Date: 2008 12/11 (Thursday), 12 (Friday)
Place: Meiji University, Ikuta campus, Univ. Bldg. 2 Annex A, Multimedia room A401
Abstract : Joint work with Eva Miranda. When geometric quantization using a real polarization is applied to a "nice enough" manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. Subsequently, several authors have taken this as motivation for counting Bohr-Sommerfeld fibres when studying the quantization of manifolds that are less "nice." One example of a "less nice" polarization is a completely integrable system, with the singular foliation given by the fibres of the moment map, which (if it is compact) will always have singularities. Nondegenerate singularities appearing in integrable systems were classified by Eliasson and Miranda into combinations of three basic types, called elliptic, hyperbolic, and focus-focus. Because of these singularities, Sniatycki's theorem doesn't apply, but we would still like to quantize these systems using this singular polarization. We study an approach to the quantization of systems with elliptic or hyperbolic singularities, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization in some cases.
Abstract : Joint work with Yuichi Nohara and Kazushi Ueda. Gelfand-Cetlin systems are completely integrable Hamiltonian systems on generalized flag manifolds of type A. Their momentum polytopes are called Gelfand-Cetlin polytopes, whose integral points are in bijection with Gelfand-Cetlin bases in representation theory. In this talk, we will define the notion of toric degeneration of an integrable system and show that Gelfand-Cetlin systems have such degenerations. As an application, some results about Lagrangian torus fibers of Gelfand-Cetlin systems will be shown.
Abstract : Joint work with Mikio Furuta and Hajime Fujita. For a prequantizable closed symplectic manifold a Riemann-Roch number is defined to be the index of a spin^c Dirac operator with coefficients in the prequatization line bundle. In this talk we show that if a prequantizable symplectic manifold admits a structure of a singular Lagrangian fibration, the Riemann-Roch number is localized on nonsingular Bohr-Sommerfeld fibers and singular fibers. The technique used here is the localization of Witten-type.
Abstract : This is a preliminary report on joint work in progress with Gregory D. Landweber and Reyer Sjamaar. Let $X$ be a $G$-space, where $G$ is a compact connected Lie group acting on a topological space $X$. Atiyah proved that the $G$-equivariant $K$-theory of $X$ is a direct summand of the $T$-equivariant $K$-group of $X$, where $T$ is a maximal torus of $G$; however, he did not tell whether this direct summand is determined by the Weyl group action on $K_T(X)$. In our work, we show that the action of the Weyl group $W$ extends to an action of a ring ${¥mathcal D}$ generated by divided difference operators. These operators were first introduced in the context of Schubert calculus by Demazure; ours is a generalization to general $G$-spaces. The ring ${¥mathcal D}$ contains a left ideal $I({¥mathcal D})$ we call the ``augmentation left ideal'' in analogy with the augmentation ideal of the group ring of $W$. Our main theorem is that $K_G(X)$ is isomorphic to the subring of $K_T(X)$ annihilated by $I({¥mathcal D})$.
Abstract : Discrete subgroups of 2-dimensional Möbius transformations are known Kleinian groups. In this talk, I will introduce some subgroups of 3-dimensional Möbius transformations and its figure of limit sets.
Abstract : To Legendrian submanifolds of contact manifolds, one may associate their (relative) contact homology. The contact manifolds may occur as boundaries of symlectic spaces. In such a case, it is natural to consider Lagrangian surfaces with a Legendrian boundary. It has been known/expected that such Lagrangians (if exact) induce maps between the contact homologies of their boundary components. In joint work with T. Ekholm and K. Honda, we wrote down explicitly some of these maps in the case when the Legendrians are one-dimensional. This can be used to distinguish between various Lagrangians spanning the same Legendrian.