Tuesday Seminar on Topology (April -- July, 2009)

[Japanese]   [Past Programs]
16:30 -- 18:00 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:00 -- 16:30 Common Room

Last updated June 30, 2009
Information :　
Toshitake Kohno
Nariya Kawazumi

April 21 -- Room 056, 16:30 -- 18:00

Ivan Marin (Univ. Paris VII)

Some algebraic aspects of KZ systems

Abstract: Knizhnik-Zamolodchikov (KZ) systems enables one to construct representations of (generalized) braid groups. While this geometric construction is now very well understood, it still brings to attention, or helps constructing, new algebraic objects. In this talk, we will present some of them, including an infinitesimal version of Iwahori-Hecke algebras and a generalization of the Krammer representations of the usual braid groups.

April 28 -- Room 056, 16:30 -- 18:00

Kengo Hirachi (The University of Tokyo)

The ambient metric in conformal geometry

Abstract: In 1985, Charles Fefferman and Robin Graham gave a method for realizing a conformal manifold of dimension n as a submanifold of a Ricci-flat Lorentz metric on a manifold of dimension n+2, which is now called the ambient space. Using this correspondence, one can construct many examples of conformal invariants and conformally invariant operators. However, if n is even, their construction of the ambient space is obstructed at the jet of order n/2 and thereby the application of the ambient space was limited. In this talk, I'll recall basic ideas of the ambient space and then explain how to avoid the difficulty and go beyond the obstruction. This is a joint work with Robin Graham.

May 12 -- Room 056, 16:30 -- 18:00

Yoshifumi Matsuda (The University of Tokyo)

Discrete subgroups of the group of circle diffeomorphisms

Abstract: Typical examples of discrete subgroups of the group of circle diffeomorphisms are Fuchsian groups. In this talk, we construct discrete subgroups of the group of orientation-preserving real analytic cirlcle diffeomorphisms which are not topologically conjugate to finite coverings of Fuchsian groups.

May 19 -- Room 056, 16:30 -- 18:00

Mark Hamilton (The University of Tokyo, JSPS)

Geometric quantization of integrable systems

Abstract: The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities.

Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nontheless too singular for Sniatycki's theorem to apply.

In this talk we will explore one approach to the quantization of integrable systems, 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.

This is joint work with Eva Miranda.

May 26 -- Room 056, 16:30 -- 18:00

Keiichi Sakai (The University of Tokyo)

Configuration space integrals and the cohomology of the space of long embeddings

Abstract: It is known that some non-trivial cohomology classes, such as finite type invariants for (long) 1-knots (Bott-Taubes, Kohno, ...) and invariants for codimension two, odd dimensional long embeddings (Bott, Cattaneo-Rossi, Watanabe) are given as configuration space integrals associated with trivalent graphs. In this talk, I will describe more cohomology classes by means of configuration space integral, in particular those arising from non-trivalent graphs and a new formulation of the Haefliger invariant for long 3-embeddings in 6-space, in relation to Budney's little balls operad action and Roseman-Takase's deform-spinning. This is in part a joint work with Tadayuki Watanabe.

June 2 -- Room 056, 16:30 -- 18:00

Alexander Voronov (University of Minnesota)

Graph homology: Koszul duality = Verdier duality

Abstract: Graph cohomology appears in computation of the cohomology of the moduli space of Riemann surfaces and the outer automorphism group of a free group. In the former case, it is graph cohomology of the commutative and Lie types, in the latter it is ribbon graph cohomology, that is to say, graph cohomology of the associative type. The presence of these three basic types of algebraic structures hints at a relation between Koszul duality for operads and Poincare-Lefschetz duality for manifolds. I will show how the more general Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This is a joint work with Andrey Lazarev.

June 9 -- Room 056, 16:30 -- 18:00

Kiyonori Gomi (Kyoto University)

A finite-dimensional construction of the Chern character for twisted K-theory

Abstract: Twisted K-theory is a variant of topological K-theory, and is attracting much interest due to applications to physics recently. Usually, twisted K-theory is formulated infinite-dimensionally, and hence known constructions of its Chern character are more or less abstract. The aim of my talk is to explain a purely finite-dimensional construction of the Chern character for twisted K-theory, which allows us to compute examples concretely. The construction is based on twisted version of Furuta's generalized vector bundle, and Quillen's superconnection. This is a joint work with Yuji Terashima.

June 16 -- Room 056, 16:30 -- 18:00

Masatoshi Sato (The University of Tokyo)

The abelianization of the level 2 mapping class group

Abstract: The level d mapping class group is a finite index subgroup of the mapping class group of an orientable closed surface. For d greater than or equal to 3, the abelianization of this group is equal to the first homology group of the moduli space of nonsingular curves with level d structure. In this talk, we determine the abelianization of the level d mapping class group for d=2 and odd d. For even d greater than 2, we also determine it up to a cyclic group of order 2.

June 23 -- Room 056, 16:30 -- 18:00

Yusuke Kuno (The University of Tokyo)

The Meyer functions for projective varieties and their applications

Abstract: Meyer function is a kind of secondary invariant related to the signature of surface bundles over surfaces. In this talk I will show there exist uniquely the Meyer function for each smooth projective variety. Our function is a class function on the fundamental group of some open algebraic variety. I will also talk about its application to local signature for fibered 4-manifolds

June 30 -- Room 056, 16:30 -- 18:00

Takahiro Kitayama (The University of Tokyo)

Torsion volume forms and twisted Alexander functions on character varieties of knots

Abstract: Using non-acyclic Reidemeister torsion, we can canonically construct a complex volume form on each component of the lowest dimension of the $SL_2(\mathbb{C})$-character variety of a link group. This volume form enjoys a certain compatibility with the following natural transformations on the variety. Two of them are involutions which come from the algebraic structure of $SL_2(\mathbb{C})$ and the other is the action by the outer automorphism group of the link group. Moreover, in the case of knots these results deduce a kind of symmetry of the $SU_2$-twisted Alexander functions which are globally described via the volume form.

July 14 -- Room 056, 17:00 -- 18:00

Makoto Sakuma (Hiroshima University)

The Cannon-Thurston maps and the canonical decompositions of punctured-torus bundles over the circle.

Abstract: To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy, there are associated two tessellations of the complex plane: one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra, and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group. In this talk, I will explain the relation between these two tessellations (joint work with Warren Dicks). I will also explain the relation of the fractal tessellation and the "circle chains" of double cusp groups converging to the fiber group (joint work with Caroline Series). If time permits, I would like to discuss possible generalization of these results to higher-genus punctured surface bundles.