Symplectic geometry of the moduli spaces
Indranil Biswas
In the lecture in conference at Takamatsu, which was based in [3] and [4], the symplectic geometry of the moduli spaces of framed logarithmic and parabolic connections,on a Riemann surface with marked points, were described. The ﬁrst aim in the lecture in
Osaka is to describe the symplectic geometry of the moduli spaces of framed meromorphic
Higgs bundles on a Riemann surface with marked points; this is based on [1] and [2]. The
second aim is to cast the symplectic geometry of moduli spaces of framed logarithmic
and parabolic connections and the symplectic geometry of the moduli spaces of framed
meromorphic Higgs bundles in the common framework of uniformization of bundles introduced by Tsuchiya, Ueno and Yamada [6]. We show that all these symplectic structures
have a common origin when approached from the point of view of uniformization. In fact,
they all are given by the Liouville symplectic form on a suitable space associated to theuniformization. This part is based on [5].

References[1] I. Biswas, M. Logares and A. Peón-Nieto, Symplectic geometry of a moduli space of framed
Higgs bundles, Int. Math. Res. Not. (2021), no. 8, 5623–5650.
[2] I. Biswas, M. Logares and A. Peón-Nieto, Moduli spaces of framed G-Higgs bundles andsymplectic geometry, Comm. Math. Phys. 376 (2020), 1875–1908.
[3] I. Biswas, M.-a. Inaba, A. Komyo and M.-H. Saito, On the moduli spaces of framed logarithmic connections on a Riemann surface, Comp. Ren. Math. 359 (2021), 617–624.
[4] I. Biswas, M.-a. Inaba, A. Komyo and M.-H. Saito, Moduli spaces of framed logarithmic
and parabolic connections on a Riemann surface, preprint (2023).
[5] I. Biswas, M.-a. Inaba, A. Komyo and M.-H. Saito, Conference proceedings.
[6] A. Tsuchiya, K. Ueno and Y. Yamada, Conformal ﬁeld theory on universal family of stable
curves with gauge symmetries, Integrable systems in quantum ﬁeld theory and statisticalmechanics, Advanced Studies in Pure Mathematics, Vol. 19, 459–566, Academic Press,
Boston, MA, 1989.
(I. Biswas) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,
Mumbai 400005, India
Email address: indranil@math.tifr.res.inOrthogonal ring patterns and discrete surfaces
Alexander Bobenko

Abstract
We introduce orthogonal ring patterns consisting of pairs of concentric circles. They
generalize orthogonal circle patterns which can be treated as conformal limit. It is shownthat orthogonal ring patterns in euclidean and hyperbolic planes and in a sphere aregoverned by integrable equations, in particular by the discrete master equation Q4 . We
deliver variational principles which are used to prove existence and uniqueness results, andalso to compute ring patterns with classical boundary conditions. The later are used togenerate discrete cmc surfaces. Relation to minimal surfaces in S 3 and AdS 3 is discussed.
Numerous virtual and printed models as well as animation movies will be demonstrated.
(A. Bobenko) Technical University of Berlin, Institute of Mathematics MA 8-4, 10623 Berlin,
Germany
Email address: bobenko@math.tu-berlin.de

The author was supported by DFG via SFB-TRR 109 ”Discretization in Geometry and Dynamics”.

Higher order generalizations of harmonic maps
Volker Branding

Abstract
Harmonic maps are one of the most famous geometric variational problems for maps
between Riemannian manifolds. The harmonic map equation is a second order semilinear
elliptic partial diﬀerential equation for which many results on questions of existence andqualitative behavior of solutions have been obtained in the literature.
Recently, many researchers got attracted in higher order variants of harmonic maps.
In the ﬁrst part of the talk we will give an overview and present a number of recent
results on polyharmonic maps which constitute one possible higher order generalizationof harmonic maps. Finally, we will focus on a second higher order version of harmonic
maps which was initially proposed by Eells and Sampson in 1964 and present several
recent results on the latter.
This is joint work with Stefano Montaldo, Cezar Oniciuc and Andrea Ratto.

References[1] V. Branding, S. Montaldo, C. Oniciuc, and A. Ratto. Higher order energy functionals. Adv.
Math., 370:107236, 60, 2020.
[2] Volker Branding. On ﬁnite energy solutions of 4-harmonic and ES-4-harmonic maps. J.
Geom. Anal., 31(8):8666–8685, 2021.
[3] Volker Branding. A structure theorem for polyharmonic maps between Riemannian manifolds. J. Diﬀerential Equations, 273:14–39, 2021.
[4] Volker Branding, Stefano Montaldo, Cezar Oniciuc, and Andrea Ratto. Unique continuation properties for polyharmonic maps between Riemannian manifolds. Canad. J. Math.,
to appear(V. Branding) Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090
Vienna, Austrua
Email address: volker.branding@univie.ac.at

The author was supported by the Austrian Science Fund (FWF) through the projects “Geometric
Variational Problems from String Theory” (P30749) and “Geometric Analysis of Biwave Maps ” (P34853).

Stability of harmonic maps to symmetric spaces
Fran Burstall
Harmonic maps of Riemannian manifolds are extrema of the energy functional andso it is very natural to try and understand the energy minimisers or, more general, thestable harmonic maps. In this talk, I shall describe some very old work in this area [1, 2]
for the case where the target is a Riemannian symmetric space. This will give me an
opportunity to pay tribute to Ohnita’s fundamental contributions to the topic as well as
to point to open questions.

References[1] D. Burns, F. Burstall, P. De Bartolomeis, and J. Rawnsley. Stability of harmonic
maps of Kähler manifolds. J. Diﬀerential Geom., 30(2):579–594, 1989.
[2] F. Burstall, J. Rawnsley, and S. Salamon. Stable harmonic 2-spheres in symmetricspaces. Bull. Amer. Math. Soc. (N.S.), 16(2):274–278, 1987.
(F. Burstall) Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Email address: feb@bath.ac.uk

Deformations of spectral curve data
Emma Carberry and Martin U. Schmidt(Presenter: Emma Carberry)
We combine the construction of deformations of spectral curves based on the Whithamequation [Kr-94] with the theory of deformations of complex spaces and their singularities.
Our main result is the construction of a universal deformation of spectral curves for thecommonly occuring case of a pair of meromorphic diﬀerentials. The universal deformation
is parameterised by a ﬁnite-dimensional manifold. Our construction applies for example
to the spectral curves of periodic solutions of the KdV and sinh-Gordon equation, anddoubly-periodic solutions of the KP and the Davey-Stewartson equation. We furthermore
modify our construction to produce a universal deformation for a special case of a tripleof meromorphic diﬀerentials, which applies for example to constant mean curvature toriin three-dimensional space forms.

References[Kr-94] I. M. Krichever: The τ -function of the universal Whtham hierarchy, matrix models andtopological ﬁeld theories. Comm. on pure and appl. Math. 47, 437-475 (1994).
(Emma Carberry) School of Mathematics and Statistics, Carslaw Building FO7, The University
of Sydney, NSW 2006, Australia
Email address: emma.carberry@sydney.edu.au
webpage: https://www.sydney.edu.au/science/about/our-people/academic-staff/emma-carberry.html(Martin Ulrich Schmidt) Mathematics Chair III, Universität Mannheim, D-68131 Mannheim,
Germany
Email address: schmidt@math.uni-mannheim.de
webpage: https://www.wim.uni-mannheim.de/schmidt/

2020 Mathematical Subject Classiﬁcation. 32G15, 14H15,14H70
Keywords and Phrases. spectral curves, integrable systems, universal deformation, Whitham deformations

Singularities of mean curvature ﬂow
Qing-Ming Cheng

Abstract
Study on singularities of mean curvature ﬂow is one of the most important subjects
in study on mean curvature ﬂow. It is well-known that self-shrinkers describe all blow-up
at a given singularity of mean curvature ﬂow. Hence, study on self-shrinkers becomesvery important. In this talk, we would like to consider complete self-shrinkers of mean
curvature ﬂow. We will focus on classiﬁcations of n-dimensional complete self-shrinkers
of mean curvature ﬂow with the constant length of second fundamental form.
(Q. M. Cheng) Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan
Email address: cheng@fukuoka-u.ac.jp

Fano 3-folds and classiﬁcation of constantly curved
holomorphic 2-spheres of degree 6 in the complex
Grassmannian G(2, 5)
Quo-Shin Chi

Abstract
Up to now the only known example in the literature of constantly curved holomorphic
2-sphere of degree 6 in the complex G(2, 5) has been the ﬁrst associated curve of theVeronese curve of degree 4. By exploring the rich interplay between the Riemann sphereand projectively equivalent Fano 3-folds of index 2 and degree 5, we prove, up to theambient unitary equivalence, that the moduli space of generic (to be precisely deﬁned)
such 2-spheres is semialgebraic of dimension 2. All these 2-spheres are veriﬁed to have
non-parallel second fundamental form.
This is a joint work with Zhenxiao Xie at China University of Mining and Technology,
Beijing, and Yan Xu at Nankai University.
(Q. S. Chi) Department of Mathematics and Statistics, Washington University, St. Louis, MO
63130, USA
Email address: chi@wustl.edu

Totally symmetric and and ﬁnite uniton typeharmonic maps into arbitrary inner symmetric spaces
Josef F. Dorfmeister and Peng Wang
(Presenter: Josef F. Dorfmeister)

Abstract
In this paper, we develop a loop group description (DPW) of harmonic maps F : M →G/K of “ﬁnite uniton type”, from a Riemann surface M into arbitrary inner symmetricspaces (i.e. of compact or of non-compact type). Along this way we also consider harmonic
maps for which the monodromy representation is trivial (totally symmetric harmonicmaps). Altogether this extends work of Uhlenbeck, Segal, and Burstall-Guest to noncompact inner symmetric spaces. To be more concrete, we prove that every harmonic
map of ﬁnite uniton type from any Riemann surface into any compact or non-compact
inner symmetric space has a normalized potential taking values in some nilpotent Lie
subalgebra, as well as a normalized extended frame with initial condition identity. This
provides a straightforward way to construct all such harmonic maps. We also illustrate
the above results (exclusively) by Willmore surfaces, since the topic of this paper was
motivated by the study of Willmore two-spheres in spheres.
(J. F. Dorfmeister) Fakultät für Mathematik, TU-München, Boltzmannstr.3, D-85747, Garching,
Germany
Email address: dorfmeis@gmail.de(P. Wang) College of Mathematics & Informatics, FJKLMAA, Fujian Normal University, QishanCampus, Fuzhou 350117, P. R. China
Email address: pengwang@fjnu.edu.cn

2020 Mathematical Subject Classiﬁcation. MSC(2010): 58E20; 53C43; 53A30; 53C35
Keywords and Phrases. harmonic maps of ﬁnite uniton type; non-compact inner symmetric spaces;
normalized potential; Willmore surfaces.

On the existence of holomorphic curves in compact
quotients of SL(2, C)
Lynn Heller

Abstract
In my talk I will report on recent joint work with I. Biswas, S. Dumitrescu and S.
Heller showing the existence holomorphic maps from a compact Riemann surface of genus
g > 1 into a quotient of SL(2, C) modulo a cocompact lattice which is generically injective.
This gives an aﬃrmative answer to a question raised by Huckleberry and Winkelmann andby Ghys. The proof uses The proof uses ideas from harmonic maps into the hyperbolic
3-space, WKB analysis, and the grafting of real projective structures.
(L. Heller) Beijing Institute for Mathematical Sciences and Applications, Yanqi Island, Huairou
District, Beijing 101408
Email address: lynn@bimsa.cn

New minimal Lagrangian surfaces in CP 2
Sebastian Heller

Abstract
A surface f : Σ → CP 2 is called minimal Lagrangian if it is both, a minimal surface
with respect to the Fubini study metric and Lagrangian with respect to the Kähler form.
Besides the real projective plane and plenty of minimal Lagrangian tori, the only known
compact examples have been constructed by Haskins and Kapouleas [2]. In this talk, I will
explain the construction of new compact minimal Lagrangian surfaces using loop groupfactorization methods. Note that there are no compact embedded oriented Lagrangiansurfaces in CP 2 , and the newly constructed examples in CP 2 are only immersed.
(Branched) minimal Lagrangian surfaces from a Riemann surface Σ can be obtained fromholomorphic families of ﬂat SL(3, C)-connections
λ ∈ C∗ 7→ λ−1 Φ1,0 + ∇ + λΦ0,1which satisfy the following properties:
1. Φ1,0 is an endomorphism-valued (1, 0)-form on Σ (harmonicity);
2. ∇λ is unitary for λ ∈ S 1 and trivial for λ = 1 (intrinsic and extrinsic closing);
3. ∇λ and ∇ϵλ are gauge equivalent by a λ-independent gauge, where ϵ = exp( 2πi)3(conformality);
4. ∇λ and ∇−λ are dual to each other (Lagrangian).
In my talk, I explain how to construct families of ﬂat connections satisfying these properties on certain Riemann surfaces of large genus. Furthermore, I will show that thecorresponding minimal Lagrangian surfaces are immersions, and derive some of their geometric properties. For example, I show that they lift to special Legendrian surfaces (SL)in S5 → CP 2 , giving the ﬁrst proof of existence of compact SLs of arbitrarily large evengenus. If time permits, I will explain the relationship between our approach and theconstruction of local minimal Lagrangians via twisted potentials [1].This talked is basedon joint work [3] with Charles Ouyang and Franz Pedit.

References[1] J. F. Dorfmeister, H. Ma, Minimal Lagrangian surfaces in CP 2 via the loop group method
Part I: the contractible case, Journal of Geometry and Physics, Volume 161, (2021).
[2] M. Haskins, N. Kapouleas, Special Lagrangian cones with higher genus links, Invent. Math.
167, No. 2, 223–294 (2007).
[3] S. Heller, Ch. Ouyang, F. Pedit, New minimal Lagrangian surfaces in CP 2 , in preperation.
(S. Heller) Beijing Institute for Mathematical Sciences and Applications, Yanqi Island, Huairou
District, Beijing 101408
Email address: sheller@bimsa.cn

Symmetry breaking
Udo Hertrich-Jeromin

Abstract
Symmetry plays a key role in geometry and physics alike: in both disciplines the notion
has been studied and was formalized around the turn of the 20th century, most notably by
F Klein (1872 and 1893) in geometry [3] and by P Curie (1894) in physics [1]. In physics,
these developments were accompanied by the observation and the study of “symmetry
breaking”; in geometry, this notion has received little to no attention. However, also ingeometry, there are phenomena that display a sort of “symmetry breaking”, in a way thatwas described by F Klein in his Erlangen programme and that features some similarities
to what is considered “symmetry breaking” in physics.
We shall investigate some of these phenomena in geometry, some related to integrable
reductions of the corresponding diﬀerential equations; discuss similarlties and diﬀerences
to the notion of symmetry breaking in physics; and attempt a working deﬁnition of“symmetry breaking” in geometry based on Klein’s work.
This presentation is based on long-term joint work with numerous colleagues and, inparticular, on a joint paper with A Fuchs and M Pember [2].

References[1] P Curie: Sur la symétrie dans les phénomnes physiques. Symétrie d un champ électriqueet d un champ magnétique; J Phys, 3rd series, 3, 393–417 (1894), see also excerpt in
K Brading, E Castellani (eds) Symmetries in physics, Cambridge Univ Press, Cambridge(2003) 311–314[2] A Fuchs, U Hertrich-Jeromin, M Pember: Symmetry breaking in geometry; EPrint
arXiv:2206.13401 (2022) https://arxiv.org/abs/2206.13401[3] F Klein: Vergleichende Betrachtungen über neuere geometrische Forschungen; Math Ann
43, 63–100 (1893), see also EPrint arXiv:0807.3161 (2008) https://arxiv.org/abs/0807.
3161(U Hertrich-Jeromin) Institute of discrete mathematics and geometry, University of Technology
Vienna, Wiedner Hauptstrasse 8-10/104, 1040 Wien (Austria)
Email address: udo.hertrich-jeromin@tuwien.ac.atGenaration of aesthetic shapes by integrable
geometry
Kenji Kajiwara
In this talk, we consider a class of plane curves called log-aesthetic curves (LAC) andtheir generalizations which have been developed in industrial design as the curves obtainedby extracting the common properties among thousands of curves that car designers regard as aesthetic. We consider these curves in the framework of similarity geometry andcharacterize them as invariant curves under the integrable deformation of plane curves
governed by the Burgers equation. We propose a variational principle for these curves,
leading to the stationary Burgers equation as the Euler-Lagrange equation.
We then extend the LAC to space curves by considering the integrable deformationof space curves under similarity geometry. The deformation is governed by the coupledsystem of the modiﬁed KdV equation satisﬁed by the similarity torsion and a linear
equation satisﬁed by the curvature radius. The curves also allow the deformation governedby the coupled system of the sine-Gordon equation and associated linear equation. The
space curves corresponding to the travelling wave solutions of those equations would give
generalization of LAC to space curves.
We also consider the surface constructed by the family of curves obtained by the integrable deformation of such curves. A special class of surfaces corresponding to the constantsimilarity torsion yields quadratic surfaces (ellipsoid, one/two-sheeted hyperboloids andparaboloid) and their deformations, which may be regarded as a generalization of LAC
to surface.
We discuss the construction of such curves and surfaces together with their mathematical properties, including integration scheme of the frame by symmetries, and present
various examples of curves and surfaces.

References[1] Inoguchi, J., Kajiwara, K., Miura, K.T., Sato, M., Schief, W.K., Shimizu, Y.: Log-aesthetic
curves as similarity geometric analogue of Euler’s elasticae. Comput. Aided Geom. Des. 61,
1–5 (2018). https://doi.org/10.1016/j.cagd.2018.02.002[2] Inoguchi, J., Jikumaru, Y., Kajiwara, K., Miura, Schief, W.K.: Log-Aesthetic Curves:
Similarity Geometry, Integrable Discretization and Variational Principles, preprint,
arXiv:1808.03104.
(K. Kajiwara) Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku,
Fukuoka 819-0395, Japan.
Email address: kaji@imi.kyushu-u.ac.jp

This work was supported by JST CREST JPMJCR1911 and JSPS Kakenhi 21K03329.

Eigenspaces and minimal surfaces
Robert Kusner

Abstract
We develop sharp bounds for the dimensions of ﬁrst Steklov (or Laplacian) eigenspaces
on surfaces, using this to study the geometry of free boundary (or closed) minimal surfaces
in the round ball (or sphere).
(R. Kusner) Department of Mathematics, University of Massachusetts, Amherst, MA 01003,
USA
Email address: kusner@umass.edu

Periodic Darboux transforms
Katrin Leschke
In classical diﬀerential geometry, geometric transformations have been used to create
new curves and surfaces from simple ones: the aim is to solve the underlying deﬁning
compatibility equations of curve or surface classes by ﬁnding solutions to a simpler systemof diﬀerential equations arising from the transforms. Classically, the main concern was a
local theory. In modern theory, global questions have led to a renewed interest in classicaltranformations. For example, in the case of a torus, the investigation of closing conditions
for Darboux transforms naturally leads to the notion of the spectral curve of the torus.
In this talk we discuss closing conditions for smooth and discrete polarised curves,
isothermic surfaces and CMC surfaces. In particular, we obtain new explicit periodic
discrete polarised curves, new discrete isothermic tori and new explicit smooth CMC
cylinder.

References[1] J. Cho, K. Leschke, Y. Ogata, Periodic discrete Darboux transforms, Submitted.
[2] J. Cho, K. Leschke, Y. Ogata, New explicit CMC cylinders and same-lobed CMC multibubbletons, Submitted.
[3] J. Cho, K. Leschke, Y. Ogata, Generalised Bianchi permutability for isothermic surfaces,
Ann. Global Anal. Geom. 61, pages 799-829, 2022.
[4] K. Leschke, Links between the integrable systems of CMC surfaces, isothermic surfaces andconstrained Willmore surfaces, Preprint.
(K. Leschke) School of Computing and Mathematical Sciences, University of Leicester, Leicester
LE17RH, United Kingdom
Email address: k.leschke@leicester.ac.uk

The author was partially supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located at Kyoto University, and the CREST-ED 3 GE project at
Kyushu University.

Construction of minimal Lagrangian surfaces in CP 2
via integrable system
Hui MaIn this talk I will discuss the construction of minimal Lagrangian surfaces in thecomplex projective plane by loop group method. This is based on the joint work with
Josef Dorfmeister and Shimpei Kobayashi.

References[1] J. Dorfmeister and S.-P. Kobayashi and H. Ma, Ruh-Vilms theorems for minimal surfaces
without complex points and minimal Lagrangian surfaces in CP 2 , Math. Z. 296 (2020), no.
3-4, 1751–1775.
[2] J. Dorfmeister and H. Ma, Minimal Lagrangian surfaces in CP 2 via the loop group method:
Part I: The contractible case, J. Geom. Phy. 161 (2021), 104016.
[3] J. Dorfmeister and H. Ma, Minimal Lagrangian surfaces in CP 2 via the loop group method
Part II: The general case, Preprint, 2023.
(H. Ma) Department of mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
Email address: ma-h@tsinghua.edu.cn

Review of my research
Reiko Miyaoka
I give a survey talk, looking back on my research including a joint work with Y. Ohnita.
First, I mention my result on minimal surfaces [1] motivated by Lawson’s work related
to Calabi’s rigidity theorem. This is because Calabi is 100 years old this year. Much later,
I found the subject to be related to the Aﬃne Toda equation [6].
Next, I talk about Dupin hypersurfaces and the Lie sphere geometry, started from
U. Pinkall’s thesis. The most important result of mine is the discovery of the “Lie curvature” [3], that is a key tool to resolve the Cecil-Ryan conjecture [2], [4]. I continue tostudy the Lie contact structure, and solved H. Sato’s problem using the Cartan-Tanaka
connection [5].
Finally, I’ve focussed on the theory of isoparametric hypersurfaces [7], and using themas a subject, we got some results on Floer homology with Ohnita et al. [8].
When time allows, I’ll show you a few pictures for fun.

References[1] R. Miyaoka, Some results on minimal surfaces with the Ricci condition, Minimal Submanifolds and Geodesics (ed. by M. Obata) Kaigai Publ. (1978), 121-142.
[2] R. Miyaoka, Compact Dupin Hypersurfaces with 3 Principal Curvatures, Math.Zeit. 187(4)(1984), 433-452.
[3] R. Miyaoka, Dupin hypersurfaces and a Lie invariant, Kodai M. J. 12(2) (1989), 228-256.
[4] R. Miyaoka and T. Ozawa, Construction of taut embeddings and Cecil-Ryan conjecture,
Geometry of Manifolds (ed. by K. Shiohama) Academic Press (1989),181-189.
[5] R. Miyaoka, Lie contact structures and normal Cartan connections, Kodai M. J. 14(1)(1991), 13-41.
[6] R. Miyaoka, The family of isometric superconformal harmonic maps and the aﬃne Toda
equations, J. Reine Angew. Math. 481, (1996), 1-25.
[7] R. Miyaoka, Isoparametric hypersurfaces with (g, m) = (6, 2), Ann. of Math. vol.177(2013),
53-110. Errata of “Isoparametric hypersurfaces with (g, m) = (6, 2)”, Ann. of Math. vol 183(2016), 1057—1071.
[8] H. Irie, H. Ma, R. Miyaoka and Y. Ohnita, Hamiltonian non-displaceability of Gauss imagesof isoparametric hypersurfaces, Bull. London Math. Soc. 48 (5): (2016), 802-812.
(R. Miyaoka) Tohoku University, Mathematical Institute, 6-3, Aoba-ku, Sendai, 980-8578, Japan
Email address: r-miyaok@tohoku.ac.jp

The author is partly supported by Kakenhi C 21K03214.

Transforms of minimal surfaces
Katsuhiro Moriya
Harmonic maps are one of Ohnita’s research subjects. That made his interest in
integrable systems. A harmonic map is minimal if it is an isometric immersion. In thistalk, I shall deﬁne transforms of minimal surfaces.
(K. Moriya) Mathematical Science Laboratoy, Department of Material Science, University of
Hyogo, 2167, Shosha, Himeji, Hyogo 671-2280, Japan
Email address: m905k019@guh.u-hyogo.ac.jp

The author was partly supported byJSPS KAKENHI Grant Number JP22K03293.

The Gaiotto correspondence, quantization, andconnections deﬁned over Q
Motohico Mulase

1

Abstract

The talk is aimed at presenting a simple mathematical description of the conformal
limit construction of Davide Gaiotto [3, 4], and its potential implications in problems on
Q-geometry and mirror symmetry. The talk is based on my ongoing collaboration with
Alex Cruz Morales, Olivia Dumitrescu, and Jon Erickson.
The Gaiotto construction assigns an oper, a connection on a Riemann surface C witha complex variation of Hodge structure, to a Higgs bundle on C equipped with an SL(2)structure. This process can be thought of as a quantization of a Hitchin spectral curve toa Schrödinger operator [1, 2].
Starting from the classical Riemann surface theory, such as projective structures, theSchwarzian derivative, holomorphic quadratic diﬀerentials, and the constant curvature
metric on a compact Riemann surface of positive genus, we can see the interplay between
Kostant’s Three-Dimensional Subgroups in a simple complex algebraic group G, quantization processes, and variations of Hodge structure, via analyzing the Gaiotto construction.
This construction also deﬁnes a biholomorphic correspondence between a complex Lagrangian in the Dolbeault moduli space of G-Higgs bundles and a complex Lagrangian in
the de Rham moduli space of G-connections. Through the hyperkähler rotation and Langlands duality, the Gaiotto correspondence is expected to play a key role in understanding
the mirror symmetry of these moduli spaces.
The talk is dedicated to Professor Yoshihiro Ohnita at his retirement.

References[1] Olivia Dumitrescu and Motohico Mulase, An invitation to 2D TQFT and quantization of
Hitchin spectral curves, Banach Center Publications 114, 85–144 (2018).
[2] Olivia Dumitrescu and Motohico Mulase, Interplay between opers, quantum curves, WKB
analysis, and Higgs bundles, SIGMA 17, 036, 53 pages (2021).
[3] Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo, Motohico Mulase, and Andrew Neitzke, From the Hitchin section to opers through nonabelian Hodge,
Journal of Diﬀerential Geometry 117 (2), 223–253 (2021).
[4] Davide Gaiotto, Opers and TBA, arXiv:1403.6137 [hep-th] (2014).
Department of Mathematics, University of California, Davis
One Shields Avenue, Davis, California 95616, U.S.A.
Email address: mulase@math.ucdavis.edu
The author’s research is partly supported by NSF-FRG (DMS-2152257).

Global geometry of q-Painlevé equations
Yousuke Ohyama
We study connection problem of q-Painlevé equations. As the same as classicalPainlevé diﬀerential equations, q-Painlevé equations are also represented by a connection preserving deformation of a q-linear diﬀerence equation. And a q-analogue of the
Riemann-Hilbert correspondence plays an important role to study q-Painlevé functions.
We review connection problems of q-linear diﬀerence equations at ﬁrst. Then we study
q-Riemann-Hilbert problem on the q-Painlevé VI equation [1, 2]. We show that the spaceof connection is a two dimensional algebraic surface, an open subvariety of the Segre
surface.

References[1] Ohyama, Y., Ramis, J.-P., Sauloy, J.; The space of monodromy data for the Jimbo 窶鉄 akaifamily of q-diﬀerence equations, Ann. Fac. sci. Toulouse Math., (6) 29 (2020) , 1119–1250.
[2] Nalini J., Pieter R.; On the monodromy manifold of q-Painlevé VI and its Riemann-Hilbert
problem, arXiv:2202.10597.
[3] Y., Ramis, J.-P., Sauloy, J.; Geometry of the space of monodromy data, arXiv:2301.04083.
(Y. Ohyama) Department of Mathematicall Sciences, Tokushima University, 2-1 Minamijyousanjimacho, Tokushima 770-8506, Japan
Email address: ohyama@tokushima-u.ac.jp

The author was partly supported by JSPS KAKENHI Grant Number JP19K03566.

Snakes, falling cats and ﬁber bundles
Oliver Gross, Felix Knöppel, Marcel Padilla, Ulrich Pinkall, Peter SchröderPresenter: Ulrich Pinkall

1

Locomotion of Snakes

Imagine a snake of length L moving on the plane R2 in such a way that all parts of its
body stay in contact with the ground. Then at each time t the position of the snake canbe modeled by an element γ(t) of the space
M = γ ∈ C ∞ ([0, L], R2 ) |γ ′ | = 1
of all arclength-parametrized plane curves of length L. The group SE(2) of orientationpreserving Euclidean motions of R2 acts on M. At each time t the shape of the snake isgiven by the orbit
St = {g ◦ γ(t) | g ∈ SE(2)}under this group action. This shape can be described by a certain curvature function, so
the space of all possible shapes is given by
f = M/SE(2) ' C ∞ [0, L].
M
f makes M into the total space of a principal ﬁber bundle
The projection map π : M → Mfwith structure group SE(2) over the base manifold M.

The authors were partly supported by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics the Caltech Center for Information Science and Technology, and theEinstein Foundation Berlin.

f of shapes. What is the resulting
The snake controls only the sequence t 7→ St ∈ Msequence t 7→ γ(t) ∈ M of snake positions? The mathematical model we propose is thatt 7→ γ(t) is a horizontal lift (with respect to a certain connection on the bundle) of thegiven curve t 7→ St in the base manifold. At each point γ ∈ M the horizontal space of thisconnection is the orthogonal complement of the tangent space to the ﬁber through γ withrespect to a certain Riemannian metric on M. This metric assigns to each velocity vectorγ̇ ∈ Tγ M the corresponding energy dissipation 21 hγ̇, γ̇i due to friction with the ground.
Putting aside the question how to compute this metric, we apply Helmholtz’ principleof least dissipation: Among all possible lifts of the given curve in the base manifold thecorresponding lift γ to the total space minimizes dissipation, i.e. the Riemannian energyof γ. In other words: the resulting snake motion γ is a horizontal lift of the given curve
S in shape space.

2

Falling Cats

Imagine a cat initially at rest in a position with its feet pointing upward. Afterbeing released, the cat starts changing its shape and ﬁnally achieves that its feet point
downward. We can ignore gravity because with respect to the center of mass the resulting
motion will be the same as in the corresponding situation with no gravity.
This situation resembles the one of the snake: now the group SE(3) acts on the space
f is the corresponding quotient space of M.
M of cat positions and the shape space M
This time, the Riemannian geometry of M is given by the kinetic energy of the cat andthe variational principle we apply is the principle of least action.

The same mathematical model also applies to shape-changing bodies under water.
The only diﬀerence is that here we also have to take into account the kinetic energy ofthe motion of the surrounding ﬂuid (enforced by the shape change of the body).

The above mathematical setup can be turned into a numerical algorithm that can beeﬃciently used in the context of Computer Graphics.

(U. Pinkall) Technical University of Berlin MA 3-2, Str. d. 17. Juni 136, 10623 Berlin, Germany
Email address: upinkall@gmail.com

Canonical coordinates of Moduli spaces ofConnections and Higgs bundles, Isomonodromic
deformations and diﬀerential equations of Painlevétype
Masa-Hiko Saito

1

Moduli Space of Connections and the Riemann-Hilbert correspondence

The moduli spaces of singular connections and singular Higgs bundles on curves areconstructed by geometric invariant theory. Using stable parabolic connections and Higgs
bundles, we can construct moduli spaces MDR and MH of these geometric objects assmooth quasi projective algebraic varieties. Moreover, we will denote by Mmon the moduli of monodromy representation of the fundamental group, or generalized monodromy
such as Stokes data, etc. Note that Classical Painlevé equations, classiﬁed into 8 types,
correspond to 10 types of generalized data and the equations for Mmond are given in [PS].
Under a suitable choice of parameter, we can deﬁne the Riemann-Hilbert correspondence
RH : MDR −→ Mmon
Usually, the left-hand side forms a family of moduli spaces of connections parametrizedby time variables coming from the datum of regular or irregular singular points. Notethat the right-hand side does not depend on time variables, at least locally.
Theorem 1.1 ([IIS1], [Ina], [IS1], [IS2]) We assume that all singularities of connections
are regular or unramiﬁed irregular. Fixing a time variable, RH is a proper surjective bimeromorphic holomorphic map.
In the generic case, RH is an analytic isomorphism. If Mmon has singularities, RH becomesan analytical resolution of singularities. Pulling back the constant sections on the righthand side (monodromy side) by the Reimann-Hilbert correspondence, we can deﬁne theset of ﬂows on the family of moduli spaces of connections on the right-hand side. Thisis called Isomonodromic Flows deﬁned on the family of moduli space of connections. We
can deﬁne the diﬀerential equations describing these isomonodromic ﬂows, which we callisomonodromic diﬀerential equations. By a theorem due to Inaba, Iwasaki and Saito andother people, we can show the following.
Theorem 1.2 ([IIS1], [Ina], [IS1], [IS2]) The isomonodromic diﬀerential equations coming from the moduli space of connections with regular or unramiﬁed singularities satisfy
Painlevé propety, that is, all of the movable singularities of the solutions of the isomonodromic diﬀerential equations are poles.
We remark here that it is essential to construct the moduli spaces in the category ofalgebraic varieties.
The author was partly supported by JSPS Grand-in-Aid for Scientiﬁc Research (S) 17H06127 , (A)22H00094.

2

Symplectc structures and Canonical coordinates for moduli
spaces

It is known that all moduli spaces MDR , MH , Mmon admit natural symplectic structures([IIS1], [Ina], [IS1], [IS2]). One can see that RH preserves these symplectic structures.
Moreover on a Zarisiki dense open set of moduli spaces of MDR , MH , we can introduce
canonical coodinates of MDR , MH by using apparent singularities and their duals ([SSz]).
We will explain these facts in new settings and the relation to the works of Iwasaki,
Dubrovin-Mazzocco and Krichever.

References[Ina] Michi-aki Inaba Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence, J. Algebraic Geom. 22 (2013), no. 3, 407–480.
[IIS1] Michi-aki Inaba, Katsunori Iwasaki, and Masa-Hiko Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI.
I, Publ. Res. Inst. Math. Sci. 42 (2006), no. 4, 987–1089.
[IS1] Michi-aki Inaba and Masa-Hiko Saito, Moduli of unramiﬁed irregular singular parabolic
connections on a smooth projective curve, Kyoto J. Math., 53, 2013, no. 2, 433–482.
[IS2] Michi-aki Inaba and Masa-Hiko Saito, Moduli of regular singular parabolic connections
with given spectral type on smooth projective curves., J. Math. Soc. Japan 70, 2018, no. 3,
879–894.
[PS] M. van der Put and M.-H. Saito, Moduli spaces for linear diﬀerential equations and the
Painlevé equations, Ann. Inst. Fourier (Grenoble), 59, 2009, no. 7, 2611–2667.
[SSz] Masa-Hiko Saito and Szilard Szabo, Apparent singularities and canonical coordinates formoduli of parabolic connections and parbolic Higgs bundles, in preparation.
(Masa-Hiko, Saito), Faculty of Bussiness Administrations, Kobe Gakuin University, 1-1-3, Minatojima, Chuo-ku, Kobe, 650-8586, Japan.
Email address: mhsaito@ba.kobegakuin.ac.jp

On two questions of I. M. James
Zizhou Tang
65 years ago, I. M. James raised two fundamental questions about octonionic Stiefel
spaces. The prime objective of this talk is to give partial answers to both of them. Thisis based on the joint work with Chao Qian and Wenjiao Yan.
(Z. Tang) Chern Institute of Mathematics, Nankai University, Tianjin, 300071, China
Email address: zztang@nankai.edu.cn

Characterizations of Ricci ﬂat metrics andLagrangian submanifolds in terms of the variational
problem
Tetsuya Taniguchi

1

Abstract

Given the pair (P, η) of (0, 2) tensors, where η deﬁnes a volume element, we consider
a new variational problem varying η only. We then have Einstein metrics and slant
immersions as critical points of the 1st variation. We may characterize Ricci ﬂat metrics
and Lagrangian submanifolds as stable critical points of our variational problem.

References[1] Tetsuya Taniguchi and Seiichi Udagawa, Characterizations of Ricci ﬂat metrics and Lagrangian submanifolds in terms of the variational problem, Glasgow Math. J. 57(2015)643-651.
(T. Taniguchi) Nihon University School of Medicine, Liberal Arts, 30-1 Oyaguchi Kamicho,
Itabashi-ku, Tokyo, 173-8610, Japan
Email address: taniguchi.tetsuya@nihon-u.ac.jp

The author was partly supported by 21H04412.

Solving the sine-Gordon equation and its
discretization
Seiichi Udagawa

Abstract
The sine-Gordon equation has played an important role in the study of integrable
systems and there are many works related to it.
1. Its (see [2]) obtained the solution of SG-equation in terms of Riemann θ-function.
2. The sine-Gordon equation is discretized by Ryogo Hirota (see [4]). The solution interms of the Riemann θ-function was obtained by Bobenko-Pinkall (see [1]).
3. On the other hand, a semi-discrete version can be considered in which position coordinates are discretized with one independent variable as a continuous time parameter. Thisis recently attracting attention as an equation that describes the equation of motion of
the Kaleidocycle (see [5], [8]).
In this talk, 1. First, we obtain the elliptic function solution of the sine-Gordon equation and conﬁrm that it can be expressed in the form of the solution of the expression of
the Riemann θ-function by Its.
2. Next, we obtain the elliptic function solution for the semi-discrete sine-Gordon equation, and again describe the relationship with the expression by the Riemann θ-function.
3. Moreover, we obtain the elliptic function solution for the discrete sine-Gordon equationand describe its relationship with the expression by the Riemann θ-function by BobenkoPinkall.
4. The ultimate goal is to describe the motion of the Kaleidcycle using elliptic functions.
Recently, Shigetomi (Kyushu University) described it in terms of Jacobi θ-functions.
This is a joint work with J. Inoguchi and K. Kajiwara.

References[1] A. I. Bobenko and U. Pinkall, Discrete surfaces with constant negative Gaussian
curvature and the Hirota equation, J. Diﬀerential Geometry 43(1996), 527-611.
[2] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its and V. B. Matveev,
Algebro Geometric Approach to Nonlinear Integrable Equations, Springer series in Nonlinear Dynamics, (1994), Springer-Verlag.
[3] J. D. Fay, Theta functions on Riemann surfaces, Lect. Notes in Math. 352(1973), SpringerVerlag.
[4] R. Hirota, Nonlinear partial diﬀerence equations III. discrete sine-Gordon equation, J.
Phys. Soc. Japan 43(1977), 2079-2086.
[5] J. Inoguchi, K. Kajiwara, N. Matsuura and Y. Ohta, Motion and Bäcklund transformations of discrete plane curves, Kyushu J. Math. 66(2012), 303-324.
The author is partly supported by JSPS Grant-in-Aid for Scientiﬁc Research （C） No. 21K03235.

[6] J. Inoguchi, T. Taniguchi and S. Udagawa, Finite gap solutions for horizontal minimal
surfaces of ﬁnite type in 5-sphere, Journal of Integrable Systems 1(2016),1-34.
[7] J. Inoguchi and S. Udagawa, Aﬃne spheres and ﬁnite gap solutions of Tzitzéica equation, Journal of Physics Communications 2(2018),115020.
[8] S. Shigetomi and K. Kajiwara, Explicit formula for isoperimetric deformations ofsmooth and discrete elasticae, JSIAM Letters 13(2021), 80-83.
[9] K. Sogo, Variational discretization of Euler’s elastica problem, J. Phys. Soc. Japan
75(2006), 064007.
(S. Udagawa) Nihon University, Itabashi 173-0032, Tokyo, Japan
Email address: udagawa.seiichi@nihon-u.ac.jp

Ray Structures on Teichmüller Space
Huiping Pan and Michael Wolf
(Presenter: Mike Wolf)

Abstract
There are several classical ray structures on Teichmüller space, including the Teichmüller geodesics and the Thurston geodesics. The former refer to the conformal structureof a Riemann surface and the latter to the hyperbolic structure. There are also harmonic
map rays, which depend on a ﬁxed conformal domain and varying hyperbolic targets:
how do these ﬁt into the family of ray structures. We depict harmonic map ray structures
on Teichmüller space as a geometric transition between Teichmüller ray structures andThurston geodesic ray structures.
As a consequence, while there may be many Thurston metric geodesics between a pairof points in Teichmüller space, we ﬁnd that by imposing an additional energy minimization
constraint on the geodesics, thought of as limits of harmonic map rays, we select a uniqueThurston geodesic through those points. Extending the target surface to the Thurston
boundary yields, for each point Y in Teichmüller space, an “exponential map” of raysfrom that point Y onto Teichmüller space with visual boundary the Thurston boundaryof Teichmüller space.
(M. Wolf) School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA
Email address: mwolf40@gatech.edu

The author was partly supported by the US NSF-DMS005551.

Symplectic structures on almost abelian Lie algebras
Luis Pedro Castellanos Moscoso

Abstract
We are interested in the classiﬁcation or ﬁnding conditions for the existence of leftinvariant symplectic structures on Lie groups. Only some classiﬁcations are known, specially in low dimensions. We approach this problem by studying the “moduli space ofleft-invariant nondegenerate 2-forms”, which is a certain orbit space in the set of all nondegenerate 2-forms on a Lie algebra. In this talk we consider this problem in the case ofalmost abelian Lie algebras, that is Lie algebras that contain a codimension 1 abelian subalgebra. Almost abelian Lie algebras can be described in terms of linear operators. The
isomorphism classes of almost abelian Lie algebras are related to the similarity classes ofthese linear operators. Then, we can show that, in this setting, the problem of existence
and classiﬁcation of left-invariant symplectic structures can be reduced to a known matrix
equation involving the Jordan normal form of a matrix and a corresponding equivalence
relation, respectively. We study the solution of these equations for several particular
examples.

References[1] Avetisyan, Zhirayr.: The structure of almost Abelian Lie algebras. Internat. J. Math. Vol.
33, No. 08, 2250057 (2022).
[2] Baues, O. and Cortés, V. : Symplectic Lie groups. Astérisque 379 (2016).
[3] Castellanos Moscoso, L. P.: Left-invariant symplectic structures on diagonal almost abelian
Lie groups. Hiroshima Math. J. 52 (3) 357–378 (2022).
(Luis Pedro Castellanos Moscoso) Osaka Central Advanced Mathematical Institute, 3-3-138,
Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Email address: caste3.1416@gmail.com

The author was partly supported by the Osaka Central Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics).

Darboux transformations in the Lorentz-Minkowski
plane
Masaya Hara
This talk concerns Lorentz-Möbius geometry, and Darboux transformations in that
context. We construct the basic structure of Lorentz-Möbius geometry by using the notion of the generalized point sphere complex, and we show some unique properties ofDarboux transformations in the Lorentz-Minkowski plane R1,1 , especially with regard to
singularities and unboundedness. Additionally, we describe suggestive Darboux transformations of type-changing curves in R1,1 .

1Darboux transformations

Let R1,1 denote a 2-dimensional pseudo-Riemannian space with signature (+−) andinner product ⟨·, ·⟩1,1 , that is, for x, y ∈ R1,1 ,⟨x, y⟩1,1 = ⟨(x1 , x2 )t , (y1 , y2 )t ⟩1,1 = x1 y1 − x2 y2 ,and we use | · |2 to denote the inner product of a vector with itself, i.e. the length squaredin R1,1 .
We can deﬁne the Darboux transformations of curves in R1,1 by the Riccati-type
equation as follows:()
2
Deﬁnition 1.1 ([1, 2]). Let x(t) : I, dtm → R1,1 be a smooth curve polarized bym(t) : I → R\{0}. For some nonzero constant µ ∈ R\{0}, a curve x̂(t) deﬁned byx̂′ = µ|x̂ − x|2 x∗ ′ − 2µ⟨x̂ − x, x∗ ′ ⟩1,1 (x̂ − x)is called a Darboux transformation with parameter µ of x(t), where x∗ ′ = −x′.
m|x′ |2

In Figure 1, the blue line is the initial line and an orange curve is the Darboux
transformation with the black point as initial condition. The spectral parameter µ is
positive (resp. negative) in the left picture (resp. right picture).

2

Properties

Let x̂ be a Darboux transformation of a spacelike curve x. We show some properties
of such Darboux transformations:• x̂ can be lightlike at some points and the common circle congruence S becomesdegenerate there.
• If x is singularity free, x̂ is also singularity free.
• If x and x̂ have the common arc-length polarization m, x̂ is unbounded if and only
if S is degenerate.
• x can be unbounded with ﬁnite limiting curvature.

-

-

+

+

Figure 1: Examples of Darboux transformations of a spacelike line.

References[1] Joseph Cho, Zero mean curvature surfaces of Bonnet-type, Ph.D. thesis, Kobe University,
2019.
[2] Joseph Cho, Wayne Rossman, Kosuke Naokawa, Yuta Ogata, Mason Pember, and Masashi
Yasumoto, Discrete Isothermic Surfaces in Lie Sphere Geometry, preprint.
(M. Hara) Department of Mathematics, Graduate School of Science, Kobe University, 1-1,
Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
Email address: mhara@math.kobe-u.ac.jp

On some Lie-theoretic solutions of the tt*-Toda
equations
Yudai Hateruma
The tt* equations are an integrable system of partial diﬀerential equations. They wereintroduced by Cecotti and Vafa in their work on N = 2 supersymmetric quantum ﬁeld
theories. From the physical point of view, Cecotti and Vafa made several conjectures
regarding solutions of these equations, in particular the solutions, whose Stokes matrices
have integer entires. Guest, Its and Lin studied the tt* equations of ‘Toda type’ for thegroup SL(n + 1, C). They described all global solutions and some of their properties.
From the Lie theoretic point of view, the set of the global solutions is in one-to-one
correspondence with the Fundamental Weyl Alcove. However describing properties ofthe ‘integer Stokes solutions’ is still diﬃcult. In this talk, we describe the following two
results of these solutions. First, we describe all integer Stokes solutions in a certain one
dimensional subspace of the Fundamental Weyl Alcove. Second, we compare the resultsof Guest-Its-Lin and the predictions of Cecotti-Vafa. We ﬁnd that, in the case of thett*-Toda equations, these two methods agree, at least for low values of n. This is joint
work with Yoshiki Kaneko.

References[CV] S. Cecotti and C. Vafa, On classiﬁcation of N = 2 supersymmetric theories, Comm. Math.
Phys. 158 (1993), 569-644.
[GIL] M. A. Guest, A. Its and C. S. Lin, Isomonodromy aspects of the tt* equations of Cecotti
and Vafa III. Iwasawa factorization and asymptotics, Commun. Math. Phys. 374 (2020),
923-973.
[HK] Y. Hateruma and Y. Kaneko, On some Lie-theoretic solutions of the tt*-Toda equations
with integer Stokes data, preprint.
(Y. Hateruma) Department of Pure and Applied Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
Email address: wuyh2.17-1@asagi.waseda.jp

On the truss structures in architectural design
based on discrete isothermic surfaces
Yoshiki Jikumaru, Kazuki Hayashi, Kentaro Hayakawa,Yohei Yokosuka, and Kenji Kajiwara(Presenter: Yoshiki Jikumaru)

Abstract. A classical truss structure discovered by A. G. M. Michell [3] is
known as a structure which transmits loads eﬃciently with minimum possible
members. In this talk, we discuss a truss structure analogous to the Michelltruss constructed from discrete isothermic surfaces. We show that our proposed structure has not only nice mechanical properties and constructibility,
but also a deep connection with log-aesthetic curves.

1

Mechanical properties of discrete isothermic surfaces [1, 5]

Let us consider a map r : Z2 → R3 , where Z2 is a planar lattice and R3 is the 3dimensional Euclidean space. For a vertex (n, m) in Z2 , denote r = r(n, m), r (1) =r(n + 1, m), r (2) = r(n, m + 1), and r (12) = r(n + 1, m + 1). In the following, we assume
all the quadrilaterals [r, r (1) , r (12) , r (2) ] are planar and concircular. Such a map r is often
called a discrete curvature net or circular net. We also assume that the internal forces
F 1 , F 2 , F 1(1) and F 2(2) act tangentially at the midpoints of the edges, see Fig. 1.

(a)

(b)

(c)

Figure 1: (a): Internal forces act tangentially on a quadrilateral. (b): The force balancing
condition induces another (closed) quadrilateral. (c): Axial forces acting on the diagonals.
Theorem 1.1 (Schief [5], Theorem 3.2). Assume that purely tangential internal forces
act at the midpoints of the edge of a discrete curvature net r. Then r is in equilibrium if
and only if r constitutes a discrete isothermic surface, and the internal forces are encodedin the discrete Christoﬀel transform r.

The author was partly supported by JST CREST JPMJCR1911.

2

Michell truss-like structures [3, 2]

Under the assumption of Theorem 1.1, by replacing the quadrilaterals with their diagonals, we obtain a truss structure in equilibrium transmitting only by their axial forces,
see Fig. 1 (c). In particular, we consider a planar discrete isothermic surface which iscalled an (integrable) discrete holomorphic function√ or discrete conformal map [1, 4].
Let us consider a map r(n, m) = exp(ρn + −1κm), where ρ and κ are constant. A
direct calculation shows that r is discrete holomorphic if and only if
sinh2

ρ
κ
= sin2⇐⇒ cosh ρ + cos κ = 2.
2
2

(2.1)

holds. If ρ and κ satisfy this condition, then we call r discrete exponential function.
Moreover, the Christoﬀel transform r of r is given as follows:r(n, m) = −√1
exp(−ρn+−1κm).
4 sinh2 (ρ/2)(2.2)

We can check that the structure is certainly minimizing the sum of the products ofmember 1
member 2(a)member 4
member 3(b)

Figure 2: (a): A discrete exponential function and its Christoﬀel transform. The axial
forces acting on the inner member of the structure is greater. (b): The result of structural
analysis. The loading is applied in the direction of twisting the circumference.
absolute value of axial force and the member length numerically, which satisﬁes Michell’s
original problem [3] as mentioned. Through the inverse of the stereographic projection,
we obtain a truss structure on the unit sphere as shown in Fig. 3.
We ﬁnally remark that discrete curves obtained from the diagonals of the discrete exponential function are integrable discretization of log-aesthetic curves. The notion of logaesthetic curves (LAC) has been originally proposed by extracting the common properties
from thousands of plane curves that car designers regard as aesthetic.
It was shown that LAC can be characterized by the invariant
curves of integrable deformation of plane curves in similarity
geometry [2]. An integrable discretization of LAC (dLAC)
is proposed by Inoguchi et al. [2] as follows. A sequence ofpoints in the plane γn ∈ C (n ∈ Z) is called a discrete curve
and the angle κn = ∠(γn − γn−1 , γn+1 − γn ) is called theturning angle. We put qn := |γn+1 − γn | and un := qn+1 /qn . Figure 4: Notations for a
discrete curve γn
un is called the discrete similarity curvature.
Deﬁnition 2.1 ([2]). A discrete curve γn ∈ C is called adiscrete log-aesthetic curve (dLAC) of slope α if the following conditions are satisﬁed:

.Economy of Material in Frame-structures.

597

equal and directed to a point on the line of" action of thethird outside the triangle formed by the points of applicationof the forces. Fig. 4 is an example.
The sectors DCG ~, FCE ~ undergo uniform bulk compression, the remainder of the field pure shear as in fig. 2.
The volume of the frame is

1_)

1},

where a is the angle ECG.
4. (Fig. 5.) Equal and opposite couples applied at points
A~ B on the straight line AB.
Fig. 5.

mirrormember 1t]member 2

member 3
member 4(a)(b)

The minimum frame consists of the series of rhumb-lines
inclined a~ 45 ~ to the meridians of the sphere having its
poles at A and B.
All the rhumb-lines of one series and all similar lines on
spheres concentric with the spherical frame are uniformly
and equally extended, all the lines orthogonal to these on thespheres equally compressed.
The third system of orthogonals~ viz., radii from the centreof the sphere, are unchanged in length and simply rotatedabout the axis AB.
The minimum ~olume of the frame is(c)

Figure 3: (a): A truss structure on the unit sphere by the inverse of the streographic
projection. The loading is applied in the direction of twisting the equator. (b): A
structure attached along the equator. (c): A structure described2L in [3].
1~ l~ tan ( 4 § 2 ) • (~- -{- ~),1. γn has constant turning angle κn = κ.

where L is the moment of the transmitted couple, 2a----AB,and X is the latitude of small circles about each polecorresponding to the'circle of radius r0 in Case 1.
Melbourn%April 19, 1904.

2. un satisﬁes the equation
1
(
) α−1
(α − 1)λκun = 1 +if α ̸= 1,(α − 1)λκn + 1

un = eλκ if α = 1,

(2.3)

where λ is a constant.
The discrete curve γn obtained from the diagonals of a discrete exponential function r
can be written in the form√
γn = r 0 exp((ρ ± −1κ)n), n ∈ Z,
(2.4)
where r 0 ∈ C is a ﬁxed point. Then we can show the following result:
Proposition 2.2. The discrete curve γn in the form (2.4) has a constant turning angle
κ, and satisﬁes un = eρ . In other words, γn is dLAC of slope 1.
An LAC of slope 1 corresponds to a logarithmic spiral in the smooth setting. Therefore a
discrete curve deﬁned in the form (2.4) can be regarded as a discrete logarithmic spiral.

References[1] A. Bobenko, and U. Pinkall, Discrete isothermic surfaces, J. reine angew. Math., Vol. 30,
4 (1996), 187–208.
[2] J. Inoguchi, Y. Jikumaru, K. Kajiwara, K. T. Miura, and W. K. Schief, LogAesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles, arXiv:1808.03104.
[3] A. G. M. Michell, The Limits of Economy of Material in Frame Structures. Philosophical
Magazine, Series 6, 8 (1904), 589-597.
[4] F. Nijhoﬀ, and H. Capel, The discrete Korteweg-de Vries equation. Acta Appl. Math. 39(1995) 133–158.
[5] W. K. Schief, Integrable structure in discrete shell membrane theory, Proc. R. Soc. A, Vol.
470 (2014), 20130757.
(Y. Jikumaru) Institute of Mathematics for Industry, Kyushu University, 744, Motooka, Nishiku, Fukuoka, 819-0395, Japan
Email address: y-jikumaru@imi.kyushu-u.ac.jp

A nonclassical algebraic solution of a 3-variable
irregular Garnier system
Arata Komyo

This talk is based on [4]. So a nonclassical algebraic solution of a 3-variable irregular
Garnier system is constructed. Diarra–Loray have studied classiﬁcation of algebraic solutions of irregular Garnier systems ([1]). There are two type of the algebraic solutions:classical type and pull-back type. They have shown that there are exactly three nonclassical algebraic solutions for N -variables irregular Garnier systems with N > 1. Explicit
forms of two of the three solutions are already given. However, an explicit form of theremaining algebraic solution is not given in [1]. This is because the corresponding explicit irregular Garnier system had not been known. On the other hand, there are studies
of explicit description of isomonodromic deformations of second order linear diﬀerential
equations with irregular singularities by using apparent singularities ([2, 3]). By applying
these studies, we construct an explicit form of the remaining algebraic solution.

References[1] K. Diarra, F. Loray, Classiﬁcation of algebraic solutions of irregular Garnier systems. Compos. Math. 156 (2020), no. 5, 881–907.
[2] K. Diarra, F. Loray, Normal forms for rank two linear irregular diﬀerential equations andmoduli spaces. Period Math Hung (2021).
[3] A. Komyo, Description of generalized isomonodromic deformations of rank two linear differential equations using apparent singularities, to appear in Publications of the ResearchInstitute for Mathematical Sciences (arXiv:2003.08045).
[4] A. Komyo, A nonclassical algebraic solution of a 3-variable irregular Garnier system, toappear in Funkcialaj Ekvacioj (arXiv:2205.14979).
(A. Komyo) Center for Mathematical and Data Sciences, Kobe University, 1-1 Rokkodai-cho,Nada-ku, Kobe, 657-8501, Japan
Email address: akomyo@math.kobe-u.ac.jp

The author was partly supported by Japan Society for the Promotion of Science KAKENHI Grant
Numbers 22H00094 and 19K14506.

Deformation space of circle patterns
Wai Yeung Lam

Abstract
William Thurston proposed regarding the map induced from two circle packings withthe 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 this theory 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 with complex 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 report
updates on the conjecture and explain the extension to inﬁnite circle patterns on open
disks.

References[1] Lam, W.Y. Quadratic diﬀerentials and circle patterns on complex projective tori. Geom.
Topol.. 25, 961-997 (2021)(W. Y. Lam) University of Luxembourg, Department of Mathematics, Maison du nombre, 6
avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg.
Email address: wai.lam@uni.lu

The author was partly supported by CoSH, FNR grant O20/14766753.

Moduli space of rank three logarithmic connections
on the projective line with three poles
Takafumi Matsumoto

Abstract(1)∗

An A2 -surface is a smooth projective rational surface in Sakai’s classiﬁcation ofPainlevé equations. Arinkin and Borodin noted that moduli space of rank three logarithmic connections on the projective line with three poles is isomorphic to the surface(1)∗
from an A2 -surface minus an anti-canonical divisor from the viewpoint of linear difference equations and the Mellin transform. In this talk, I will explain how to describe
the moduli space directly. This method is based on the approach to the Painlevé VIcase by Inaba-Iwasaki-Saito. Using this method, we can prove that moduli space of certain parabolic ϕ-connections, which is a compactiﬁcation of moduli space of logarithmic(1)∗
parabolic connection, is isomorphic to an A2 -surface.
(T. Matsumoto) Departmant of Mathematics, Graduate School of Science, Kobe University,
Kobe, Rokko 657-8501, Japan
Email address: tmatumt@math.kobe-u.ac.jp

The author was partly supported by JSPS KAKENHI Grant Numbers 22J10695.

Generalizations of the Hermitian-Einstein equation
for cyclic Higgs bundles
Natsuo Miyatake
We introduce some generalizations of the Hermitian-Einstein equation for diagonalharmonic metrics on cyclic Higgs bundles, including a generalization using (quasi) subharmonic functions. When the coeﬃcients are all smooth, we prove the existence, uniqueness, and convergence of the solution of their heat equations with Dirichlet boundary
conditions. We also generalize two fundamental inequality estimates for solutions of theHermitian-Einstein equation of cyclic Higgs bundles. This talk is base on the content of
the paper [1].

References[1] N. Miyatake, Generalizations of Hermitian-Einstein equation of cyclic Higgs bundles, theirheat equation, and inequality estimates, preprint, arXiv:2301.01584.
(N. Miyatake) Institute of Mathematics for Industry, Kyushu University 744 Motooka, Fukuoka
819-0395, Japan
Email address: n-miyatake@imi.kyushu-u.ac.jp
Email address: natsuo.m.math@gmail.com

Nilpotent Lie algebras obtained by quivers and Ricci
solitons
Fumika Mizoguchi

Abstract
Nilpotent Lie groups with left-invariant metrics provide non-trivial examples of Ricci
solitons. A quiver is a directed graph where loops and multiple arrows between two
vertices are allowed. In this talk, we introduce a new method for obtaining nilpotent Liealgebras from ﬁnite quivers without cycles. For all of these Lie algebras, we prove that thecorresponding simply connected nilpotent Lie groups admit left-invariant Ricci solitons.
This constructs a large family of examples of Ricci soliton nilmanifolds with arbitrarily
high nilpotency steps.

References
[1]
[2]
[3]
[4]
[5]
[6]
[7]

Böhm, C. and Lafuente, R., Non-compact Einstein manifolds with symmetry,
arXiv:2107.04210.
Dani, S. G. and Mainkar, M., Anosov automorphisms on compact nilmanifolds associatedwith graphs, Trans. Amer. Math. Soc. 357 (2005), no. 6, 2235–2251.
Lauret, J., Ricci soliton homogeneous nilmanifolds, Math. Ann. 319 (2001), no. 4, 715–733.
Lauret, J., Ricci soliton solvmanifolds, J. Reine Angew. Math. 650 (2011), 1–21.
Lauret, J. and Will, C., Einstein solvmanifolds: existence and non-existence questions,Math. Ann. 350 (2011), no.1, 199–225.
Nikolayevsky, Y., Einstein solvmanifolds and the pre-Einstein derivation, Trans. Amer.
Math. Soc. 363 (2011), no.8, 3935–3958.
Tamaru, H., Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds,
Math. Ann. 351 (2011), no. 1, 51–66.

(F. Mizoguchi) Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Email address: m21sa032@st.osaka-cu.ac.jp

This work was partly supported by Osaka Metropolitan University Advanced Mathematical Institute(MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).

Minimal PF submanifolds in Hilbert spaces
with symmetries
Masahiro Morimoto
As a generalization of submanifolds in Euclidean spaces, one can consider submanifolds
in Hilbert spaces. In 1988, R. S. Palais and C.-L. Terng [5, 7] introduced a suitable classof submanifolds in Hilbert spaces, which are called proper Fredholm (PF) submanifolds.
It follows that the shape operators of PF submanifolds are compact self-adjoint operators.
Moreover one can apply the inﬁnite dimensional diﬀerential topology and Morse theory
to PF submanifolds. Palais and Terng [5, 7] gave examples of PF submanifolds which areorbits of the gauge transformations. Later, U. Pinkall and G. Thorbergsson [6] extended
those examples and Terng [8] eventually formulated them in the framework of path group
actions. Those actions are understood through an equivariant Riemannian submersion
called the parallel transport map [9] and moreover they have close relations to the inﬁnite
dimensional symmetric spaces called aﬃne Kac-Moody symmetric spaces [8, 1]. In thistalk, I will explain foundations of PF submanifolds in Hilbert spaces, introduce my results
concerning minimal PF submanifolds with symmetries [2, 3, 4], and discuss some future
works related to integrable systems.

References[1] E. Heintze, Toward symmetric spaces of aﬃne Kac-Moody type, Int. J. Geom. Methods
Mod. Phys. 3 (2006), no. 5–6, 881–898.
[2] M. Morimoto, Austere and arid properties for PF submanifolds in Hilbert spaces, Diﬀerential Geom. Appl., 69 (2020) 101613, 24 pp.
[3] M. Morimoto, On weakly reﬂective PF submanifolds in Hilbert spaces, Tokyo J. Math. 44(2021), no. 1, pp. 103–124.
[4] M. Morimoto, Curvatures and austere property of orbits of path group actions induced byHermann actions, Transform. Groups (2022). https://doi.org/10.1007/s00031-022-09732-w[5] R. S. Palais, C.-L. Terng, Critical Point Theory and Submanifold Geometry, Lecture Notes
in Mathematics, 1353. Springer-Verlag, Berlin, 1988.
[6] U. Pinkall, G. Thorbergsson, Examples of inﬁnite dimensional isoparametric submanifolds,
Math. Z. 205 (1990), no. 2, 279–286.
[7] C.-L. Terng, Proper Fredholm submanifolds of Hilbert space. J. Diﬀerential Geom. 29(1989), no. 1, 9–47.
[8] C.-L. Terng, Polar actions on Hilbert space. J. Geom. Anal. 5 (1995), no. 1, 129–150.
[9] C.-L. Terng, G. Thorbergsson, Submanifold geometry in symmetric spaces. J. Diﬀerential
Geom. 42 (1995), no. 3, 665–718.
(M. Morimoto) Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Email address: morimoto.mshr@gmail.com
This work was partly supported by the Grant-in-Aid for Research Activity Start-up (No. 20K22309)
and by Osaka Central Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP06192178499), Osaka Metropolitan University.

Wall-crossing formula for framed quiver moduli
Ryo Ohkawa

Abstract. We study wall-crossing phenomena of moduli of framed quiver
representations. These spaces are expected to be very useful to capture representation theoretic nature of special functions for integrable systems. Amongthese moduli spaces, we have type A ﬂag manifold, type A aﬃne Laumon
spaces, Nakajima quiver variety, and framed moduli of sheaves on blow-ups.
In particular, we study wall-crossing formulas for integrals of the Euler classes
of weighted sums of tautological bundles and tangent bundles of these moduli
spaces.

1

Introduction

Framed quiver representation is a quiver representation with a framing. Moduli offramed quiver representations are studied by Reineke [10]. It is expected to be very useful to capture representation theoretic nature of special functions for integrable systems.
Among these moduli spaces, we have type A ﬂag manifold, type A aﬃne Laumon spaces,
Nakajima quiver variety, and framed moduli of blow-ups.
Mochizuki [3] studied wall-crossing formula for moduli of parabolic sheaves on surfaces.
Nakajima-Yoshioka [6] translated it to quiver language from the context of Nekrasov partition functions.
Nekrasov’s conjecture states that these partition functions give deformations of the
Seiberg-Witten prepotentials for N = 2 SUSY Yang-Mills theory. This conjecture isproven in Braverman-Etingof [1], Nekrasov-Okounkov [4] and Nakajima-Yoshioka [5] independently.
On the other hand, another types of wall-crossing formulas are also studied in [7]where walls coincide with hyperplanes perpendicular to imaginary roots. In this paper,
we summarize these techniques in general settings of framed quiver moduli.

2

Main results

Let Q = (Q0 , Q1 , Q2 ) be a quiver with relations, where Q0 is the set of vertices, Q1 isthe set of arrows, and Q2 consists of linear combinations of paths with the same beginningand ending vertices in Q0 . Here path is a composable sequence of arrows. For each arrowa ∈ Q1 , we write by out(a) and in(a) the beginning and the ending vertices in Q0 .

The author was partly supported by Grant-in-Aid for Scientiﬁc Research 21K03180 and 17H06127,
JSPS, Osaka Central Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849, and by the Research Institute for Mathematical
Sciences, an International Joint Usage/Research Center located in Kyoto University.

We consider a ﬁnite dimensional Q0 -graded vector space V =representation B on V is a collection of linear mapsY
B = (Bh ) ∈HomC (Vout(h) , Vin(h) )Lv∈Q0 Vv .

A Q-

a∈Q1

satisfying relations deﬁned from Q2 . In this talk, we assume that there exists a vertex∞ ∈ Q0 such that dim V∞ = 1. We call ∞ a framing vertex, and put I = Q0 \ {∞}.
We take stability parameter ζ = (ζi )i∈I ∈ RI deﬁning stability conditions for Qrepresentations in a certain way. For the dimension vector α = (dim Vv )v∈Q0 ∈ (Z≥0 )Q0 of
V , we write by M ζ (α) = MQζ (V ) moduli of ζ-semistable Q-representations on V .
We take another dimension vector β = (βi )i∈I ∈ (Z≥0 )I and a stability parameter ζ̄from the hyperplane()
X
β ⊥ = ζ = (ζi )i∈I ∈ RI
ζi βi = 0i∈Iwhich is not perpendicular to any other direction of dimension vectors. We have twochambers whose boundaries containing ζ̄, and write by C+ the one whose element ζ satisfy(ζ, β) < 0, and by C− the other one.
+
For ζ ± ∈ C± , we develop method to compare equivariant integrals over M ζ (α) and−M ζ (α), and derive wall-crossing formula among them generalizing the previous works[7], [8] and [9] based on [3], [6]. Furthermore, we apply this formula to the Euler classes
of weighted sums of tautological bundles and tangent bundles.

References[1] A. Braverman and P. Etingof, Instanton counting via aﬃne Lie algebras II: from Whittakervectors to the Seiberg-Witten prepotential, In: Bernstein J., Hinich V., Melnikov A., (eds),
Studies in Lie theory, Progr. Math. 243, 61–78, Birkhauser, Boston (2006)[2] M. Bershtein and A. Shchechkin, Bilinear equations on Painlevé τ functions from CFT,
Comm. Math. Phys. 339 (2015), no. 3, 1021–1061.
[3] T. Mochizuki, Donaldson Type Invariants for Algebraic Surfaces: Transition of Moduli
Stacks, Lecture Notes in Math. 1972, Springer, Berlin, 2009.
[4] N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, In: Etingof
P., Retakh V. S., Singer, I.M., (eds), The unity of mathematics, Progr. Math. 244, 525–596,
Birkhauser Boston, Boston, MA, (2006)[5] H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge
theory, Invent. Math. 162 (2005), no. 2, 313–355.
[6] H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blowup. III. Blow-up formulafrom wall-crossing, Kyoto Journal of Mathematics, Vol. 51, No. 2 (2011), 263–335.
[7] R. Ohkawa, Wall-crossing between stable and co-stable ADHM data, Lett. Math. Phys. 108(2018), no. 6 1485–1523.
[8] R. Ohkawa, Functional Equations of Nekrasov Functions Proposed by Ito, Maruyoshi, andOkuda, Moscow Math. J. 20 (2020), no. 3, 531–573.
[9] R. Ohkawa and Y. Yoshida, Wall-crossing for vortex partition function and handsaw quivervarierty[10] M. Reineke, Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9 (2010),
no. 3, 653–667.

(R. Ohkawa) Osaka Central Advanced Mathematical Institute, 3-3-138, Sugimoto, Sumiyoshiku, Osaka, 558-8585, Japan
Email address: ohkawa.ryo@omu.ac.jp

Constructing isometric tori with the same curvatures
Andrew O. Sageman-Furnas

Abstract
Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic
mean curvature function. However, there are exceptions. These may arise in a family likethe isometric family of vanishing mean curvature surfaces transforming a catenoid into ahelicoid, or as a so-called Bonnet pair of surfaces.
For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and nonconstant mean curvature function determine at most one immersion with genus zero, but
at most two compact immersions (compact Bonnet pairs) for higher genus.
In this talk, we discuss our recent construction of the ﬁrst examples of compact Bonnet
pairs. It uses a local classiﬁcation by Kamberov, Pedit, and Pinkall in terms of isothermic
surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic
surfaces and Bonnet pairs led to this discovery.
The smooth theory is joint work with Alexander Bobenko (Technical University of
Berlin, Germany) and Tim Hoﬀmann (Technical University of Munich, Germany) andthe discrete theory is joint work with Tim Hoﬀmann and Max Wardetzky (University ofGöttingen, Germany).
(A.O. Sageman-Furnas) Department of Mathematics, North Carolina State University, Raleigh,NC 27607, USA.
Email address: asagema@ncsu.edu

Some submanifolds of the associative Grassmann
manifold
Yuuki Sasaki
A 4-dimensional subspace of the octonions O satisfying the associative law is called an
associative subspace and the associative Grassmann manifold GH (O) is the set of all associative subspaces. The associative Grassmann manifold is introduced by Harvey-Lawson[3]. The associative Grassmann manifold is a type G irreducible compact symmetric space.
Moreover, the associative Grassmann manifold has a quaternionic Kähler structure andbecomes a compact quaternionic symmetric space.
In recently, Enoyoshi-Tsukada construct a harmonic map from a Lagrangian submanifold of S 6 from GH (O) [2]. In this talk, we introduce a construction of a totally complex
harmonic immersion from an almost complex submanifold of S 6 into GH (O) by using a
Gauss map. Using similar arguments, we introduce a construcion of a CR immersion
from a 3-dimensional CR submanifold of S 6 into GH (O).
Moreover, if time permits, we will introduce the following results. Enoyoshi-Tsukada
prove that a polar of GH (O) is an image of a totally complex immersion, where a polaris a connected component of the ﬁxed point set of a geodesic symmetry [1]. It is known
that a polar is obtained as an orbit of the action of the isotropy group of the isometrygroup. In this talk, we observe some properties of each orbit of the isotropy group action
with respect to the quaternionic Kähler structure. In particular, we observe that many
orbits are an image of a CR immersion which has similar properties of a totally complex
immersion.

References[1] K. Enoyoshi, K. Tsukuda, Examples of transversally complex submanifolds of the associative Grassmann manifold, Tsukuba J. Math. vol.43, No.1(2019), 23-36[2] K. Enoyoshi, K. Tsukada, Lagrangian submanifolds of S 6 and the associative Grassmann
manifold, Kodai Math. J. 43(2020), 170-192[3] R. Harvey, H. B. Lawson, Calibrated geometries, Acta Math. 148(1982), 47-157(Yuuki Sasaki) National Institute of Technology, Tokyo College, 1220-2, Kunugida-machi, Hachiojishi, Tokyo, 193-0997, Japan
Email address: y sasaki@tokyo.kosen-ac.jp

Integrable systems in four independent variables
related to contact structures in dimension three
Artur Sergyeyev
We start with a review of our construction [1] (cf. also [2]) that gave rise to a large newclass of nonlinear partial diﬀerential systems in four independent variables integrable inthe sense of soliton theory using a certain contact structure in dimension three. We thenproceed to explore the possibility of extending this construction to more general contact
structures in the same dimension.

References[1] A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math.
Phys. 108 (2018), no. 2, 359-376 (arXiv:1401.2122)[2] A. Sergyeyev, Integrable (3+1)-dimensional system with an algebraic Lax pair, Appl. Math.
Lett. 92 (2019), 196-200 (arXiv:1812.02263)(A. Sergyeyev) Mathematical Institute, Silesian University in Opava, Na Rybnı́čku 1, 74601
Opava, Czech Republic
Email address: artur.sergyeyev@math.slu.cz

On a heat equation of SL(2, R), and a future directionof its q-deformation
Masafumi Shimada

Abstract
In this talk, we discuss several formulae of the heat kernel on SL(2, R), and a futureapplication of these formulae to a q-deformed heat equation on a quantum group. In
previous studies on the heat kernel in a unimodular Lie group of type I, including thecase of SL(2, R), we use a noncommutative-harmonic-analytic technique, called generalized noncommutative Fourier transform. Our formula is, with the help of weighted Maass
Laplacians, another explicit form of the heat kernel on SL(2, R), which explains an intrinsic structure of the heat kernel in terms of hyperbolic geometry.
We recall the fact that, given a parabolic partial diﬀerential equation on a Riemannian
manifold, its heat kernel plays an important role in its initial value problem. Motivatedby this fact, we hopefully apply a q-deformation of our heat kernel formula to a study
on a q-diﬀerence heat equation of a quantum group SLq (2) in the context of quantum
Riemannian geometry.
(M. Shimada) Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuokashi,Fukuoka, 819-0395, Japan
Email address: m.shimada.a90@s.kyushu-u.ac.jp

Geometry of anisotropic double crystals andclassiﬁcation of some examples
Eriko Shinakwa, Miyuki Koiso(Presenter: Eriko Shinkawa)
There was a long-standing conjecture which was called the double bubble conjecture.
It says that the standard double bubble provides the least-perimeter way to enclose andseparate two given volumes, here the standard double bubble is consisting of three spherical caps meeting along a common circle at 120-degree angles. This conjecture had been
believed since about 1870 and was proved in 2002 by M. Hutchings et al. in R3 , and a
student of F. Morgan extended it to higher dimensions.
Double bubbles are a mathematical model of soap bubbles. The energy functional
is the total area of the surface. On the other hand, when we think about a mathematical model of anisotropic substance like crystals, we need to consider the energy density function γ : S n → R+ depending on the normal direction ν of the surface, whereS n = {X ∈ Rn+1 | |X| = 1} is the n-dimensional unit sphere in Rn+1 . γ is called ananisotropic energy density function, and its sum (integral) over the surface is called ananisotropic energy. The surface is a constant anisotropic mean curvature (CAMC) surface if it is a critical point of the anisotropic energy for all volume-preserving variations.
CAMC surfaces are a generalization of constant mean curvature surfaces.
In this study, we extend the double bubble problem to a double crystal (DC) problem,that is, we minimize the anisotropic energy instead of the surface area. The solutions are amathematical model of multiple crystals. There were some previous studies relating to the
DC problem. G. R. Lawlor ([1]) determined the energy-minimizer for the case where eachenergy density function γi (i = 0, 1, 2) is constant. For n = 1, F. Morgan et al. studied theshapes of the minimizers of the total anisotropic energy of curves among curves enclosingprescribed areas. Especially, they determined the shapes of all minimizers for the case ofγi (ν1 , ν2 ) = |ν1 | + |ν2 | (i = 0, 1, 2) ((ν1 , ν2 ) ∈ S 1 ) ([2]).
There is a unique hypersurface that minimizes the anisotropic energy among all closed
hypersurfaces enclosing the same volume, and this hypersurface is known as the Wulﬀ
shape. In this study, we assume that the Wulﬀ shape is smooth. We will derive the ﬁrst
variation formula for the anisotropic energy and obtain the conditions for a hypersurface
to be a double crystal ([3], [4]). In particular, for n = 1, for a kind of special energy
density functions, some of which approximate the energy studied in [2], we classify the
double crystals in terms of symmetry and the given areas.

References[1] G. R. Lawlor, Double Bubbles for Immiscible Fluids in Rn J. Geom. Anal. 24 (1), 190–204(2014).
This work was partly supported by JST CREST Grant Number JPMJCR1911, JSPS KAKENHI
Grant Numbers JP18H04487, JP20H04642 and JP20H01801.

[2] F. Morgan, C. French and S. Greenleaf, Wulﬀ Cluster in R2 , J. Geom. Anal. 8 (1), 97–115(1998).
[3] M. Koiso and E. Shinkawa, Geometry of anisotropic double crystals, to appear in JSIAM
Letters (2023).
[4] M. Koiso and E. Shinkawa, Geometry of anisotropic double crystals and classiﬁcation ofsome examples, in preparation.
(E. Shinkawa) Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira,Aoba-ku, Sendai 980-8577, Japan
Email address: eriko.shinkawa.e8@tohoku.ac.jp(M. Koiso) Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku,
Fukuoka 819-0395, Japan
Email address: koiso@imi.kyushu-u.ac.jp

Smooth and discrete constrained elastic curves inspace forms
Gudrun Szewieczek
Euler’s famous elastic curves in the Euclidean plane are obtained as the critical pointsof the bending energy while ﬁxing the length of the curve. If, additionally, also the sector
area is constrained, then the curve is called constrained elastic. Alternative characterizations in terms of the curvature [6] or the existence of a special ’axis’ [1, 4] allow for
natural generalizations to space forms.
Recent interest in constrained elastic curves in space forms is caused by the fact thatthose arise as planar or spherical curvature lines of many integrable surface classes: forexample, on constrained Willmore tori [3, 5], isothermic surfaces with planar curvaturelines [2] and, more generally, on Lie applicable surfaces [4].
In this talk we shall give an overview on smooth constrained elastic curves in space
forms and discuss an integrable discretization. The latter is joint work with Tim Hoﬀmann
and Jannik Steinmeier.

References[1] G. Arreaga, R. Capovilla, C. Chryssomalakos and J. Guven, Area-cosntrained planar elastica. Physical Review E 65:031801 (2002)[2] A.I. Bobenko, T. Hoﬀmann and A.O. Sageman-Furnas, Compact Bonnet Pairs: isometric
tori with the same curvatures. arXiv:2110.06335 [math.DG] (2021)[3] C. Bohle, G. Peters and U. Pinkall, Constrained Willmore surfaces. Calculus of Variations
32:263–277 (2008)[4] J. Cho, M. Pember and G. Szewieczek, Constrained elastic curves and surfaces with spherical curvature lines. Indiana University Mathematics Journal, accepted (2022)[5] L. Heller, Constrained Willmore tori and elastic curves in 2-dimensional space forms. Communications in Analysis and Geometry 22(2), 343–369 (2014)[6] J. Langer and D. Singer, The total squared curvature of closed curves. Journal of Diﬀerential
Geometry 20(1):1–22 (1984)(G. Szewieczek) Technical University of Berlin, Institute of Mathematics MA 8-4, 10623 Berlin,
Germany
Email address: szewieczek@math.tu-berlin.de

The author was supported by DFG via SFB-TRR 109 ”Discretization in Geometry and Dynamics”.

A maximal element of a moduli space of Riemannian
metrics
Yuichiro Taketomi

1Introduction

Let X be a connected smooth manifold, and denote by M(X) the set of all smooth
Riemannian metrics on X. Deﬁne a equivalent relation ∼ on M(X) as follows:g ∼ h :⇔

There exists some c > 0 such that two Riemannian manifolds(X, cg) and (X, h) are isometric with each other.

Denote by [g] the equivalent class of g ∈ M(X) with respect to ∼. Now deﬁne a preorder≺ on the moduli space M(X)/∼ as follows:[g] ≺ [h] :⇔For all g ′ ∈ [g] there exists some h′ ∈ [h] such that
Isom(X, g ′ ) ⊂ Isom(X, h′ ).

Note that the preorder ≺ is not a partial order in general. Namely, the order ≺ satisﬁes
reﬂexivity and transitivity, but does not satisfy asymmetry.
Deﬁnition (T.). A Riemannian metric g ∈ M(X) is said to be maximal if the equivalentclass [g] is a maximal element of the preordered space (M(X)/∼ , ≺) (i.e. [g] ≺ [h] implies[g] = [h]).
One can see that a Riemannian metric g ∈ M(X) is maximal if and only if
Isom(X, g) ⊂ Isom(X, h) ⇒ [g] = [h].

(♯)

Now we consider a metric evolution equation ∂t gt = R(gt ) on X. Here, R is a map from
M(X) to the set of all symmetric (0, 2)-tensor on X. A typical example is a Ricci ﬂow(i.e. R = −2Ric). A solution {gt }t∈[0,T ) of a metric evolution equation is said to be
self-similar if [g0 ] = [gt ] for all t ∈ [0, T ). By the property (♯), one has
Proposition. Let ∂t gt = R(gt ) be a metric evolution equation, and {gt }t∈[0,T ) be a solutionof the equation whose initial metric g0 is a maximal metric. If the solution {gt }t∈[0,T )preserves the isometry group in the sense that Isom(X, g0 ) ⊂ Isom(X, gt ) for all t ∈ [0, T ),then the solution {gt }t∈[0,T ) is self-similar.
It is known that a solution of the Ricci ﬂow whose initial metric is complete andhas bounded curvature exists uniquely ([1, 3]), and hence preserves the isometry group.
Hence a complete maximal metric with bounded curvature (in particular, a homogeneous
maximal metric) is a Ricci soliton. Maximal metrics give examples of self-similar solutions
not only for the Ricci ﬂow but also for any geometric evolution equations which preserve
isometry groups.
The author was partly supported by JSPS KAKENHI Grant Number JP22K13916.

2

Examples of maximal metrics

Recall that a Riemannian manifold (X, g) is called an isotropy irreducible if theisotropy representation Isom(X, g)p ↷ Tp X is an irreducible representation for eachp ∈ X. Many geometer have studied the classiﬁcation problem of isotropy irreducible
spaces (e.g. [7, 8]), and in the end, complete connected isotropy irreducible spaces havebeen classiﬁed by Wang-Ziller ([5]). By the Schur’s lemma, one has
Proposition. For a complete connected isotropy irreducible space (X, g), the metric g isa maximal metric.
Conversely, for the compact case, maximal metrics are isotropy irreducible metrics:
Theorem (T.). Let (X, g) be a compact connected Riemannian manifold. If g is maximal
then (X, g) is isotropy irreducible.
On the other hand, one can show that there are many examples of maximal metrics
on the noncompact manifold Rk . For examples,
Proposition (T.). For W = (w2 , w3 , . . . , wn ) ∈ Rn−1 , the following Riemannian metricgW ∈ M(Rn ) is a maximal metric:gW := (dx1 )2 + e−2w2 x1 (dx2 )2 + · · · + e−2wn x1 (dxn )2 .
The metric gW is isotropy irreducible if and only if w2 = w3 = · · · = wn .
Note that the Riemannian metric gW is isometric to some left-invariant metric onsome simply connected solvable Lie group. Two metrics gW , gW ′ ∈ M(Rn ) are isometric if
and only if the two subspaces W, W ′ ∈ Rn−1 coincide under some permutation σ ∈ Sn−1 .
Hence the family {gW }W ∈Rn−1 gives a continuous family of maximal metrics.
Let G be a simple ﬁnite graph with p vertices and q edges. Denote by V := {v1 , v2 , . . . , vp }and E := {ep+1 , ep+2 , . . . , ep+q } the set of (numbered) vertices and the set of (numbered)edges of G, respectively. For j ∈ {p + 1, . . . , p + q}, deﬁne sj , tj ∈ {1, 2, . . . , p} as follows:
two vertices vsj and vtj are jointed by the edge ej , where sj < tj .
Then we deﬁne a Riemannian metric hG ∈ M(Rp+q ) as follows:
hG =

∑
i≤p

(dxi )2 +∑1
1(dxj + xtj dxsj − xsj dxtj )2 .
2
2
j≥p+1

Note that the Riemannian metric hG is isometric to some left-invariant metric on some
simply connected nilpotent Lie group ([2]).
Proposition (T.). Let G = (V, E) be a simple ﬁnite edge-transitive graph with the vertex
set V = {v1 , v2 , . . . , vp } and the edge set E = {ep+1 , ep+2 , . . . , ep+q }. Then the metrichG ∈ M(Rp+q ) is a maximal metric which is not isotropy irreducible.
Here, a simple graph G = (V, E) is called an edge-transitive graph if the automorphismgroup Aut(G) acts on the edge set E transitively. Note that two metrics hG and hG ′ areisometric if and only if the two graphs G and G ′ are isomorphic as graph ([2, 6]). Hencethe family {hG }G gives inﬁnitely many examples of maximal metrics.

References[1] B.-L. Chen and X.-P. Zhu, Uniqueness of the Ricci ﬂow on complete noncompact manifolds,
J. Diﬀerential Geom. 74 (2006), no. 1, 119–154.
[2] M. G. Mainkar, Graphs and two-step nilpotent Lie algebras, Groups Geom. Dyn. 9 (2015),
no. 1, 55–65.
[3] W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Diﬀerential Geom.
30 (1989), no. 1, 223–301.
[4] Y. Taketomi, A maximal element of a moduli space of Riemannian metrics,
arXiv:2210.01483 (2022).
[5] M. Wang and W. Ziller, On isotropy irreducible Riemannian manifolds, Acta Math. 166(1991), no. 3-4, 223–261.
[6] E. N. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (1982),
no. 3, 337–346.
[7] J. A. Wolf, The goemetry and structure of isotropy irreducible homogeneous spaces, Acta
Math. 120 (1968), 59–148.
[8] J. A. Wolf, Correction to: “The geometry and structure of isotropy irreducible homogeneous spaces” [Acta Math. 120 (1968), 59–148; MR 36 #6549], Acta Math. 152 (1984),
no. 1-2, 141–142.
(Y. Taketomi) Osaka Central Advanced Mathematical Institute, 3-3-138, Sugimoto, Sumiyoshiku, Osaka, 558-8585, Japan
Email address: v21255l@omu.ac.jp

Immersed special Lagrangians and mean curvature
ﬂow
Albert Wood

Abstract
PDE gluing constructions have become an indispensable tool in modern geometricanalysis. To give just a couple of examples, Joyce utilised gluing constructions bothto provide the ﬁrst examples of compact G2 manifolds and to desingularise special Lagrangians with conical singularities. Recent pioneering work of Brendle-Kapouleas describes a parabolic gluing construction for Ricci ﬂow, giving an example of an ancient
ﬂow beginning at a singular torus quotient.
This talk is based on upcoming joint work with Chung-Jun Tsai and Wei-Bo Su, inwhich we develop a parabolic gluing construction for Lagrangian submanifolds in CalabiYau manifolds, inspired by the work of Joyce and Brendle-Kapouleas. Explicitly, we will
demonstrate that any special Lagrangian submanifold in a Calabi-Yau manifold L ⊂ Mwith a single immersed point x ∈ L is the inﬁnite-time singular limit of a mean curvaturee → M , t ∈ [Λ, ∞).
ﬂow of Lagrangian submanifolds Ft : L(A. Wood) Hsiu-Chi House, No 16-1 Siyuan Street, Zhongzheng District, Taipei City 100, Taiwan
Email address: awood@ncts.ntu.edu.tw
webpage: www.albert-maths.co.uk

The author was partly supported by National Taiwan University, Taipei.