10 Atsushi Katsuda (Kyushu Univ.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
An extension of the Bloch-Floquet theory to the Heisenberg group and its applications
18:30 - 19:30 Konrad Polthier (FU Berlin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Discrete minimal surfaces for generating 3D ﬁllet structures2

Program
09 Jan. (Sun.)
14:10 - 15:10 Kazutoshi Inoue (Tohoku Univ.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Classification of grain boundaries in Euclidean space
15:30 - 16:30 Lin Yong (Tsinghua Univ.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Normalized discrete Ricci flow and community detection
16:50 - 17:50 Na Lei (Dalian Univ. of Tech.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Structural Mesh Generation Based on Computational Conformal Geometry

17:50 - 17:55 Hisashi Naito (Chiar of scientific committee, Nagoya Univ.)
Closing3

Broadening discrete surface classes
Wayne Rossman
Kobe University

Discrete surface theory is a central topic within the more general
newly-developing research ﬁeld called discrete diﬀerential geometry, abbreviated as DDG, and one of the important goals in this topic is toexpand the classes of surfaces that can be studying while still maintaining the underlying mathematical structures.
We will mention a number of approaches to this goal, and then, in
more detail, describe one particular approach, as follows: while discrete
isothermic surfaces have been deﬁned and have an underlying integrablesystems-based mathematical structure, we can extend this to include discrete surfaces that are not themselves isothermic, but do have associatedsphere congruences which are isothermic in a certain sense, an extensionthat still allows us to keep the underlying mathematical structure.
This topic has origins in the work of Darboux, Bianchi, Guichard
and Demoulin in the early 20th century, leading to progress in smoothsurface geometry that we now recognize has close relations to integrable
systems. Isothermic surfaces were introduced by Bour, Guichard surfaces by Guichard, and Omega surfaces by Demoulin. All of these have
dual surfaces, which then guide us to their rich transformation theory as
developed by Bianchi, Calapso, Darboux, Eisenhart and Guichard. Further, we ﬁnd they all ﬁt naturally into the more symmetric settings ofMoebius and Lie sphere geometries, where the surfaces can be regarded
as Legendre maps and Legendre immersions.
From this smooth theory of Omega surfaces and their subclasses,
we can then obtain a satisfying theory for discrete Omega surfaces that
replicates the smooth case in almost every detail, as we will describe inthis talk.
This talk is based on joint work with Fran Burstall, Joseph Cho, Udo
Hertrich-Jeromin and Mason Pember.

4Deformation space of circle patterns
Wai Yeung Lam
Tsinghua University

William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic
function. A discrete analogue of the Riemann mapping is deduced from
Koebe-Andreev-Thurston theorem. One question is how to extend thistheory to Riemann surfaces and relate classical conformal structures todiscrete conformal structures. Since circles are preserved under complex
projective transformations, we consider circle packings on surfaces withcomplex projective structures. Kojima, Mizushima and Tan conjectured
that for a given combinatorics the deformation space of circle packings
is diﬀeomorphic to the Teichmueller space. In this talk, we explain how
the cotangent Laplacian is used to prove the conjecture for the torus caseand its connection to Weil ‒ Petersson geometry.

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A surface generated by a smooth motion of a line in Eulidean 3-space R3 iscalled a ruled surface, and if its Gaussian curvature vanishes identically, thenit is called developable. Developable surfaces are locally isometric to R2 withthe standard metric such as planes, cones, cylinders and tangential surfaces.
This property gives a natural idea for discretizing developable surfaces. In
fact, a ‘discrete’ motion of a line, that is, a sequence of lines in R3 is calleda dicrete developable surface if any adjacent two lines of the sequence lie ina plane in R3 , as in the following figures:

Fig. Disrete versions of a plane, cylindrical surface, cone and tangential
surface, resplectively.
In this talk, we give several results related to topologies and singularities
of discrete developable surfaces. This project is based on a joint work withChirstian Müller (TU-Wien).

Discrete timelike minimal surfaces
Masashi Yasumoto
Institute of Mathematics for Industry, Kyushu University
744 Motooka, Nishi-ku, Fukuoka 819-0395, Japanm-yasumoto@imi.kyushu-u.ac.jp
Based on integrable systems approach, Bobenko and Pinkall described a
discrete version of isothermic surfaces. In the smooth case, isothermic surfaces
include important classes of surfaces such as quadrics, surfaces of revolution,
minimal surfaces, and nonzero constant mean curvature surfaces. So discrete
minimal surfaces in the 3-dimensional Euclidean space were originally studied
as special class of discrete isothermic surfaces, and various links to differential
geometry, complex analysis, and integrable systems were found.
In this talk we focus on discrete timelike surfaces in 3-dimensional Minkowski
space. They have similar properties to discrete isothermic surfaces, and in thespecial case there are several properties that do not appear in the case of discrete
isothermic surfaces. Using them, we introduce that discrete timelike minimal
surfaces admit two types of Weierstrass-type representations. As an application,
we can show all the discrete timelike minimal surfaces that are not necessarily
discrete timelike can be described by a discrete version of null curves.

On a discretization of confocal quadrics:
Geometric parametrizations a integrable systems and incircular nets
Alexander Bobenko
TU Berlin

We propose a discretization of classical confocal coordinates. It isbased on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. The
theory is illustrated with a variety of examples in two and three dimensions. These include confocal coordinate systems parametrized in termsof Jacobi elliptic functions. Connections with incircular nets and elliptic
billiards are established.

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Convergence for discrete harmonic maps
between nonpositively curved surfaces
Jonah Gaster
University of Wisconsin-Milwaukee

Roughly speaking, a map between Riemannian manifolds is harmonic
if it is unreasonably well balanced, in that values at points are closer
than they should be to averages of values at nearby points. Harmonic
maps have proven to be enormously useful in many diﬀerent branchesof mathematics, in particular in the setting of hyperbolic surfaces andTeichmüller theory. I’ll (brieﬂy) discuss some of the background in thesmooth setting, and focus on the problem of discretization in the settingof Riemannian surfaces.
Obtaining convergence of discrete harmonic maps to a smooth harmonic map is a subtle task, which we achieve in the setting of nonpositively curved surfaces together with some mild conditions on thediscretization regime. Curiously, the proof of convergence involves twopieces of analysis which are absent in the smooth setting: 1. a uniform
estimate of strong convexity for the discrete energy functional, and 2. animprovement of the usual comparison between the L2 - and L∞ -metrics
on the space of discrete maps that holds near the discrete harmonic map.
This is joint work with Brice Loustau and Léonard Monsaingeon.

9Percolation on crystal lattices and covering
monotonicity of percolation clusters
Tatsuya Mikami∗
Percolation theory is a branch of probability theory that describes the behavior of clusters. This theory has its origin in applied problems as exemplified
by immersion in porous stone, represented by the following model: each edge(bond) in a lattice 𝑋 is assumed to be open with the same probability 𝑝 ∈ [0, 1],
independently of all other edges. This model is called the bond percolation modeland has been of great interest regarding the critical probability 𝑝 𝑐 (𝑋) ∈ [0, 1],
the point at which an infinite cluster of open edges appears. As one method
for evaluating the critical probability, the paper [1] gives the covering monotonicity of the critical probabilities: for a free action 𝐺 ! 𝑋 of a group 𝐺, thecomparison 𝑝 𝑐 (𝑋) ≤ 𝑝 𝑐 (𝑋/𝐺) holds for the quotient graph 𝑋/𝐺.
In order to observe the relationships between lattices and the shape of clusters in detail, this talk will consider the first passage percolation (FPP) model
on a periodic realization Φ : 𝑋 → R𝑑 of a 𝑑-dimensional crystal lattice 𝑋. Thismodel is a time evolution version of the bond percolation model: each edge 𝑒in 𝑋 is assigned a random passage time 𝑡 𝑒 ≥ 0 independently, and consideration
is given to the behavior of the percolation region 𝐵(𝑡), which consists of thosevertices that can be reached from the origin within a time 𝑡 > 0. As a generalization of the result in the cubic lattice model [2], the normalized region 𝐵(𝑡)/𝑡converges to some deterministic set B, called the limit shape.
In this talk, the covering monotonicity of the limit shapes will be given
as follows: Let 𝑃 : R𝑑 → 𝑊 be the orthogonal projection onto some suitable
subspace 𝑊 ⊂ R𝑑 . Then the projection 𝑃◦Φ(𝑋) coincides with the image Φ1 (𝑋1 )of a periodic realization Φ1 : 𝑋1 → 𝑊 of some crystal lattice 𝑋1 obtained asa quotient graph of 𝑋 (see [4, Section 7.2]). Under this setting, the following
comparison holds for the two limit shapes B1 , B of 𝑋1 , 𝑋, respectively.
Theorem 1. B1 ⊂ 𝑃(B).
This talk is based on the paper [3].

References[1] I. Benjamini and O. Schramm, Percolation beyond Z𝑑 , many questions and a few answers,
Electron. Comm. Probab. 1 (1996), no. 8, 71–82, DOI 10.1214/ECP.v1-978. MR1423907[2] J. T. Cox and R. Durrett, Some limit theorems for percolation processes with necessary
and sufficient conditions, Ann. Probab. 9 (1981), no. 4, 583–603. MR624685[3] T. Mikami, Covering monotonicity of the limit shapes of first passage percolation on
crystal lattices (2021), available at arxiv:2009.11679.
[4] T. Sunada, Topological crystallography, Surveys and Tutorials in the Applied Mathematical
Sciences, vol. 6, Springer, Tokyo, 2013. With a view towards discrete geometric analysis.
MR3014418∗ Department of Mathematics, Kyoto University, E-mail:mikami1642@gmail.com

On the large deviation principle for persistence diagramsof random cubical filtration
Shu Kanazawa ∗Persistent homology, which is a main tool in the growing field of topological data
analysis, allows us to describe the multiscale topological features in various data.
For example, given a (high-dimensional) grayscale image data, we can construct
a filtration of cubical sets (i.e., the union of pixels or higher-dimensional voxels)
in an appropriate way. The persistent homology can capture the birth and deathtimes of high-dimensional holes such as loops and cavities in the filtration. Byplotting each birth-death pair into two-dimensional parameter space, we get a usefuldescriptor of the multiscale topological features in the grayscale image data, the socalled persistence diagram. In application, it is important to examine the effect ofrandomness on persistence diagrams since data contain some noise.
With the above motivation in mind, we consider persistence diagrams of random
cubical filtrations. A random cubical filtration is an increasing family of random cubical sets, which are the union of randomly generated higher-dimensional unit cubeswith integer coordinates. The objective of this work is to investigate the asymptotic
behavior of the (random) persistence diagrams of a random cubical filtration modelas the window size tends to infinity. Recently, the strong law of large numbers forthe persistence diagrams was proved by Hiraoka, Miyanaga, and Tsunoda, whichstates that the persistence diagram converges vaguely to a deterministic measurealmost surely.
In this talk, we are interested in the decay rate of the probability that thepersistence diagram is far from the deterministic limiting measure. We show large
deviation principles for Betti numbers, persistent Betti numbers, and the histograms
generated by counting the birth-death pairs falling in each fine rectangular region.
The key tool for the proofs is a general large deviation principle for regular nearly
additive processes, established by Seppäläinen and Yukich [1]. Time permitting, we
will also discuss the ongoing work on how to provide the large deviation principle
for the persistence diagrams themselves.
This talk is based on joint work with Yasuaki Hiraoka, Jun Miyanaga, andKenkichi Tsunoda.

References[1] T. Seppäläinen and J. Yukich, Large deviation principles for Euclidean functionals and other nearly additive processes, Probab. Theory Related Fields 120(2001), 309–345.

∗

Kyoto University Institute for Advanced Study, Kyoto University.
E-mail address: kanazawa.shu.2w@kyoto-u.ac.jp

Tension tensors of harmonic nets
Hiroki Kodama
Tohoku University / iTHEMS, RIKEN

For a given harmonic net on n-dimensional Euclidean space, we definea square matrix of order n. We call it the tension tensor. We introduce
the properties of the tension tensor. We describe when a harmonic net
becomes standard by using the tension tensor. We also show that thecovariance of a random walk on a harmonic net is consistent with thetension tensor.

12The energy minimizing discrete harmonic maps
and its application to closed surface
Toru Kajigaya
Tokyo Sci University

A map from a weighted ﬁnite graph into a smooth Riemannian manifold is called a discrete harmonic map if it is a stationary point of thediscrete Dirichlet energy. The notion of discrete harmonic maps was introduced in the study of geodesic triangulations of surfaces and a mathematical study of crystal in nature, and is still studied in various directions. In this talk, we will consider the existence of least energy discrete
harmonic maps. In particular, we will discuss the (non-)existence of local minimizer in a simply-connected positively curved manifold, and theexistence of a solution of the ”double” minimization problem for hyperbolic closed surfaces. A part of this talk is based on a joint work with
Ryokichi Tanaka (Kyoto).

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An extension of the Bloch-Floquet theory to the
Heisenberg group and its applications
Atsushi Katsuda
Faculty of Mathematics, Kyushu University
The Bloch-Floquet theory are popular tools for the investigation of materials withperiodic structures. For example, we can show that the spectrum of periodic Schrödinger
operators have band structures. In the context of this talk, this was applied to thefollowing problems in the case of abelian extensions:(1) A geometric analogue of the Chebotarev density theorem for prime closed geodesics
in a compact Riemannian manifold with negative curvature(2) A long time asymptotic expansion of the heat kernels of covering manifolds of compact Riemannian manifolds.
In this talk, we shall extend the above results by applying our version of the BlochFloquet theory for the Heisenberg group. Our method is based on a combination of therepresentation theory for discrete Heisenberg groups especially due to Pytlik and that of
the Heisenberg Lie group.
As a by-product, we also give another mathematical explanation of the semi-classicalasymptotic expansion formula for the spectrum of the Harper operator due to Wilkinson,
which is originally done by Helffer-Sjöstrand.
The spectrum of the Harper operator is described by the following figure, which iscalled the celebrated Hofstadter butterfly,

Figure 1: the Hofstadter’s butterfly (created by Hisashi Naito, Special thanks to Nogizaka
46)
1Discrete minimal surfaces for generating 3D fillet structures
Konrad Polthier
FU Berlin

Modelling volumetric shapes with surface meshes is a core problem in
3d-manufacturing. In this presentation we combine two centrals aspects,
the topological structure of 3d-volumes and the geometric filling withsurface meshes based on discrete minimal surfaces.

15Classification of grain boundaries in Euclidean
space
Kazutoshi Inoue1
Advanced Institute for Materials Research, Tohoku University(joint work with Kazuaki Kawahara2 , Mitsuhiro Saito2 , Motoko Kotani1 ,and Yuichi Ikuhara1,2 )Solid-state materials have been used by polycrystalline form, and their macroscopic
properties strongly depend on defects by which crystalline structures are disordered.
Grain boundary is especially important among them, which is an interface formed by
two adjacent crystal grains.
For an n-dimensional lattice L in Rn , R ∈ SO(n) is said to be a coincidence rotation
if the intersection L ∩ RL forms a sublattice of full rank. The sublattice L ∩ RL is called
the CSL if the index Σ := [L : L ∩ RL] ∈ Z is finite. The CSL is the maximal sublatticecontained in both L and RL. For a coincidence isometry R and an affine hyperplaneΠ passing through the origin with a normal vector n, Π is called a grain boundary for
L1 ∪L2 if L1 ={ p ∈ L; $p, n% < 0 } and L2 ={ p ∈ RL; $p, n% ! 0 } with respect to the
Euclidean inner product $, %.
Although CSL grain boundaries have been classified by the index Σ, the current
method may fail to list all the grain boundaries. In this talk, we propose a systematicclassification method for grain boundaries by using the 2-dimensional lattice matchingwhich is useful in designing grain boundaries. In the 2-dimensional case, the CSL theory
can be formulated by the theory of quadratic field. The problem of 2-dimensional lattice
matching can be deduced to the calculation of ideals of integer rings.

1Advanced Institute for Materials Research, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai;

980-8577
2Institute of Engineering Innovation, School of Engineering, The University of Tokyo, 2-11-16,
Yayoi, Bunkyo-ku, Tokyo; 113-8656
1

Normalized discrete Ricci ﬂow and community detection
Lin Yong
Tsinghua University

We introduce the Ricci ﬂow equations of Ollivier-Lin-Lu-Yau curvature deﬁned on weighted graphs. We prove the existence and uniquenesstheorem for solutions to a continuous time normalized Ricci ﬂow. We
also do the graph partition using the discrete Ricci ﬂow method.

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Structural Mesh Generation Based on Computational Conformal Geometry
Na Lei
Dalian University of Tech.

Generating meshes with regular structure plays a fundamental rolein computational mathematics. Regular hexahedral mesh generation iscalled the holy grid problem in computational mechanics. Intensive research eﬀorts have been spent on it for tens of years. Although there
are many heuristic methods in practice, the theoretic foundation still remains widely open. Recently, we have established a theoretic frameworkfor quadrilateral mesh generation based on conformal geometry. Basically, we have discovered the intrinsic relation between quad-meshes andmeromorphic diﬀerentials on Riemann surfaces. This framework is simple, elegant but powerful. It can answer many fundamental problems,that no other methods could shed a light. For examples, it can show theexistence of quad-meshes with special properties, estimate the dimensionof quad-meshes with constraints, specify the geometric relations among
the singular vertices of quad-meshes. More importantly, it gives a simple algorithm for high quality quad-mesh generation based on RiemannRoch and Abel-Jacobi theorems. Furthermore, the quad-meshes based on
Strebel diﬀerential can leads to hexahedral mesh generation for volumes.

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