Last update : August 11, 2009
| 日時： | 2009年6月30日(火)14:00 - 7月2日(木)14:35 |
| 場所： |
奈良女子大学 理学部会議室（理学部A棟1階） ただし7月1日（水）午後のみ A201講義室（理学部A棟2階） |
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組織委員会： 小磯深幸 (奈良女子大学，JSTさきがけ) 長澤壯之 (埼玉大学) Bennett Palmer (Idaho State University, USA) 片桐民陽 (奈良女子大学) 招待講演者： Jaigyoung Choe (Korea Institute for Advanced Study, Korea) 藤森祥一 (福岡教育大学) Randall D. Kamien (University of Pennsylvania, USA) 川久保哲 (福岡大学) 村井実 (龍谷大学) 長澤壯之 (埼玉大学) 岡部真也 (岩手大学) Bennett Palmer (Idaho State University, USA) Paolo Piccione (Universidade de Sao Paulo, Brazil) Wayne Rossman (神戸大学) 酒井高司 (首都大学東京) 高橋太 (大阪市立大学) CONTACT :
SUPPORTED BY : 科学技術振興機構「数学と諸分野の協同によるブレークスルーの探索」さきがけ 科学研究費補助金・基盤研究(C)No.19540217 Poster[PDF] (1.3MB) |
| 6月30日(火) | |
| 14:00-14:05 | 開会挨拶 |
| 14:05-15:05 | Randall D. Kamien （University of Pennsylvania, USA） |
| "Films and Layers: Nematics and Smectics" | |
| 15:15-16:15 | Bennett Palmer （Idaho State University, USA） |
| "Capillary Surfaces for Anisotropic Surface Energies" | |
| 16:30-17:30 | 藤森 祥一 （福岡教育大学） |
| "A Construction Method for Triply Periodic Minimal Surfaces" | |
| 7月1日(水) | |
| 9:00-10:00 | Wayne Rossman （神戸大学） |
| "A (Limited) Survey on the Foundations of Discrete Differential Geometry" | |
| 10:10-11:10 | 村井 実 （龍谷大学） |
| "On the Equilibrium States of Elastic Rings" | |
| 11:30-12:30 | 川久保 哲 （福岡大学） |
| "Traveling Wave Solutions of the Localized Induction Hierarchy in Three-dimensional Space Forms" | |
| 13:50-14:50 | 酒井 高司 （首都大学東京） |
| "Absolutely Area-Minimizing Cones over Some Minimal Submanifolds in S^n" | |
| 15:00-16:00 | Paolo Piccione （Universidade de Sao Paulo, Brazil） |
| "Bifurcation of Constant Mean Curvature Tori in Euclidean Spheres" | |
| 16:20-17:20 | 高橋 太 （大阪市立大学） |
| "Morse Indices and the Number of Maximum Points of Some Solutions to a Two-dimensional Elliptic Problem" | |
| 7月2日(木) | |
| 9:00-10:00 | Randall D. Kamien （University of Pennsylvania, USA） |
| "Smectic Topology, Topography, and Tomography" | |
| 10:10-11:10 | Jaigyoung Choe （Korea Institute for Advanced Study, Korea） |
| "Capillary Surfaces in a Convex Cone" | |
| 11:20-12:20 | 岡部 真也 （岩手大学） |
| "The Variational Problem for a Certain Action Functional Defined on Closed Curves" | |
| 13:30-14:30 | 長澤 壯之 （埼玉大学） |
| "Constrained Gradient Flows of Bending Energy for Plane Curves: Total-length Constraint Versus Local-length Constraint" | |
| 14:30-14:35 | 閉会挨拶 |
| Jaigyoung Choe (Korea Institute for Advanced Study, Korea) |
| "Capillary Surfaces in a Convex Cone" |
| We show that a compact embedded hypersurface $S\subset\mathbb{R}^{n+1}$ with constant higher-order mean curvature in a convex piecewise smooth cone $C$ which is perpendicular to $\partial C$ is part of a hypersphere. Also we prove that an embedded disk type constant mean curvature surface $S\subset\mathbb{R}^3$ in a convex polyhedral cone $C$ which makes constant contact angles with $\partial C$ is a spherical cap if $C$ has at most 5 faces. This condition on the number of faces can be dropped if $C$ is a right cone over a regular $n$-gon and the contact angles are the same on $\partial S$. |
| Shoichi Fujimori (藤森祥一，福岡教育大学) |
| "A Construction Method for Triply Periodic Minimal Surfaces" |
| This is joint work with Matthias Weber. We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean three-space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces. |
| Randall D. Kamien (University of Pennsylvania, USA), I |
| "Films and Layers: Nematics and Smectics" |
| We discuss problems that arise in liquid crystalline systems. We describe a new type of foam with embedded nematic order with special emphasis on the physics of liquid crystals. We then move on to multiple layers, that is, foliations of space, known as smectic liquid crystals. This will serve as an introduction for further talks. |
| Randall D. Kamien (University of Pennsylvania, USA), II |
| "Smectic Topology, Topography, and Tomography" |
| We develop topological methods to calculate the values of energy functionals in smectic liquid crystals and revisit the theory of topological defects in ordered media. We show that the use of homotopy theory is problematic and discuss a new approach which studies critical points of maps and graphs. |
| Satoshi Kawakubo (川久保哲，福岡大学) |
| "Traveling Wave Solutions of the Localized Induction Hierarchy in Three-dimensional Space Forms" |
| The localized induction equation is an evolution equation of curves modeled on the motion of thin vortex filaments in ideal fluids in the three-dimensional Euclidean space. This equation is known as an infinite dimensional completely integrable Hamiltonian system, and the associated infinite hierarchy of evolution equations is called the localized induction hierarchy. In this talk, we construct traveling wave solutions of the localized induction hierarchy in three-dimensional space forms, and investigate explicit expressions of the solutions. |
| Minoru Murai (村井実，龍谷大学) |
| "On the Equilibrium States of Elastic Rings" |
| We will consider the problem of equilibrium states of an elastic inextensible ring under a uniform external pressure. On this problem, Tadjbakhsh and Odeh proposed a variational problem. In this talk, we will show the uniqueness of its minimizer. |
| Takeyuki Nagasawa (長澤壯之，埼玉大学) |
| "Constrained Gradient Flows of Bending Energy for Plane Curves: Total-length Constraint Versus Local-length Constraint " |
| Consider gradient flows of bending energy for plane curves under constraints of "enclosed area and total-length" and "enclosed area and local-length." Our research group and Okabe have investigated these independently. We discuss relationship and differences between them, and propose a new gradient flow unified both gradient flows. In other words we regard gradient flows constrained total-length-constrain and local-length as special cases of the unified flow. |
| Shinya Okabe (岡部真也，岩手大学) |
| "The Variational Problem for a Certain Action Functional Defined on Closed Curves" |
| We consider a variational problem for a certain space-time functional defined on planar closed curves. The functional is appeared in the action minimization problem for stochastic Allen-Cahn equation. The variational problem is stated as follows: ``Let $\Gamma_0$ and $\Gamma_1$ be given planar closed curves and $T$ be a given positive constant. Then minimize the space-time functional over families of planar closed curves, which change from $\Gamma_0$ to $\Gamma_1$ in time $T$." In this talk, we focus on an existence of non-radial symmetry critical point of the variational problem, in particular a solvability of an initial final value problem. |
| Bennett Palmer (Idaho State University, USA) |
| "Capillary Surfaces for Anisotropic Surface Energies" |
| A surface energy is anisotropic if it depends on the direction of the surface. This type of energy arises when considering an interface between ordered and disordered phases of materials. We will discuss the stability of solutions to free boundary problems when the free energy is anisotropic. The boundary terms contain wetting and line tension. This is a report on joint work with Professor Miyuki Koiso. |
| Paolo Piccione (Universidade de Sao Paulo, Brazil) |
| "Bifurcation of Constant Mean Curvature Tori in Euclidean Spheres" |
| We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus. (Joint work with L. Alias) |
| Wayne Rossman (神戸大学) |
| "A (Limited) Survey on the Foundations of Discrete Differential Geometry" |
| In this talk we will consider discrete differential geometry for surface theory, and in particular the developments in this field that have recently come from researchers at (or closely connected to) Berlin Technical University. The goal is to find ways to discretize surfaces so that integrable systems properties are preserved, thus allowing for discrete versions of transformations, such as Christoffel and Calapso and Darboux transformations, and thus also allowing for a notion of discrete constant mean curvature. |
| Takashi Sakai (酒井高司，首都大学東京) |
| "Absolutely Area-Minimizing Cones over Some Minimal Submanifolds in $S^n$" |
| The cone over a minimal submanifold $M$ in $S^n$ is also minimal in $\mathbb{R}^{n+1}$. It may be absolutely area-minimizing among all submanifolds with boundary $M$ in $\mathbb{R}^{n+1}$. In this talk, we discuss two methods to construct area-minimizing cones. First we deal with calibration. We can construct special Lagrangian cones from austere submanifolds in the sphere via twisted normal bundle. Second, we show that cones over some minimal orbits of $s$-representations are absolutely area-minimizing, constructing area-nonincreasing retractions concretely. |
| Futoshi Takahashi (高橋太，大阪市立大学) |
| "Morse Indices and the Number of Maximum Points of Some Solutions to a Two-dimensional Elliptic Problem" |
| In this talk, we consider the positive solutions of the equation $-\Delta u = u^p$ on a bounded domain in $\mathbb{R}^2$ with the Dirichlet boundary condition, where $p>1$. We prove that if $u_p$ is a solution (with some assumption) of Morse index $m$, then the number of maximum points of $u_p$ is less than or equal to $m$ when $p$ large. Also we prove that if the domain is convex, then any solution of Morse index one (which satisfies some assumption) has a unique critical point and the level sets are star-shaped for $p$ sufficiently large. |