The 21st Century COE Program Constitution of wide-angle mathematical basis focused on knots Department of Mathematics and Physics Graduate School of Science Osaka City University
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As a project of OCAMI, we shall start the seminar on differential geometry in the wide sense of including the areas related to geometric analysis, topology, algebraic geometry, mathematical physics, integrable systems, information sciences etc.

 Contact to : Yoshihiro Ohnita Shin Kato Department of Mathematics Osaka City University Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, JAPAN TEL: 06-6605-2617（Ohnita） 06-6605-2616（Kato） e-mail: ohnita@sci.osaka-cu.ac.jp shinkato@sci.osaka-cu.ac.jp

 Differential Geometry Seminar（2007） （2006）
 Speaker ：Motohico Mulase (UC Davis, RIMS ) Title ：Sp-Hitchin Systems and Sp-invariant KP Systems （Abstract） Date ：March 28 (Fri.) 13:30～15:00 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yuichi Nohara ( Dept. of Math., Tohoku Univ. ) Title ：Toric degeneration of flag manifolds and the Gelfand-Cetlin system （Abstract） Date ：February 21 (Thu.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Hiroshi Iriyeh (Tokyo Denki Univ.) Title ：On the local minimality of geodesics of Hofer's metric. （Abstract） Date ：February 21 (Thu.) 13:00～14:30 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Ryoichi Kobayashi ( Dept. of Math., Nagoya Univ. ) Title ：Ricci flow ancient solutions arising from the natural collapsing of the twistor space of positive quaternion K\"ahler manifolds （Abstract） Date ：February 7 (Thu.) 10:40～12:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yu Kawakami (Dept. of Math., Nagoya Univ., Osaka City University Advanced Mathematical Institute ) Title ：Width and Ricci flow （Abstract） Date ：February 6 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Takumi Yamada ( Dept. of Math., Shimane Univ. ) Title ：Lattice of compact complex-parallelizable pseudo-Kaehler manifolds （Abstract） Date ：January 30 (Wed.) 17:15～18:45 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yoshihiro Ohnita ( Dept. of Math.,Osaka City Univ. ) Title ：On the deformation of certain 3-dimensional minimal Legendrian submanifolds （Abstract） Date ：January 23 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Kazuyoshi Kiyohara ( Dept. of Math., Okayama Univ. ) Title ：The conjugate loci and an asymptotic property of conjugate points on ellipsoids （Abstract） Date ：January 16 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Sadahiro Maeda ( Department of Math., Saga Univ. ) Title ：Homogeneous closed curves on geodesic spheres in a complex projective space from the viewpoint of submanifold theory （Abstract） Date ：December 19 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yoo Chul-Moon ( Yukawa Institute for Theoretical Physics, Kyoto Univ. ) Title ：Gravitational Lensing in a Clumpy Universe （Abstract） Date ：December 12 (Wed.) 17:15～18:15 Place ：Dept. of Mathematics, Sci. Bldg., 3068 Top Speaker ：Yasuo Kurita ( Osaka City University Advanced Mathematical Institute ) Title ：Thermodynamics of five-dimensional black holes with squashed horizons （Abstract） Date ：December 12 (Wed.) 16:10～17:10 Place ：Dept. of Mathematics, Sci. Bldg., 3068 Top Speaker ：Masanori Kobayashi ( Dept. of Math., Tokyo Metro. Univ. ) Title ：Elliptic fibration on compact Spin(7)-manifolds （after K.Aga) （Abstract） Date ：November 28 (Wed.) 17:00～18:30 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Shouhei Honda ( Dept. of Math., Kyoto Univ. ) Title ：Measure theory on the limit spaces of Riemannian manifolds （Abstract） Date ：October 24 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Daisuke Yamakawa ( Dept. of Math., Kyoto Univ. ) Title ：Moduli of parabolic connections on a Riemann surface with marked points and representations of quiver （Abstract） Date ：October 17 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yukinori Yasui ( Dept. of Physics., Osaka City Univ. ) Title ：Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness （Abstract） Date ：October 10 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Miyuki Koiso ( Dept. of Math., Nara Women's Univ. ) Title ：Rolling construction of plane curves and its application to surface theory （Abstract） Date ：July 18 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Tobias Muehlenbruch ( OIST Institute, Okinawa) Title ：Transfer operators for the geodesic billiard for Hecke triangle groups and the dynamics of continued fractions （Abstract） Date ：June 27 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yumiko Kitagawa ( Dept. of Math., Nara Women's Univ. ) Title ：On subriemannian contact manifolds （Abstract） Date ：June 13 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Atsushi Kasue ( Dept. of Math., Kanazawa Univ. ) Title ：Convergence of metric graphs and energy forms （Abstract） Date ：June 6 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yohei Komori ( Dept. of Math., Osaka City Univ. ) Title ：Drawing Bers embeddings of the 1-dimensional Teichmuller space （Abstract） Date ：May 23 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yasufumi Nitta ( Dept. of Math., Osaka Univ. ) Title ：Generalized Calabi-Yau structures and Duistermaat-Heckman theorem （Abstract） Date ：May 16 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Dr. Nahid Sultana (Osaka City University Advanced Mathematical Institute) Title ：Constant mean curvature surfaces of revolution in spherically symmetric 3-manifolds, and their stability （Abstract） Date ：April 18 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top Speaker ：Yoshihiro Ohnita (Dept. of Math., Osaka City Univ.) Title ："Classification of homogeneous Lagrangian submanifolds in complex hyperquadrics" （Abstract） Date ：April 11 (Wed.) 14:40～16:10 Place ：Dept. of Mathematics, Sci. Bldg., 3040 Top

 Abstracts

 Speaker： Motohico Mulase (UC Davis, RIMS ) Title： Sp-Hitchin Systems and Sp-invariant KP Systems

In this informal talk I explain a concrete example of mirror symmetric pairs of complex manifolds that exhibits Strominger-Yau-Zalsow geometric mirror symmetry. These are obtained as a family of Prym varieties and its Fourier-Mukai dual family. The total space of the family is an algebraically completely integrable Hamiltonian system, and the Hamiltonian vector fields are determined by the (seemingly new) Sp-invariant KP flows.

 Speaker： Yuichi Nohara ( Dept. of Math., Tohoku Univ. ) Title： Toric degeneration of flag manifolds and the Gelfand-Cetlin system

It is well known that a polarized toric variety is related to a convex polytope in two different ways: the moment map and monomial basis. For a flag manifold $U(n)/T$, we can also associate a certain polytope $\Delta$ in similar ways: the Gelfand-Cetlin system, a completely integrable system; the Gelfand-Cetlin basis, a basis of an irreducible representation of $U(n)$. Furthermore, the flag manifold degenerates into the toric variety corresponding to $\Delta$. Kogan-Miller proved that the Gelfand-Cetlin basis can be deformed into a monomial basis on the toric variety under the degeneration. In this talk we prove that the Gelfand-Cetlin system can be deformed into the moment map of the toric variety.

 Speaker： Hiroshi Iriyeh (Tokyo Denki Univ.) Title： On the local minimality of geodesics of Hofer's metric.

Hofer's metric on the Hamiltonian diffeomorphism groupof a symplectic manifold $M$ is a kind of Finsler metric which is defined by Hamiltonian functions on $M$.

In this talk, we define geodesics of Hofer's metric as critical points of Hofer's length functions and give a characterization by Hamiltonian functions, in the case of Hamiltonian diffeomorphism groups (Lalonde-McDuff), in the case of the space of Lagrangian submanifolds (joint work with T.Otofuji).

Next we explain the local minimality following Polterovich's idea via Floer homology in the case of Hamiltonian diffeomorphism groups.

 Speaker： Ryoichi Kobayashi ( Dept. of Math., Nagoya Univ. ) Title： Ricci flow ancient solutions arising from the natural collapsing of the twistor space of positive quaternion K\"ahler manifolds

The first part of this lecture is an introduction of Perelman's No Local Collapsing Theorem (background, proof idea and conclusions) with emphasis on the Ricci flow "ancient solutions" arising from singularities of the Ricci flow at finite time. Such ancient solution is extremely important because it includes all information of the singularity under question. In the second part, we show that there exists a 2 parameter family of the Ricci flow ancient solutions on the twistor space of positive quaternion K\"ahler manifolds. Each of these solutions is (after scaling) asymptotically K\"ahler-Einstein as $t \to -\infty$ and extinct in finite time where the extinction (after scaling) corresponds to one of the three types of the natural collapsing of the twistor space. We discuss on application of these ancnent solutions to the LeBrun-Salamon conjecture.

 Speaker： Yu Kawakami (Dept. of Math., Nagoya Univ., Osaka City University Advanced Mathematical Institute ) Title： Width and Ricci flow

In this talk, we reveal the relationship between "Width" and "Ricci flow", prove the result of the finite extinction time for the Ricci flow on homotopy 3-spheres by Tobias H. Colding and William P. Minicozzi II.
【References】
1. T. H. Colding and W. P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman, J Amer. Math. Soc. 18 (2005), 561-569.
2. T. H. Colding and W. P. Minicozzi II, Width and finite extinction time of Ricci flow, preprint, arXiv:0707.0108.
3. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint, math.DG/0307245.

 Speaker： Takumi Yamada ( Dept. of Math., Shimane Univ. ) Title： Lattice of compact complex-parallelizable pseudo-Kaehler manifolds

A complex manifold is called complex-parallelizable if the holomorphic tangent　bundle of it is trivial. It is well known that a compact complex-parallelizable manifold is complex homogeneous. In the present talk we introduce a necessary condition that a compact compact complex-parallelizable manifold has a pseudo-Kaehler structure.

 Speaker： Yoshihiro Ohnita ( Dept. of Math.,Osaka City Univ. ) Title： On the deformation of certain 3-dimensional minimal Legendrian submanifolds

A minimal Legendrian submanifold in a Sasakian manifold is by definition a Legendrian submanifold in a Sasakian manifold which is a minimal submanifold in the sense that its mean curvature vector field with respect to its Sasakian metric vanishes. The minimal Legendrian deformation means a smoooth family of minimal Legendrian submanifolds. In this talk, after we explain the fundamental results on a Legendrian submanifold in a Sasakian manifold such as Legedrian deformation, Legendrian stability, relations with Lagrangian submanifolds etc., we will discuss minimal Legendirian deformations for two examples of $3$-dimensional compact minimal Legendrian submanifolds embedded in the $7$-dimensional standard $\eta$-Einstein Sasakian manifolds.

 Speaker： Kazuyoshi Kiyohara ( Dept. of Math., Okayama Univ. ) Title： The conjugate loci and an asymptotic property of conjugate points on ellipsoids

This is a joint work with Jin-ichi Itoh.
It is well known that the geodesic flow of any ellipsoid is completely integrable in the sense of hamiltonian mechanics. In this talk, we would like to discuss much finer properties of the behavior of geodesics. In particular, we shall show that the conjugate locus of a general point contains just three connected components of singularities, each of which is a cuspidal edge (n>2). This result is a higher dimensional version of "the last geometric statement of Jacobi", which asserts that the conjugate locus of a general point on any two-dimensional ellipsoid contains just four cusps. Also, we shall show that the distribution of the conjugate points along a general geodesic possesses an interesting asymptotic property. The above results also hold for some Liouville manifolds.

 Speaker： Sadahiro Maeda ( Department of Math., Saga Univ. ) Title： Homogeneous closed curves on geodesic spheres in a complex projective space from the viewpoint of submanifold theory

It is known that every geodesic on each geodesic sphere in a complex projective space is a homogeneous curve, that is, every geodesic is an orbit under a certain one-parameter subgroup of the isometry group of the geodesic sphere. In this talk, we give a family of homogeneous non-geodesic closed curves on this geodesic sphere through submanifold theory.

 Speaker： Yoo Chul-Moon ( Yukawa Institute for Theoretical Physics, Kyoto Univ. ) Title： Gravitational Lensing in a Clumpy Universe

According to recent observation and theoretical studies, 99% of the mass in our universe is not luminous. Therefore, in order to investigate the all of the inhomogeneities in our universe, we have to know the distribution of the "dark components". Consequently, gravitational lensing effects are focused on as a means of observation of the dark matter distribution. Gravitational lensing effects are General relativistic effects on light rays which are induced by the matter distribution on the ray path. In this talk, first, I will give an explanation of these effects with a focus on the geometry of the ray bundle. Then, I will talk about our work in which we have studied multiple lensing effects on the light rays from point sources.

 Speaker： Yasuo Kurita ( Osaka City University Advanced Mathematical Institute ) Title： Thermodynamics of five-dimensional black holes with squashed horizons

In the five-dimensional Einstein gravity, there exist black hole solutions with squashed horizons. In general, black holes have thermodynamical nature because they are characterized by the irreversibility. After giving a brief review of black holes and thermodynamics from the viewpoint of theoretical physics, I will talk about thermodynamics of the five-dimensional black holes with squashed horizons.

 Speaker： Masanori Kobayashi ( Dept. of Math., Tokyo Metro. Univ. ) Title： Elliptic fibration on compact Spin(7)-manifolds （after K.Aga)

Compact SU(4)-manifold with elliptic fibration interested in from the point of view from string theory. In this talk we treat compact Spin(7)-manifolds, which do not have complex structure, nevertheless with elliptic fibration mainly after the master thesis of Ki-ichiro Aga (TMU).

 Speaker： Shouhei Honda ( Dept. of Math., Kyoto Univ. ) Title： Measure theory on the limit spaces of Riemannian manifolds

M.Gromov showed; the set of Riemannian manifolds whose Ricci curvatures have a definite lower bound, is precompact with respect to the Gromov-Hausdorff distance. Recently, many important results for the Gromov-Hausdorff limit spaces of such Riemannian manifolds were shown by J.Cheeger, T.H.Colding. In this talk, we study some measure theoritic results for such spaces with related to the works of them.

 Speaker： Daisuke Yamakawa ( Dept. of Math., Kyoto Univ. ) Title： Moduli of parabolic connections on a Riemann surface with marked points and representations of quiver

In this talk, I will describe a moduli space of parabolic connections on a Riemann surface with marked points as a certain analogue of a quiver variety. This variety is constructed as a "group-valued" holomorphic symplectic quotient and has many similarities to a quiver variety.

 Speaker： Yukinori Yasui ( Dept. of Physics., Osaka City Univ. ) Title： Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness

Higher dimensional black hole solutions have attracted renewed interests in the recent developments of supergravity and superstring theories. Recently, the d-dim. Kerr NUT-de Sitter metrics were constructed by Chen-Lu-Pope. All the known vacuum type D black hole solutions are included in these metrics. In this talk we study spacetimes with a conformal Killing-Yano tensor. It is shown that the Kerr-NUT-de Sitter spacetime is the only spacetime admitting a rank-2 closed conformal Killing-Yano tensor with a certain symmetry.

 Speaker： Miyuki Koiso ( Dept. of Math., Nara Women's Univ. ) Title： Rolling construction of plane curves and its application to surface theory

We give a new geometric description of the rolling curve of a general plane curve and apply it to the generating curves of anisotropic Delaunay surfaces. For example, the rolling curve of the generating curve of an anisotropic unduloid or nodoid S is obtained as a type of dual curve' of a mean curvature profile' of S. Here, a mean curvature profile of S is a curve whose curvature is equal to twice the mean curvature of S. This generalizes the classical construction for rotationally symmetric surfaces with constant mean curvature due to Delaunay.

 Speaker： Tobias Muehlenbruch ( OIST Institute, Okinawa) Title： Transfer operators for the geodesic billiard for Hecke triangle groups and the dynamics of continued fractions

After introducing Hecke triangle groups and the associated geodesic billiard, I show a discretization, relating periodic orbits on the geodesic billiard to periodic continued fraction expansions of points. This allows us to describe the dynamics on the billiard system by understanding the discrete dynamics related to the continued fractions. The continued fractions are the so called nearest $\lambda$-integer continued fractions. Hurwitz already studied these fractions for $\lambda =1$.

 Speaker： Yumiko Kitagawa ( Dept. of Math., Nara Women's Univ. ) Title： On subriemannian contact manifolds

A subriemannian manifold (M,D,g) is a differential manifold M equipped with a subbundle D of the tangent bundle TM of M and a riemannian metric g on D. In particular, it is called a subriemannian contact manifold if D is a contact structure, i.e., a subbundle of codimension 1 and non-degenerate. In this talk we study the structure of infinitesimal automorphisms of a subriemannian contact manifold.

 Speaker： Atsushi Kasue ( Dept. of Math., Kanazawa Univ. ) Title： Convergence of metric graphs and energy forms

Some aspects of infinite networks will be discussed from a view point of convergence of finite networks with respect to the topology of both the Gromov-Hausdorff distance and variational convergence called $\Gamma$-convergence.

 Speaker： Yohei Komori ( Dept. of Math., Osaka City Univ. ) Title： Drawing Bers embeddings of the 1-dimensional Teichmuller space

We present a computer-oriented method of producing pictures of Bers embeddings of the 1-dimensional Teichmuller space. Our algorithm consists of two steps; we first compute the monodromy representation of Schwarzian differential equation by numerical integral. Then we decide whether the monodromy group is discrete by applying Jorgensen's theory on the quasi-Fuchsian space of once-punctured tori. This is a joint work with Toshiyuki Sugawa (Hiroshima), Masaaki Wada(Nara) and Yasushi Yamashita(Nara).

 Speaker： Yasufumi Nitta ( Dept. of Math., Osaka Univ. ) Title： Generalized Calabi-Yau structures and Duistermaat-Heckman theorem

This talk is about generalized complex structures. This is a new kind of geometrical structure introduced by Hitchin and developed by Gualtieri, which contains complex and symplectic geometry as its extremal special cases. In the present talk, we shall introduce the reduction theorem for a group action on generalized complex manifolds. Moreover we shall explain that the Duistermaat-Heckman theorem holds in the case of generalized Calabi-Yau manifolds.

 Speaker： Dr. Nahid Sultana (Osaka City University Advanced Mathematical Institute) Title： Constant mean curvature surfaces of revolution in spherically symmetric 3-manifolds, and their stability

We compute explicit conformal parametrizations of Delaunay surfaces in $\mathbb{R}^3$, $\mathbb{S}^3$ and $\mathbb{H}^3$ by using the generalized Weierstrass type representation for CMC surfaces. By using these explicit parametrizations, we introduce the explicit area formula for the fundamental pieces of Delaunay surfaces. When Delaunay surfaces in $\mathbb{S}^3$ close to become tori, we can study their Morse index. The Morse index of general closed CMC surfaces of revolution in $\mathbb{S}^3$ is still unknown. Hence, we compute lower bounds for the Morse index and nullity of CMC tori of revolution in $\mathbb{S}^3$. To test the sharpness of the lower bounds, we numerically compute the eigenvalues of the Jacobi operator. Furthermore, we study the stability properties of CMC surfaces of revolution in general simply-connected spherically symmetric $3$-spaces, and in particular case a positive-definite $3$-dimensional slice of Schwarzschild space.

 Speaker： Yoshihiro Ohnita (Dept. of Math., Osaka City Univ.) Title： "Classification of homogeneous Lagrangian submanifolds in complex hyperquadrics"

This talk is based on my recent joint work with Hui Ma (Tsinghua University, Peking). The $n$-dimensional complex hyperquadric $Q_{n}({\bold C})$ is a compact complex algebraic hypersurface defined by the quadratic equation $z_{0}^{2}+z_{1}^{2}+\cdots+z_{n}^{2}+z_{n+1}^{2}=0$ in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented $2$-dimensional vector subspaces of ${\bold R}^{n+2}$. It is a compact Hermitian symmetric space of rank $2$. In this talk we provide a classification of compact homogeneous Lagrangian submanifolds, i.e. Lagrangian orbits of compact Hamiltonian group actions, in complex hyperquadrics by using the moment map technique from the viewpoint of homogeneous isoparametric hypersurface geometry in spheres. We showed that any compact homogeneous Lagrangian submanifold in complex hyperquadrics is obtained as the Gauss image of a homogeneous isoparametric hypersurface in a sphere or certain one-parameter families of Lagrangian orbits in $Q_{n}({\bold C})$.