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# Differential Geometry and Integrable Systems

 DateF February 13 (Sat)- February 17 (Wed), 2016 PlaceF Osaka City University (Building E of Faculty of Science, Lecture Room E408) and Kobe University

 OrganizersF Yoshihiro OhnitaiOCU, OCAMI Director), Wayne Rossman (Kobe University), Martin Guest (Waseda University & Visiting Professor of OCAMI), Masashi Yasumoto (Kobe University), Kentaro Saji (Kobe University), Shoichi Fujimori (Okayama University)

 SponsorsF 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)@ @

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 Invited speakers F Professor Franz Pedit (UMASS Amherst, USA)* Professor Christoph Bohle (University of Tuebingen, Germany)* Professor Sebastian Heller (University of Tuebingen, Germany)* Professor Lynn Heller (University of Tuebingen, Germany)* Professor Tim Hoffmann (Technische Universitaet Muenchen, Germany)* Professor Peng Wang (Tongji University, P.R.China)* Professor Martin Guest (Waseda University and OCAMI, Japan)* Professor Shyuichi Izumiya (Hokkaido University, Japan)* Doctor Masashi YasumotoiKobe University and OCAMI, Japan)* Doctor Toshiaki OmoriiTokyo University of Science, Japan)* Professor Shimpei Kobayashi (Hokkaido University, Japan)* Professor Katsuhiro MoriyaiUniversity of Tsukuba, Japan)* Professor Satoshi Kawakubo (Fukuoka University, Japan)* Professor Junichi InoguchiiUniversity of Tsukuba, Japan) Professor Shoichi FujimoriiOkayama University, Japan)* Professor Yoshinori TeshimaiChiba Institute of Technology, Japan)* Professor Atsufumi HondaiMiyakonojo Nat. Coll. Tech., Japan)* Doctor Kosuke Naokawa (Kobe University, Japan)* Mr. Shintaro Akamine (Kyushu University)* Mr. Asahi Tsuchida (Hokkaido University)* etc. (*: a confirmed speaker )

 Title and Abstract of Talks F abstract (in preparation) Franz Pedit (UMASS Amherst) (1) Harmonic maps and the self-duality equations (2) HIlbert's 21st problem for loop groups with applications to surface geometry (3) Constrained Willmore surfaces and conformal Willmore gradient flow (at MORITO Meeting in Tokyo on February 18 (Thu.)) Abstract: (1) In this talk we will collect various ideas which play a role in the study of harmonic maps of compact Riemann surfaces. Starting from gauge theory we will overview some results of non-abelian Hodge theory and discuss how these play a role in the construction of harmonic maps on higher genus Riemann surfaces. This talk will not contain new results but should be viewed as background material for more research oriented lectures on this topic during the conference. (2) The classical 21st Hilbert problem asks whether any GL(r) representation of the fundamental group of a k-punctured sphere can be realized by a Fuchsian differential equation. The analogous problem for the loop group LGL(r) and its solution gives rise to a description of genus zero constant mean curvature surfaces with k Delauney ends. This approach can also be used to describe minimal Lagrangian genus zero surfaces with Delaunay-like ends and elliptic affine spheres modeled on a k-punctured sphere. (3) The constrained Willmore problem asks for the minimizers of the Moebius invariant Willmore energy, the total squared mean curvature over the surface, in a given conformal class. Utilizing the spinorial description, the space of surfaces of fixed genus and conformal type is viewed as a submanifold in the space of Dirac potentials. The gradient of the distance function on this submanifold gives rise to a flow which decreases the Willmore energy and thus can be used to obtain candidates for the minimizers. Since our flow does not loose derivatives it can be analyzed using a filtered version of the Picard-Lindelov theorem. Martin Guest (Waseda University) Title: Convexity for a certain space of solutions to the Hitchin equations Abstract: In previous work with Alexander Its and Chang-Shou Lin we have solved a special case of the 2D Toda equations. This special case has various interpretations: harmonic maps of a surface, the topological-antitopological fusion equations of Cecotti-Vafa and Dubrovin, the Hitchin equations (Higgs bundles). We shall review this example, and describe/explain a convexity property of the solutions, which arises from the third point of view, i.e. moduli spaces of flat connections. This is joint work with Nan-Kuo Ho. Christoph Bohle (University of Tuebingen) Title: Multi--component KP and the Differential Geometry of Surfaces (and Curves) Abstract: In my talks I give an elementary introduction to the multi--component version of the Kadomtsev--Petviashvili (KP) hierarchy. I plan to focus on some geometric aspects that don't seem to be as well known as they should be. The aim is to explain how several fundamental ingredients of the quaternionic holomorphic approach to surface theory appear naturally within KP theory. Sebastian Heller (University of Tuebingen) and Lynn Heller (University of Tuebingen) (1) CMC surfaces and moduli spaces of flat connections (by Sebastian) (2) Spectral data for higher genus CMC surfaces (by Sebastian) (3) Deformation theory of spectral data (by Lynn) Abstract: In this series of talks we report on recent progress in the Integrable Systems approach to higher genus CMC surfaces in space forms. (1) We first study CMC immersions from compact Riemann surfaces via associated family of flat connections. The elliptic PDE $H= const$ hereby translates into a system of ODEs. As has already been observed in the 1990s, associated families of flat connections can be constructed by loop group factorization methods. We use this idea and develop a theory reducing the investigation of compact CMC surfaces to the study of certain holomorphic curves into the moduli space of flat connections. (2) In the second talk, we deal with the following question: How can, in a computationally accessible way, holomorphic curves into the moduli space of flat connections (which are associated to CMC surfaces) be described. For this purpose we develop an abelianization procedure for flat connections on (symmetric) Riemann surfaces. The abelianization is related to the famous Hitchin system, but differs in crucial parts. As an outcome of the theory, we are able to obtain holomorphic curves into the moduli space in terms of families of flat line bundle connections which are parametrized by spectral data as in the case of CMC tori. (3) We develop a deformation theory for spectral data of CMC surfaces. In particular, we introduce a flow from CMC tori towards higher genus CMC surfaces. We explain the relation of our theory to the classical deformation theory via the Jacobi operator. Albeit not all theoretical questions for our flow and for the deformation theory are solved by now, we can use the theory for a systematic numerical investigation of the moduli space of embedded CMC surfaces with symmetries. Tim Hoffmann (Technische Universitaet Muenchen) (1) A discrete parametrized surface theory in R^3 (2) On circular k-nets, 3d-consistency, and DPW (3) s-conical minimal and cmc nets (at MORITO Meeting in Tokyo on February 18 (Thu.)) Abstract: TBA Peng Wang (Tongji University) Title: On Willmore surfaces with symmetries via loop groups Abstract: In this talk, we will first introduce the loop group description of Willmore surfaces. Then we will discuss the symmetries of Willmore surfaces in terms of the loop group data. Some applications include the existence of (holomorphic) meromorphic invariant potential for (non-) compact Willmore surfaces, descriptions of equivariant Willmore $RP^2$ in $S^4$, characterizations of (compact) homogeneous surfaces and some new concrete examples of Willmore surfaces. This talk is mainly based on joint work with Prof. Josef Dorfmeister (TU Munich). Shyuichi Izumiya (Hokkaido University) Title: The Darboux frame along a cuspidal edge Abstract: In this talk we consider developable surfaces along the singular set of a cuspidal edge surface. We focus on typical two developable surfaces along the cuspidal edge. One of them is a developable surface which is tangent to the cuspidal edge surface and another one is normal to the cuspidal edge surface. These two developable surfaces are considered to be flat approximations of the cuspidal edge surface along the cuspidal edge. For the study of singularities of such developable surfaces, we introduce the notion of Darboux frames along cuspidal edges and new invariants. As a by-product, we introduce the notion of higher order helices which are generalizations of previous notions of generalized helices (i.e. slant helices and clad helices). We use this notion to characterize special cuspidal edge surfaces. Shoichi Fujimori (Okayama University) Title: Embedded zero mean curvature surfaces of mixed causal type in the Lorentz-Minkowski 3-space Abstract: A surface in Lorentz-Minkowski 3-space is called of mixed type if it changes causal type from space-like to time-like. In this talk, we construct several families of properly embedded zero mean curvature surfaces of mixed type in Lorentz-Minkowski 3-space. Msaashi Yasumoto (Kobe University) Title: Construction of discrete constant mean curvature surfaces in Riemannian spaceforms and its applications Abstract: In this talk we will introduce a construction method for discrete constant mean curvature (CMC) surfaces in 3-dimensional Riemannian spaceforms, called the DPW method for discrete CMC surfaces. Hoffmann introduced the DPW method for discrete non-zero CMC surfaces in Euclidean 3-space using matrix-splitting formulae. Based on his work, we extend the method to discrete CMC surfaces in spherical 3-space and hyperbolic 3-space. As an application, we constructed discrete constant positive Gaussian curvature surfaces in Riemannian spaceforms obtained by taking parallel surfaces of discrete CMC surfaces, and we then analyze their singularities. This talk is based mainly on joint work with Yuta Ogata (Kobe University), and partly on joint work with Wayne Rossman (Kobe University). Toshiaki Omori (Tokyo University of Science) Title: A discrete surface theory on 3-valent embedded graphs in 3-dimensional Euclidean space Abstract: Toward a new insight into the physical properties of carbon materials of "negatively-curved" type, especially of Mackay-like carbon materials, we have developed a new discrete surface theory on 3-valent embedded graphs in 3-dimensional Euclidean space. In this talk, we shall start with the definitions of the mean curvature and the Gauss curvature of 3-valent discrete surfaces, continue with several fundamental properties of them, then introduce several basic examples. We shall also introduce an example of a discrete minimal surface which is constructed from a standardly realized (due to Kotani-Sunada) discrete surface. Additionally, if time permitted, we discuss convergence of a sequence of 3-valent discrete surfaces which are coming from the so-called Goldberg-Coxeter construction. This talk is based on joint work with Motoko Kotani (Tohoku University) and Hisashi Naito (Nagoya University). Shimpei Kobayashi (Hokkaido University) Title: Nonlinear d'Alembert formula for discrete pseudospherical surfaces Abstract: A nonlinear d'Alembert formula is a construction method of constant negative Gaussian curvature (pseudospherical) surfaces in the Euclidean 3-space by using separation of variables for the Gauss equation (sine-Gordon equation) of the surface. The heart of this formula is that an additional parameter (spectral parameter) dependence of the moving frame (extended frame) of a psudospherical surface and the loop group decomposition of the extended frame. Bobenko and Pinkall gave a discrete version of pseudospherical surfaces which have the same spectral parameter dependence as in the smooth case. In this talk I will give a discrete analogue of nonlinear d'Alembert formula for discrete pseudospherical surfaces. As an application, we give a simple algorithm to obtain discrete pseudospherical surfaces. This talk is based on a paper arXiv:1505.07189. Karsuhiro MoriyaiUniversity of Tsukuba) Title: Transforms of minimal surfaces and harmonic maps Abstract: A minimal surface in Euclidean space is a Willmore surface. A gauss map of a minimal surface and a conformal Gauss map of a Willmore surface are harmonic maps. Simple factor dressing of the Gauss map gives a new conformal harmonic map and that of the conformal Gauss map gives a new harmonic map. Then the existence of the corresponding transform of a minimal surface or a Willmore surface is expected. A $\mu$-Darboux transform of a minimal surface is a Willmore surface. A special $\mu$-Darboux transform is a minimal surface. Then we have transforms of harmonic maps which are conformal Gauss maps or Gauss maps. In this talk, I will explain the relationship of these transforms. We find an associated family of a minimal surface, the Goursat transform of a minimal surface and the Lopez-Ros deformation of a minimal surface in these transforms. This is joint work with Katrin Leschke. Satoshi KawakuboiFukuoka University ) Title: Fourth soliton curves of the localized induction hierarchy Abstract: A solution of the n-th stationary equation associated to the localized induction hierarchy is called an n-th soliton curve. In this talk, we construct periodic fourth soliton curves in three-dimensional Euclidean space. Also, by using these curves, we construct congruence solutions of a geometric evolution equation associated to the mKdV equation. Atsufumi HondaiMiyakonojo Nat. Coll. Tech.) Title: Isometric immersions with singularities between space forms of the same positive curvature Abstract: In this talk, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O'Neill and Stiel: any isometric immersion of the n-sphere into the (n+1)-sphere of the same sectional curvature is totally geodesic. Kosuke Naokawa (Kobe University) Title: Realization problem of intrinsic cross caps Abstract: We will explain the concept of intrinsic cross caps (or Whitney metrics), which are certain positive semi-definite metrics on 2-manifolds. The metric induced by a cross cap in Euclidean 3-space is a typical example of intrinsic cross caps. In this talk, we explain the isometric realization of intrinsic cross caps as formal power series. As an application, we give a countably infinite number of intrinsic invariants of cross caps. This talk is based on a joint work with Atsufumi Honda (National Institute of Technology, Miyakonojo College), Masaaki Umehara (Tokyo Institute of Technology) and Kotaro Yamada (Tokyo Institute of Technology). Shintaro Akamine (Kyushu University) Title: Causal characters of zero mean curvature surfaces of Riemann-type in the Lorentz-Minkowski 3-space Abstract: A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and we give new examples of ZMC surfaces containing lightlike lines and a zero mean curvature entire graph of mixed type. Yoshinori TeshimaiChiba Institute of Technology) Title: Development of 3D models for intuitive mathematics Abstract: TBA Asahi Tsuchida (Hokkaido University) Title: Smooth solvability of an implicit Hamiltonian system: toward a study of singular curves of a distribution Abstract: A generalized Hamiltonian system is a kind of implicit differential systems. We introduce some conditions given by Fukuda and Janeczko for which an implicit differential system to be (smoothly) solvable. We then see smooth solvability of a generalized Hamiltonian system and of an isotropic submanifold of tangent bundle on a standard symplectic space. As an application, we observe singular curves of a distribution via Pontryagin's maximal principle. ETC.

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(Provisional) Program F program (in preparation)

 2/13iSatj AM 09:30-10:00 OCAMI Registration and Opening AM 10:00-11:00 OCAMI Franz Pedit (1) AM 11:10-12:10 OCAMI Christoph Bohle (1) PM 13:30-14:30 OCAMI Sebastian Heller (1) PM 14:45-15:45 OCAMI Shoichi Fujimori PM 16:00-17:00 OCAMI Martin Guest PM 18:00- Party 2/14iSunj AM 09:20-10:20 OCAMI Franz Pedit (2) AM 10:30-11:30 OCAMI Masashi Yasumoto AM 11:40-12:40 OCAMI Toshiaki Omori Discussion & Joint Research OCAMI 2/15iMonj AM 09:30-10:30 OCAMI Christoph Bohle (2) AM 10:45-11:45 OCAMI Sebastian Heller (2) PM 13:30-14:30 OCAMI Peng Wang PM 14:40-15:40 OCAMI Katsuhiro Moriya PM 15:50-16:50 OCAMI Satoshi Kawakubo 2/16iTuej AM 09:45-10:30 Kobe Tim Hoffmann (1) AM 10:45-11:30 Kobe Tim Hoffmann (2) PM 13:00-13:45 Kobe Kosuke Naokawa PM 13:55-14:40 Kobe Atsufumi Honda PM 15:10-15:55 Kobe Shintaro Akamine PM 16:05-16:50 Kobe Shimpei Kobayashi PM 17:00-17:45 Kobe Shyuichi Izumiya 2/17iWedj AM 09:45-10:30 Kobe Lynn Heller (3) AM 10:40-11:25 Kobe Asahi Tsuchida AM 11:35-12:20 Kobe Yoshinori Teshima

Suggestion to Speakers:

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

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

 LinkF 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" MORITO One-Day Meeting on "Differential Geometry and Integrable Systems", Morito Memorial Hall, Tokyo University of Science , February 18 (Thu), 2016. Kansai Kenshu Center (KKC)

 SupportF 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)

Contactie-mailj
Yoshihiro OhnitaF ohnita (at) sci.osaka-cu.ac.jp

̂ Last updated on 15/January/2016