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Geometry Colloquium
Seminar information archive ～04/25｜Next seminar｜Future seminars 04/26～
| Date, time & place | Friday 10:00 - 11:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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Seminar information archive
2012/12/19
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
FUJIWARA, Koji (Kyoto University)
Funk metric on Weil-Petersson spaces (JAPANESE)
FUJIWARA, Koji (Kyoto University)
Funk metric on Weil-Petersson spaces (JAPANESE)
[ Abstract ]
We discuss the Funk function $F(x; y)$ on a Teichmuller space with the Weil-Petersson metric introduced by Yamada, $F(x; y)$ is an asymmetric distance and invariant by the action of the mapping class group. The Funk metric was originally studied for an open convex subset in a Euclidean space by Funk. Its symmetrization is the Hilbert metric.
We discuss the Funk function $F(x; y)$ on a Teichmuller space with the Weil-Petersson metric introduced by Yamada, $F(x; y)$ is an asymmetric distance and invariant by the action of the mapping class group. The Funk metric was originally studied for an open convex subset in a Euclidean space by Funk. Its symmetrization is the Hilbert metric.
2012/12/05
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Hiroaki Ishida (Osaka City University Advanced Mathematical Institute)
Maximal torus actions on complex manifolds (JAPANESE)
Hiroaki Ishida (Osaka City University Advanced Mathematical Institute)
Maximal torus actions on complex manifolds (JAPANESE)
[ Abstract ]
We say that an effective action of a compact torus $T$ on a connected manifold $M$ is maximal if there is an orbit of dimension $2\\dim T-\\dim M$. In this talk, we give a one-to-one correspondence between the family of connected closed complex manifolds with maximal torus actions and the family of certain combinatorial objects, which is a generalization of the correspondence between complete nonsingular toric varieties and nonsingular complete fans. As an application, we construct a lot of concrete examples of non-K\\"{a}hler manifolds with maximal torus actions.
We say that an effective action of a compact torus $T$ on a connected manifold $M$ is maximal if there is an orbit of dimension $2\\dim T-\\dim M$. In this talk, we give a one-to-one correspondence between the family of connected closed complex manifolds with maximal torus actions and the family of certain combinatorial objects, which is a generalization of the correspondence between complete nonsingular toric varieties and nonsingular complete fans. As an application, we construct a lot of concrete examples of non-K\\"{a}hler manifolds with maximal torus actions.
2012/11/28
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Shouhei Honda (Kyushu University)
Ricci curvature and angles (JAPANESE)
Shouhei Honda (Kyushu University)
Ricci curvature and angles (JAPANESE)
[ Abstract ]
Let X be the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound. In this talk we will give the definition of angles between geodesics on X. We apply this to prove there is a weakly twice differentiable structure on X and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.
Let X be the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound. In this talk we will give the definition of angles between geodesics on X. We apply this to prove there is a weakly twice differentiable structure on X and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.
2012/11/14
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Kotaro Kawai (Tohoku University)
Construction of coassociative submanifolds (JAPANESE)
Kotaro Kawai (Tohoku University)
Construction of coassociative submanifolds (JAPANESE)
[ Abstract ]
The notion of coassociative submanifolds is defined as the special class of the minimal submanifolds in G_2 manifolds. In this talk, we introduce the method to construct coassociative submanifolds by using the symmetry of the Lie group action. As an application, we give explicit examples in the 7-dimensional Euclidean space and in the anti-self-dual bundle over the 4-sphere.
The notion of coassociative submanifolds is defined as the special class of the minimal submanifolds in G_2 manifolds. In this talk, we introduce the method to construct coassociative submanifolds by using the symmetry of the Lie group action. As an application, we give explicit examples in the 7-dimensional Euclidean space and in the anti-self-dual bundle over the 4-sphere.
2012/10/31
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Kei Irie (Kyoto University)
Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits (JAPANESE)
Kei Irie (Kyoto University)
Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits (JAPANESE)
[ Abstract ]
Hofer-Zehnder (HZ) capacity is an invariant of symplectic manifolds, which is important in symplectic topology and Hamiltonian dynamics. The energy-capacity inequality (due to Hofer and many others) claims that HZ capacity of a domain is bounded from above by its dispalcement energy.
In this talk, we prove a variant of this inequality, which is applicable to nondisplaceable domains. We also give some applications, including case of disc cotangent bundles.
Hofer-Zehnder (HZ) capacity is an invariant of symplectic manifolds, which is important in symplectic topology and Hamiltonian dynamics. The energy-capacity inequality (due to Hofer and many others) claims that HZ capacity of a domain is bounded from above by its dispalcement energy.
In this talk, we prove a variant of this inequality, which is applicable to nondisplaceable domains. We also give some applications, including case of disc cotangent bundles.
2012/10/17
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Masato Mimura (Tohoku University)
p-Kazhdan constants and non-expanders (JAPANESE)
Masato Mimura (Tohoku University)
p-Kazhdan constants and non-expanders (JAPANESE)
[ Abstract ]
In study of graphs and finitely generated groups (as Cayley graphs) as metric spaces with the path metrics, one basic idea is to "linearize" them, more precisely, to embed them into certain Banach spaces in some nice way. Special attention has been paid to embeddings of graphs into Hilbert spaces or l^p spaces. It is a well-known result that a "family of expanders", namely, a family of finite graphs (of unifromly bounded degree) with uniform lower bound of spectral gaps (equivalently, of Cheeger constants), does not coarsely embed into Hilbert spaces, or l^p spaces.
In this talk, we investigate a "family of NON-expanders" coming from Cayley graphs of a family of finitely generated groups. In this setting we define l^p-version of the Kazhdan constant and of the property tau constant for groups, and study the decay rate of p-spectral gap of non-expanders in terms of them. This gives some metric geometrical information on the family. Our main example will be the family of (Cayley graphs of SL_n(Z/k_nZ)), indexed by n>2, for (k_n)_n a sequence of natural numbers>2 and with respect to standard 4-element generating sets. We will start from basic definitions, such as ones of Cayley graphs, expander families, and Kazhdan constants.
In study of graphs and finitely generated groups (as Cayley graphs) as metric spaces with the path metrics, one basic idea is to "linearize" them, more precisely, to embed them into certain Banach spaces in some nice way. Special attention has been paid to embeddings of graphs into Hilbert spaces or l^p spaces. It is a well-known result that a "family of expanders", namely, a family of finite graphs (of unifromly bounded degree) with uniform lower bound of spectral gaps (equivalently, of Cheeger constants), does not coarsely embed into Hilbert spaces, or l^p spaces.
In this talk, we investigate a "family of NON-expanders" coming from Cayley graphs of a family of finitely generated groups. In this setting we define l^p-version of the Kazhdan constant and of the property tau constant for groups, and study the decay rate of p-spectral gap of non-expanders in terms of them. This gives some metric geometrical information on the family. Our main example will be the family of (Cayley graphs of SL_n(Z/k_nZ)), indexed by n>2, for (k_n)_n a sequence of natural numbers>2 and with respect to standard 4-element generating sets. We will start from basic definitions, such as ones of Cayley graphs, expander families, and Kazhdan constants.
