International Research Network Project "SYMMETRY, TOPOLOGY and MODULI"

The 2nd OCAMI-KOBE-WASEDA Joint International Workshop on

Differential Geometry and Integrable Systems

Date March 14 (Tue)- March 17 (Fri), 2017
Place Osaka City University (Building E of Faculty of Science, Lecture Room E408)

Organizers Yoshihiro Ohnita(OCU, OCAMI Director), Wayne Rossman (Kobe University),
Martin Guest (Waseda University & Visiting Professor of OCAMI), Masashi Yasumoto (Kobe University)

Sponsors JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers  
(Oct. 2014-Mar. 2017)
"Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI"
(Osaka City University - Kobe University - Waseda University, Principal investigator: Yoshihiro Ohnita) 

Invited speakers Professor Franz Pedit (UMass Amherst, USA)*
Professor Christoph Bohle (University of Tuebingen, Germany)*
Professor Francis Burstall (University of Bath, UK)*
Professor Seiichi Udagawa (Nihon University, Japan)*
Professor Takashi Kurose (Kwansei Gakuin University, Japan)*
Professor Atsushi Fujioka (Kansai University, Japan)*
Professor Kentaro Saji (Kobe University, Japan)*
Professor Shintaro Yanagida(Nagoya University and OCAMI, Japan)*
Doctor Masashi Yasumoto(Kobe University and OCAMI, Japan)*
Doctor Atsuhira Nagano(Waseda University, Japan)*
Doctor Kohei Iwaki(Nagoya University and OCAMI, Japan)*
Doctor Hideo Takioka(OCAMI, Japan)*
Doctor Hitoshi Yamanaka(OCAMI, Japan)*
Doctor Toru Kajigaya(AIST and OCAMI, Japan)*
Doctor Kentaro Mitsui(Kobe University, Japan)*
Mister Shintaro Akamine(Kyushu University, Japan)*
Mister Joseph Cho(Kobe University, Japan)*
etc. (*: a confirmed speaker )

Title and Abstract of Talks abstract (in preparation)
Franz Pedit (1)
Title: Energy quantization for harmonic 2-spheres in non-compact symmetric spaces
Abstract: It is well known from results by Uhlenbeck, Chern-Wolfson and Burstall-Rawnsley that harmonic 2-spheres in compact symmetric spaces have quantized energies. Using the reformulation of the harmonic map equation as a family of flat connections, we construct an energy preserving duality between harmonic maps from non-compact symmetric spaces into their compact duals. Applying this construction to the conformal Gauss maps of Willmore 2-spheres in the n-sphere provides a generalization and unifying approach to existing quantization results in special cases: Bryant for n=3; Montiel for n=4; and Ejiri for Willmore 2-spheres admitting a dual Willmore surface.
Franz Pedit (2)
Title: Classification of homogeneous Willmore 2-spheres in the n-sphere
Abstract: The simplest surfaces in any ambient geometry are those which arise as orbits via a representation into the symmetry group defining the geometry. For Willmore surfaces in n-space this symmetry group is the Moebius group. We show that every Moebius homogeneous Willmore 2-sphere is either a round 2-sphere or is Moebius congruent to a Veronese minimal 2-sphere in a constant curvature 2m-sphere given by the irreducible SO(3) representation in 2m+1.
Francis Burstrall
Title: Integrable systems in conformal and Lie sphere geometry
Abstract: I shall give an overview of some geometric integrable systems that arise in conformal and Lie sphere geometry. I shall describe isothermic surfaces and Moebius-flat submanifolds (which include Guichard surfaces and conformally flat hypersurfaces) in conformal geometry and their transformation theory. Then I shall show how these are special cases of an integrable class of Lie applicable submanifolds in a Lie sphere geometry which is, itself, novel in higher codimension.
Christoph Bohle
Title: Differential Geometry and mcKP
Abstract: In my talks I want to explain how the multi-component KP hierarchy (mcKP) can be related to differential geometry. After briefly going through the general scheme behind this relation, I plan to focus on objects of classical differential geometry (like multi-dimensional conjugate nets...) and applications to mathematical physics.
Seiichi Udagawa
Title: Finite gap solutions for horizontal minimal surfaces of finite type in 5-sphere
Abstract: PDF
Takashi Kurose
Title: Miura transformation and geometry of curves
Abstract: The Miura transformation, which played an important role in the development of the theory of integrable systems, relates the Korteweg-de Vries (KdV) equation and the modified KdV equaion. On the other hand, it is known that various integral systems naturally appear in geometry of curves. In this talk, we shall review the cases of the KdV equation and of the modified KdV equaion, and shall discuss geometric counterparts of the Miura transformation.
Atsushi Fujioka
Title: Centroaffine surfaces with parallel or recurrent cubic form relative to the induced connection
Abstract: The cubic form of a hypersurface in the affine space is an important invariant in affine differential geometry. In this talk, we shall give some classification results for nondegenerate centroaffine surfaces in terms of the cubic form and the induced connection.
Kentaro Saji
Title: Geometric foliations of fronts
Abstract: In this talk, generic classification of geometric foliations near singular points of fronts and frontals will be discussed. In particular, a normal form of swallowtail will be introduced and asymptotic curves and characteristic curves near swallowtail will be given.
Masashi Yasumoto
Title: Trivalent maximal surfaces in Minkowski space
Abstract: In this talk we introduce trivalent maximal surfaces in Minkowski 3-space based on the combination of two ideas below. In particular, we see that such trivalent maximal surfaces admit associated families and they also have variational properties and certain singularities.
Lam investigated two types of trivalent surfaces in Euclidean 3-space with vanishing mean curvature, which are called trivalent minimal surfaces. The advantage of this discretization is that we can treat both integrable geometric aspects of such minimal surfaces and their variational properties, which generalize many previous works.
On the other hand, Yasumoto described quadrilateral surfaces (or, discrete surfaces) in Minkowski 3-space with mean curvature identically 0, which are called discrete maximal surfaces. Unlike the case of discrete surfaces in Euclidean 3-space with vanishing mean curvature, it is shown that discrete maximal surfaces generally have singularities, which occurs only in the case of discrete constant mean curvature surfaces in Lorentzian spaceforms.
This talk is based on joint work with Wai Yeung Lam (Brown University).
Atsuhira Nagano
Title: Toric K3 hypersurfaces and a Shimura variety
Abstract: In the 19th century, elliptic modular functions appeared in the study of elliptic curves. Special values of elliptic modular functions generate class fields over imaginary quadratic fields (Kronecker's Jugendtraum). In this talk, the speaker will present an extension of the classical result by using K3 surfaces given by hypersurfaces in toric 3-folds. Namely, K3 surfaces are 2-dimensional analogy of elliptic curves and its periods give the moduli of K3 surfaces. Moreover, canonical models of Shimura varieties give natural extensions of Kronecker's Jugendtraum. We will see a construction of an explicit model of a Shimura variety via techniques of periods of toric K3 hypersurfaces.
Toru Kajigaya
Title: On Lagrangian submanifolds with symmetries
Abstract: Recently, Lagrangian submanifolds in a symplectic manifold M are of particular interests in several contexts, e.g., the Lagrangian intersection theory, the Hamiltonian volume minimizing problem and the Lagrangian mean curvature flows. In particular, homogeneous Lagrangian submanifolds, or more generally, Lagrangian submanifolds with symmetries give nice examples of Lagrangian submanifolds with several properties (e.g., minimal, special, monotone, Hamiltonian stable etc). In this talk, I explain some known results for Lagrangian submanifolds with symmetries in a symplectic or Kahler manifold by using the notions of Hamitonian actions and moment maps. After that, I introduce the following own results: (i) A classification result of homogeneous Lagrangian submanifolds in the complex hyperbolic spaces (a joint work with Takahiro Hashinaga). (ii) A Hsiang-Lawson type theorem for minimal Lagrangian submanifolds in a Kahler manifold.
Hideo Takioka
Title: Infinitely many knots with the trivial $(2,1)$-cable $\Gamma$-polynomial
Abstract: For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\Gamma$-polynomial of a knot is the $\Gamma$-polynomial of the $(p,q)$-cable knot of the knot, where the $\Gamma$-polynomial is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. In this talk, we show that there exist infinitely many knots with the trivial $(2,1)$-cable $\Gamma$-polynomial, that is, the $(2,1)$-cable $\Gamma$-polynomial of the unknot. Moreover, we see that the knots have the trivial $\Gamma$-polynomial, the trivial first coefficient HOMFLYPT polynomial, and the distinct Conway polynomials.
Kohei Iwaki
Title: Exact WKB analysis and related topics
Abstract: Exact WKB analysis, developed by Voros, is an effective method for global study of (singularly perturbed) ordinary differential equations defined on a complex domain. After recalling fundamental facts about exact WKB analysis, I’ll talk about relationships to other research topics, such as cluster algebras, topological recursion, integrable systems of Painlev\’e type, etc.
Hitoshi Yamanaka
Title: Invariant function on GKM-representation space
Abstract: After recalling several notions, such as GKM-representations, GKM-manifolds and representation coverings, I propose a conjecture concerning the existence of torus-invariant Morse functions on GKM-manifolds. Toward solving the conjecture, I will examine the Hessian matrix of invariant functions on a GKM-representation spaces at the origin. The result reveals a Morse-theoretic nature of the GKM-conditon.
Shintaro Yanagida
Title: Turaev's skein algebra for torus and a variant of homological mirror symmetry
Abstract: Turaev introduced skein algebras for surfaces as quantization of Goldman's Lie algebras. Morton and Samuelson showed that the skein algebra for torus coincides with a specialization of (the Drinfeld double of non-extended) Hall algebra of elliptic curves. We introduce a version of Hall algebra for elliptic curves over a field with one element, and show that the skein algebra of torus is isomorphic to our Hall algebra without specialization. We shall further explain a speculation on an extension of homological mirror symmetry for torus.
Kentaro Mitsui
Title: Models of torsors under elliptic curves
Abstract: Any smooth curve of genus one over a general base field can be given as a torsor (a principal homogeneous space) under an elliptic curve. In the case of a function field of a curve C, we may define an elliptic surface as a model of the smooth curve, which is a fibration over the curve C. There are at most finitely many singular fibers. When the curve C is defined over the field of complex numbers, these fibers were classified by Kodaira. We generalize his result to the case where the curve C is defined over a general perfect field such as a finite field. We apply this classification to study rational points on torsors over local fields.
Shintaro Akamine
Title: Behavior of the Gaussian curvature of timelike minimal surfaces with singularities
Abstract: A timelike minimal surface in the 3-dimensional Lorentz-Minkowski space is a surface with a Lorentzian metric whose mean curvature vanishes identically. The shape operator of a timelike surface is not always diagonalizable. The diagonalizability of the shape operator of a timelike minimal surface is determined by the sign of the Gaussian curvature away from flat points. In this talk we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface. This talk is based on the preprint arXiv:1701.00238.
Joseph Cho
Title: Deformation and singularities of maximal surfaces with planar curvature lines
Abstract: Minimal surfaces with planar curvature lines in the Euclidean 3-space have been studied since the late 19th century, and have connections to various different subjects of differential geometry. On the other hand, the classification of maximal surfaces with planar curvature lines in the Minkowski 3-space has only recently been given. In this talk, we use an alternative method not only to refine the classification of maximal surfaces with planar curvature lines, but also to show that there exists a deformation consisting exactly of all such surfaces. Furthermore, we investigate the types of singularities that appear on maximal surfaces with planar curvature lines.


(Provisional) Program program (in preparation)

3/14(Tue) AM 09:30-10:05 OCAMI Registration and Opening
AM 10:05-10:55 OCAMI Francis Burstall (1)
AM 11:10-12:00 OCAMI Franz Pedit (1)
PM 13:30-14:20 OCAMI Hideo Takioka
PM 14:35-15:25 OCAMI Atsuhira Nagano
PM 16:00-16:50 OCAMI Masashi Yasumoto
PM 17:05-17:55 OCAMI Seiichi Udagawa
3/15(Wed) AM 10:00-10:50 OCAMI Christoph Bohle (1)
AM 11:00-11:50 OCAMI Francis Burstall (2)
PM 13:30-14:20 OCAMI Toru Kajigaya
PM 14:35-15:05 OCAMI Shintaro Akamine
PM 15:40-16:30 OCAMI Kentaro Mitsui
PM 16:45-17:35 OCAMI Atsushi Fujioka
3/16(Thu) AM 10:00-10:50 OCAMI Christoph Bohle (2)
AM 11:05-11:55 OCAMI Shintaro Yanagida
PM 13:30-14:20 OCAMI Kohei Iwaki
PM 14:35-15:05 OCAMI Joseph Cho
PM 15:40-16:30 OCAMI Hitoshi Yamanaka
PM 16:45-17:35 OCAMI Takashi Kurose
PM 18:30- Party
3/17(Fri) AM 09:30-10:20 OCAMI Francis Burstall (3)
AM 10:35-11:25 OCAMI Kentaro Saji
PM 11:40-12:30 OCAMI Franz Pedit (2)
PM 12:30- Closing

Suggestion to Speakers:

At the lecture room there are enough blackboards, the computer projector and the visualizer. Please prepare your talk using them.

Notice: This workshop is held as one of activities under the above JSPS program.

Osaka City University Advanced Mathematical Institute (OCAM I)
Department of Mathematics, Osaka City Univercity
JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers "Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI"
OCAMI-KOBE-WASEDA International Workshop on "Differential Geometry and Integrable Systems" , February 13 (Sat)- February 17 (Wed), 2016, at Osaka City University and Kobe University.
MORITO One-Day Meeting on "Differential Geometry and Integrable Systems", Morito Memorial Hall, February 18 (Thu), 2016.
Kansai Kenshu Center (KKC)

Support JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers
(Oct. 2014-Mar. 2017)
"Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI"
(Osaka City University - Kobe University - Waseda University, Principal investigator: Yoshihiro Ohnita)

Yoshihiro Ohnita: ohnita (at)

製作 のだ Last updated on 26/February/2017