第3回国際ワークショップ 「微分幾何学と幾何解析」

開催日 平成23年3月13日(日)〜15日(火)3日間
場所 大阪市立大学 大学院理学研究科 3階 数学講究室 3040室

組織委員 Yng-Ing Lee (国立台湾大学), Shu-Cheng Chang (国立台湾大学), Chang-Shou Lin (国立台湾大学, TIMS所長),
高橋 太(大阪市立大学), Martin Guest(首都大学東京, (兼)大阪市立大学客員教授)
河内 明夫(大阪市立大学, OCAMI所長), 大仁田義裕(大阪市立大学,代表)

主催 大阪市立大学 数学研究所 OCAM I 
国立台湾大学 台大数学科学中心 TIMS 

講演予定者 Professor Yng-Ing Lee (National Taiwan Univ., Taiwan)
Professor Su-Jen Kan (Academic Sinica, Taiwan)
Professor Tai-Chia Lin (National Taiwan Univ., Taiwan)
Professor Chiung-Ju Liu (National Taiwan Univ., Taiwan)
Professor Wu-yen Chuang (National Taiwan Univ., Taiwan)
Doctor Ting-Hui Chang (Academic Sinica, Taiwan)
Doctor Kuo-Wei Lee (Academic Sinica, Taiwan)
Professor Mikiya Masuda 枡田 幹也 (Osaka City Univ., Japan)
Professor Qing-Ming Cheng 成 慶明 (Saga Univ., Japan)
Professor Yasuyuki Nagatomo 長友 康行 (Kyushu Univ., Japan)
Professor Ryushi Goto 後藤 竜司 (Osaka Univ., Japan)
Professor Yoshie Sugiyama 杉山 由恵 (Tsuda College, Japan)
Doctor Yohei Sato 佐藤 洋平 (OCAMI, Japan)
Doctor Keisuke Onda 恩田 健介 (Nagoya Univ. & OCAMI, Japan)
Mr. Yohsuke Imagi 今城 洋介 (D1, Kyoto Univ., Japan)
Ms. Saki Okuhara 奥原 沙季 (D1, Tokyo Metropolitan Univ., Japan)
Mr. Hitoshi Yamanaka 山中 仁 (D1, Osaka City Univ., Japan)

講演タイトル,アブストラクト abstract
Yng-Ing Lee (National Taiwan Univ., Taiwan)
Title: "Special solutions to Lagrangian Mean Curvature Flow"
Abstract: Mean curvature flow deforms a submanifold in the direction of its mean curvature vector. When the initial submanifold is Lagrangian in a Kahler-Einstein manifold, the solution will continue to be Lagrangian whenever it is smooth. It thus becomes a nice way to construct special Lagrangians. However, finite-time singularities may occur in general and cause the main difficulties. I will report some of my works with a few collaborators on special solutions to Lagrangian mean curvature flow that is closely related to the study of singularities. Most of the talk will concentrate on examples related to Schoen-Wolfson cones, which are the obstructions to the existence of special Lagranians in two-dimension.
Su-Jen Kan(Academic Sinica, Taiwan)
Title: "Complete Ricci-flat metrics through a rescaled exhaustion"
Abstract: PDF
Tai-Chia Lin (National Taiwan Univ., Taiwan)
Title: "The Poisson-Nernst-Planck system for ion transport"
Abstract: Understanding ion transport is crucial in the study of many physical and biological problems, such as semiconductors, electro-kinetic fluids, transport of electrochemical systems and ion channels in cell membranes. One of the fundamental models for the ionic transport is the time dependent coupled diffusion-convection equations, the Poisson-Nernst-Planck (PNP) system. The PNP system consists of the electro-static Poisson and Nernst-Planck equations describing electro-diffusion and electrophoresis. In this lecture, I’ll introduce our recent results on the equilibrium of the PNP system and the linear stability problem.
Chiung-Ju Liu(National Taiwan Univ., Taiwan)
Title: "Bergman kernel on singular Riemann surfaces"
Abstract: The Bergman kernel has a asymptotic expansions on smooth polarized compact Kahler manifolds. However, some properties may not hold on singular manifolds. In this talk, we will study the information of the singularity in the expansion.
Wu-yen Chuang(National Taiwan Univ., Taiwan)
Title: "Wallcrossing in ADHM theory and Cohomology of the Hitchin system"
Abstract: I will introduce ADHM sheaf theory and present a conjectural recursive formula for the Poincare/Hodge polynomial of Hitchin system. The formula is derived from the wallcrossing of ADHM sheaf theory on curves. It establishes a connection between previous work of Hausel and Rodriguez-Villegas and refined local Donaldson-Thomas invariants.
Ting-Hui Chang (Academic Sinica, Taiwan)
Title: "On the existence of pseudoharmonic maps from pseudohermitian manifolds into Riemannian manifolds with nonpositive sectional curvature"
Abstract: In this talk, we first derive a CR Bochner identity for the pseudoharmonic map heat flow on pseudohermitian manifolds. Secondly, we are able to prove the existence of global smooth solution for the pseudoharmonic map heat flow from a closed pseudohermitian manifold into a Riemannian manifold with nonpositive sectional curvature. In particular, we prove the existence theorem of pseudoharmonic maps. This is served as the CR analogue of Eells-Sampson's Theorem for the harmonic map heat flow.
Kuo-Wei Lee (Academic Sinica, Taiwan)
Title: "The mean curvature flow of compact submanifolds in higher codimension"
Abstract: PDF
Mikiya Masuda(Osaka City Univ., Japan)
Title: "Iterated circle bundles"
Abstract: PDF    
Qing-Ming Cheng(Saga Univ., Japan)
Title: "Estimates for eigenvalues on complete Riemannian manifolds"
Abstract: In this talk, we consider the Dirichlet eigenvalue problem of the Laplacian on a bounded domain in complete Riemannian manifolds. Estimates for eigenvalues will be discussed. We will talk about the conjecture of Polya for the lower bounds for eigenvalues on a bounded domain in Euclidean spaces and the generalized conjecture of Polya for the lower bounds of eigenvalues on a bounded domain in complete Riemannian manifolds. In order to obtain the lower bounds of eigenvalues, we need to have sharper universal inequalities for eigenvalues and the recursion formula of Cheng and Yang (Math. Ann. 2007). Furthermore, we will deal with the upper bounds for eigenvalues. In particular, for lower order eigenvalues, we will discuss the conjectures of Payne, Polya and Weinberger and their generalizations.     
Ryushi Goto(Osaka Univ., Japan)
Title: "Holomorphic Poisson and generalized Kahler structures"
Abstract: First I will give a brief introduction of generalized complex and generalized Kaehler geometry. Deformations of generalized complex and generalized Kahler structures are disc ussed. Holomorphic Poisson structures gives rise to intriguing generalized Kaehler structures. Next I will construct Ricci-flat conical Kaehler metrics on crepant resolutions of Sasaki-Einstein cones in every Kaehler class. Applying the deformation method by Poisson structures in generalized geometry, I will deform ordinary Calabi-Yau structures to obtain generalized Calabi-Yau metrical structures on the crepant resolution.     
Yasuyuki Nagatomo(Kyushu Univ., Japan)
Title: "Harmonic maps into Grassmannians and its applications to isoparametric functions and moduli problems"
Abstract: PDF     
Yoshie Sugiyama (Tsuda College, Japan)
Title: "Measure valued solutions of the 2D Keller-Segel system"
Abstract: We deal with the two-dimensional Keller-Segel system describing chemotaxis in a bounded domain with smooth boundary under the nonnegative initial data. As for the Keller-Segel system, the $L^1$ norm is the scaling invariant one for the initial data, and so if the initial data is sufficiently small in $L^1$, then the solution exists globally in time. On the other hand, if its $L^1$ norm is large, then the solution blows up in a finite time. The first purpose of my talk is to construct a time global solution as a measure valued function beyond the blow-up time even though the initial data is large in $L^1$. The second purpose is to show the existence of two measure valued solutions of the different type depending on the approximation, while the classical solution is unique before the blow-up time.     
Yohei Sato(OCAM I, Japan)
Title:"The existence and non-existence of positive solutions for the nonlinear Schroedinger equations"
Abstract: PDF     
Kensuke Onda (Nagoya Univ. & OCAM I, Japan)
Title: "Sol-solitons on Lorentzian manifolds"
Abstract: J. Lauret introduced Sol-soliton and studied in the Riemannian framework. Sol-solitons have relevance to Ricci solitons, that is fixed point of the Ricci flow. We study sol-soliton theory and use it to give sol-solitons and Ricci solitons on Lorentzian manifolds.     
Yohsuke Imagi (D1, Kyoto Univ., Japan)
Title: "A Uniqueness Theorem for Gluing Special Lagrangian Submanifolds"
Abstract: Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. I talk about a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimension 3.     
Saki Okuhara (D1, Tokyo Metropolitan Univ., Japan)
"Special Lagrangian 3-folds via harmonic maps"
Abstract: Special Lagrangian cones in ${\mathbb{C}}^3$ can be constructed from certain harmonic maps into a 6-symmetric space. We will review this process which was derived by Joyce and McIntosh, and use it to give a new construction of special Lagrangian cones in ${\mathbb{C}}^3$.     
Hitoshi Yamanaka(D2, Osaka City Univ., Japan)
Title: "Intersection of stable and unstable manifolds for invariant Morse functions"
Abstract: In Witten's Morse theory, it is important to consider the negative gradient flows which connect two critical points whose Morse indices differ by 1. However there are too many examples of Morse functions which have no such pairs of critical points. In the case of an invariant Morse-Smale function, we investigate the intersection of a stable and an unstable manifold which corresponds to a pair of critical points whose Morse indices differ by 2. We also consider the group action on it.


プログラム program

3/13(日) AM 10:00-10:50 数学講究室 (3040室)  Mikiya Masuda(50分) Slides of Talk
AM 11:10-12:00 数学講究室 (3040室) Yng-Ing Lee(50分) Slides of Talk
PM 13:30-14:20 数学講究室 (3040室) Kuo-Wei Lee(50分) Slides of Talk
PM 14:30-15:20 数学講究室 (3040室) Tai-Chia Lin(50分) Slides of Talk
PM 15:40-16:30 数学講究室 (3040室) Yoshie Sugiyama(50分) Slides of Talk
PM 16:40-17:30 数学講究室 (3040室) Yohei Sato (50分) Slides of Talk
3/14(月) AM 9:30-10:20 数学講究室 (3040室) Su-Jen Kan(50分)
AM 10:30-11:20 数学講究室 (3040室) Ryushi Goto(50分)
AM 11:30-12:20 数学講究室 (3040室) Chiung-Ju Liu(50分) Slides of Talk
PM 14:00-14:50 数学講究室 (3040室) Kensuke Onda(50分) Slides of Talk
PM 15:00-15:40 数学講究室 (3040室) Hitoshi Yamanaka(40分) Slides of Talk
PM 16:00-16:40 数学講究室 (3040室) Saki Okuhara(40分) Slides of Talk
PM 16:50-17:30 数学講究室 (3040室) Yohsuke Imagi(40分)
3/15(火) AM 9:30-10:20 数学講究室 (3040室) Wu-yen Chuang(50分)
AM 10:30-11:20 数学講究室 (3040室) Yasuyuki Nagatomo(50分) Slides of Talk
AM 11:30-12:20 数学講究室 (3040室) Ting-Hui Chang(50分) Slides of Talk
PM 13:40-14:30 数学講究室 (3040室) Qing-Ming Cheng (50分) Slides of Talk

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備考 : この企画は、台湾・国立台湾大学台大数学科学中心と大阪市立大学数学研究所の間の学術研究協力協定のもとでの活動の一環として行われます。

リンク等 大阪市立大学数学研究所
大阪市立大学 大学院理学研究科 数学教室
国立台湾大学台大数学科学中心 TIMS
TIMS One-Day Workshop on Differential Geometry (April 1, 2008)
第1回大阪市立大学数学研究所-台大数学科学中心 共催 国際ワークショップ 「微分幾何学と幾何解析」 (March 9-10, 2009)
第2回台大数学科学中心-大阪市立大学数学研究所 共催 国際ワークショップ 「微分幾何学と幾何解析」 (March 21-23, 2010)
関西研修センター(Kansai Kenshu Center-KKC)

補助 科学研究費基盤研究(A) 研究課題番号:21244004
(研究代表者 Martin Guest)

大仁田 義裕 : ohnita (at)
高橋 太 : futoshi (at)

製作 のだ Last updated on 14/March/2011