The 5th Geometry Conference for Friendship of Japan and China

Abstract

• Orbifold Relative GW-Invariants and Applications
  • Bohui Chen (Sichuan University)
  • (Abstract) In this talk, we generalize the relative Gromov-Witten invariant theory to orbifolds and give some applications.
• Gevery Regularity of Subelliptic Monge-Ampere Equations in the Plane
  • Hua Chen (Wuhan University)
  • (Abstract) In this talk, we shall consider the regularity problem for a class of two-dimensional degenerate Monge-Ampere equations. We know that, in this case, the solution is smooth if the principal curvature of the solution is strictly positive and the order of degenerate point for the coefficient of the equation is finite. Here we prove that, under the same conditions, the solution would be Gevery regularity.
• The Ricci Curvature In Finsler Geometry
  • Xinyue Cheng (Chongqing University of Technology)
  • (Abstract) The Ricci curvature plays a very important role in Finsler geometry. In this talk, we introduce the Ricci curvature and its some applications in Finsler geometry, including the role of the Ricci curvature in Finsler projective geometry, the volume comparison in Finsler geometry and Ricci curvature of Randers metrics.
• Using topology to explore DNA packing
  • Yuanan Diao (University of North Carolina at Charlotte)
  • (Abstract) In this talk, we will discuss how topology (more specifically knot theory) is used to explore and to provide insights about basic properties of DNA and of its packing principles. We will also discuss various models used to describe the DNA packing geometry in many bacteriophages and some animal.
• Prescribed curvature representation and its applications
  • Qing Ding (Fudan University)
  • (Abstract) In this talk, we introduce the geometric concept of PDEs with prescribed curvature representation in the category of Yang-Mills theory, which generalizes the well-known concept of integrable equations with zero curvature representation. As the applications, we use this new geometric tool to study the dynamical properties of the nonintegrable discrete (semi-calssical) Heisenberg model ${S_n}_t={S_n} \times (S_{n+1}-S_{n-1})$ and the (classical) Schrodinger maps from the Euclidean plane to the hyperbolic 2-space $H^2$.
• A Poincare-Hopf index formula for complex vector fields
  • Huitao Feng (Nankai University)
  • (Abstract) For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincare-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincare-Hopf index formula to the case of the complex vector fields.
• The volume growth of hyperkaehler manifolds of type $A_{\infty}$
  • Kota Hattori (University of Tokyo)
  • (Abstract) Hyperkaehler manifolds of type $A_{\infty}$ were first constructed by Gibbons-Hawking ansatz due to Anderson, Kronheimer and LeBrun. These manifolds are noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius r>0 where r goes to infinity with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is r^4 (ALE space) or r^3 (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\infty}$ whose volume growth is r^c for a given 3 < c < 4.
• Ricci flow, simplicial volume, and smooth structures
  • Masashi Ishida (Sophia University)
  • (Abstract) It is conjectured by Fuquan Fang, Yuguang Zhang and Zhenlei Zhang that the existence of non-singular solutions to the normalized Ricci flow on smooth closed 4-manifolds with non-trivial Gromov's simplicial volume and negative Perelman's invariant implies the Gromov-Hitchin-Thorpe type inequality. This conjecture is still open. In this talk, we shall discuss the existence of closed topological 4-manifolds with non-trivial Gromov's simplicial volume and satisfying the Gromov-Hitchin-Thorpe type inequality, but admitting infinitely many exotic smooth structures for which Perelman's invariant is negative and there is no non-singular solution to the normalized Ricci flow for any initial metric. In particular, the main result of this talk tells us that the converse of the conjecture dose not hold in general. We use the Seiberg-Witten monopole equations to prove the main result.
• Modeling of morphogenesis in development
  • Yoh Iwasa (Kyushu University)
  • (Abstract) We start our life from a single cell, named fertilized egg, and develop to a adult body with many complex structures. I will speak on recent attempts of modeling morphogenesis in development. In particular, I will speak two examples of modeling of organ growth: limb (wing) bud formation of chick and branching pattern formation of kidney ducts. Finally I will speak on the analysis of traveling wave of gene expression in zebrafish somitogenesis.
• The symplectic volume of the moduli space of spatial polygons
  • Yasuhiko Kamiyama (Ryukyu University)
  • (Abstract) The moduli space of spatial polygons is a typical example in symplectic geometry and toric topology. The symplectic volume of the space was studied by many mathematicians using various approaches. I will survey these researches.
• A rough equivalence among partial differential equations
  • Tsuyoshi Kato (Kyoto University)
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• Real hypersurfaces in a complex space form and the generalized Tanaka-Webster connection
  • Mayuko Kon (Hokkaido University)
  • (Abstract) In this talk I will present the results for the curvature tensor and the Ricci tensor with respect to the generalized Tanaka-Webster connection of a real hypersurface in a complex space form. The generalized Tanaka-Webster connection for a real hypersurfaces of Kaehlerian manifolds was studied by J. T. Cho. It coincids with the Tanaka-Webster connection if the associated CR-structure of the real hypersurface is pseudo-Hermitian and strongly pseudo-convex.
• Random groups and scaling limit argument
  • Takefumi Kondo (Kobe University)
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• A Kazdan-Warner type identity and a class of curvature invariants $v^{2k}$
  • Haizhong Li (Tsinghua University)
  • (Abstract) In this talk, we present a Kazdan-Warner type identity involving the $v^{2k}$ curvature and a Killing vector field on a compact manifold. In the special case when the Riemannian manifold is locally conformally flat, the result reduces to the well-known result.
• Non existence of quasi-harmonic spheres
  • Jiayu Li (Chinese Academy of Sciences)
  • (Abstract) Let $M$ and $N$ be compact Riemannian manifolds. To prove the global existence and convergence of the heat flow for harmonic maps between $M$ and $N$, it suffices to show the nonexistence of harmonic spheres and nonexistence of quasi-harmonic spheres. In this talk, we prove that, if the universal covering of $N$ admits a non-negative strictly convex function with polynomial growth, then there are no quasi-harmonic spheres nor harmonic spheres. This generalizes the famous Eells-Sampson's theorem.
• On Lagrangian submanifolds in complex hyperquadrics
  • Hui Ma (Tsinghua University)
  • (Abstract) The Gauss map of any oriented isoparametric hypersurface of the sphere defines a minimal Lagrangian submanifold in the complex hyperquadric. In this talk, we determine the Hamiltonian stability of ALL compact minimal Lagrangian submanifolds embedded in complex hyperquadrics obtained as the Gauss images of homogeneous isoparametric hypersurfaces in spheres. The relation between the Gauss image construction and the conormal bundle construction for Lagrangian submanifolds in complex hyperquadrics will also be discussed. This talk is mainly based on the joint work with Professor Yoshihiro Ohnita.
• An approach to Donaldson-Tian-Yau's Conjecture
  • Toshiki Mabuchi (Osaka University)
  • (Abstract) Donaldson-Tian-Yau's Conjecture asks whether a polarized algebraic manifold $(M,L)$ admits a CSC K\"ahler metric if and only if $(M,L)$ is K-stable. In this talk, we shall give our recent approach to the existence problem for CSC K\"ahler metric.
• The moduli space of transverse Calabi-Yau structures on foliated manifolds
  • Takayuki Moriyama (Kyoto University)
  • (Abstract) We develop a moduli theory of transverse structures given by closed forms on foliated manifolds. We show that the moduli space of transverse Calabi-Yau structures is a Hausdorff and smooth manifold if the foliation is taut. In this talk, we will give some examples of transverse Calabi-Yau structures on foliated manifolds.
• Wave equations and the LeBrun-Mason twistor correspondence
  • Fuminori Nakata (Tokyo Institute of Technology)
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• Invariant geometric flows and integrable systems
  • Changzheng Qu (Northwest University)
  • (Abstract) In this talk, we shall discuss the relationship between invariant geometric flows and integrable systems. It is shown that many integrable systems are associated with the invariant geometric flows in some geometries. The geometric interpretation to properties of integrable systems such as the Backlund transformation, Miura transformation and bi-Hamiltonian structure etc are given in terms of the geometric flows.
• Finslerian models in biology
  • Sorin Sabau (Tokai University)
  • (Abstract) I will prsent two topics in this talk. Firstly, some Finslerian type models introduced in ecology, and secondly, the so-called Kosambi-Cartan-Chern theory. The second topic is a Finslerian alternative to the classical notion of Lyapunov stability of dynamical systems and it is used in various fields of the science nowadays.
• Prolongations of canonical systems on Jet spaces
  • Kazuhiro Shibuya (Hiroshima University)
  • (Abstract) We will introduce the prolongatoin procedure for differential systems given by E.Cartan to study PDEs. In this tahk, we will consider a generalization of Monster Goursat manifolds which are obtained by prolongations of canonical systems on Jet spaces. It will be important to study geometric solutions of PDEs.
• Tangle analysis of site-specific recombination
  • Koya Shimokawa (Saitama University)
  • (Abstract) Action of site-specific recombinase can be analysed using tangle model introduced by Ernst and Sumners. In this talk we apply this method and give topological characterizations of actions of several site-specific recombinases.
• A splitting theorem for weighted Alexandrov spaces
  • Takashi Shioya (Tohoku University)
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• Selection for Complexity can Induce Modularity
  • Robert Sinclair (Okinawa Institute of Science and Technology)
  • (Abstract) The animals, plants, birds and fish we know seem to belong to a progression towards ever greater complexity. We can identify the parts of the more complex organisms, and know that these parts are sometimes exchangeable by grafting or transplantation. This suggests modular design. Are the higher organisms more modular than, say, mushrooms due to more complex challenges they must face? For an abstract model, a synchronous boolean network model with only inhibitory interactions, we provide a computer proof to demonstrate that the greatest complexity for a given system size is only achievable if the maximal network is disjoint, for systems not less than a critical size. This is a mathematical approach to a biological problem which required the development of dedicated software. A number of purely mathematical conjectures arise from this work.
• Tissue invasion: early stage of tumor metastasis and its mathematics
  • Takashi Suzuki (Osaka University)
  • (Abstract) Invasion arises in the early stage of tumor metastatis and its control has been the important issue in medical science. This process is composed of three factors - actin reorganization, ECM degradation, and adhesive regulation. We shall describe two mathematical approaches to this object, inverse source identification and mathematical modeling. In the former case we adopt several new mathematical tools, homology and parallel optimization to take advances in cancer diagnosis. In the latter we show the notion of a bi-directed modeling whereby new mean field equation is formulated. Numerical simulations and mathematical analysis of these models are also presented.
• Pattern formation on surfaces -- from a biological point of view
  • Izumi Takagi (Tohoku University)
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• Wasserstein geometry of non-linear Fokker-Planck equations
  • Asuka Takatsu (Tohoku University)
  • (Abstract) In this talk, inequalities associated with the non-linear Fokker-Planck equations will be introduced in order to describe relations among functionals defined on pairs of probability measures, called generalized relative entropy, Fisher information, and Wasserstein distance. I will discuss the importance of the $q$-Gaussian measures in these inequalities.
• Isoparametric hypersurface in a Reimannian manifold
  • Zizhou Tang (Beijing Normal University)
  • (Abstract) Isoparametric hypersurfaces in various Riemannian manifolds will be studied, especially in Milnor exotic spheres.
• Kahler-Einstein metrics and compactness theorem on Fano manifolds
  • Gang Tian (Peking University and Princeton University)
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• Geometry of wave fronts
  • Masaaki Umehara (Osaka University)
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• Moebius CMC-surfaces in S^3
  • Changping Wang (Peking University)
  • (Abstract) Let x: M \to R^3 be a surface without umbilical point. Let \xi be the sphere tangent to x with radius 1/H, where H is the mean curvature of x. Then \xi: M \to S^4_1 is a surface in the de-Sitter space, called the mean curvature sphere of x. Moreover, the mean curvature vector of \xi is light-like. Conversely, any surface \xi: M \to S^4_1 with light-like mean curvature vector is the mean curvature sphere of a surface x: M \to R^3. A surface x: M \to R^3 is called Moebius CMC surface if its mean curvature sphere \xi: M \to S^4_1 is a surface with parallel mean curvature vector. It is a generalization of Willmore surfaces. In this talk, we classify all Moebius CMC-surfaces with constant Moebius curvature: they are Moebius equivalent to either minimal surfaces in R^3, or CMC-1 surfaces in H^3, or a cirular cylinder, a circular cone, a revolution torus in R^3. We classify also all surfaces x: M \to R^3 such that its mean curvature sphere \xi intersects a fixed plane with constant angle.
• The regularity of harmonic maps into spehres and applications to Bernstein problems
  • Yuanlong Xin (Fudan University)
  • (Abstract) We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.
• Harmonic metrics on unipotent bundles over quasi-compact Kaehler manifolds
  • Yihu Yang (Tongji University)
  • (Abstract) In this talk, I will show the existence of harmonic metrics on unipotent bundles over quasi-compact Kaehler manifolds.
• Towards a new sphere theorem via the Ricci flow
  • Takumi Yokota (University of Tsukuba)
  • (Abstract) The Ricci flow is one of the powerful tools in Riemannain geometry. We can deform a given positively curved Riemannain manifold to a rounder one by solving the Ricci flow equation, which is a heat-type equation for evolving Riemannian metrics. In this talk, I will talk about an approach to apply the Ricci flow to prove a new sphere theorem.
• Geometric quantization on manifolds with boundary
  • Weiping Zhang (Nankai University)
  • (Abstract) We describe our recent joint work with Xiaonan Ma where we resolve a conjecture of M. Vergne on the geometric quantization formula on noncompact manifolds with proper moment maps.
• Classification of Four-dimensional Manifolds with Positive Isotropic Curvature
  • Xiping Zhu (Sun Yat-sen University)
  • (Abstract) Around 1980, Schoen and Yau used minimal surface theory to obtain their famous classification to three-dimensional manifolds with positive scalar curvature. Nevertheless, up to now, the understanding to positively curved four-dimensional manifolds is still poor. For example, one does not know if $S^2 \times S^2$ admits a metric of positive sectional curvature (Hopf conjecture); one does not have the classification of four-dimensional Einstein manifolds with positive scalar curvature. Recently, inspired by the Hamilton-Perelman theory, Chen, Tang and the speaker used the Ricci flow with surgery to obtain a complete classification to four-dimensional manifolds with positive isotropic curvature. We proved that a compact four-dimensional manifold with a metric of positive isotropic curvature if and only if it is diffeomorphic to the standard $S^4$, or the standard $RP^4$, or a cocompact metric quotient of the standard $R \times S^3$, or a connected sum of a finite number of them. Note that it was earlier proved by Hamilton that any exotic $RP^4$ does not admits a metric of positive isotropic curvature. As a byproduct, we also obtain a complete classification to compact, locally conformally flat four-dimensional manifolds with positive Yamabe invariant. In this talk I will discuss these classification theorems.