Tuesday Seminar on Topology

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) KOHNO Toshitake, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya Tea: 16:30 - 17:00 Common Room

Seminar information archive

2010/01/26

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

On the (co)chain type levels of spaces
[ Abstract ]
Avramov, Buchweitz, Iyengar and Miller have introduced
the notion of the level for an object of a triangulated category.
The invariant measures the number of steps to build the given object
out of some fixed object with triangles.
Using this notion in the derived category of modules over a (co)chain
algebra,
we define a new topological invariant, which is called
the (co)chain type level of a space.
In this talk, after explaining fundamental properties of the invariant,
I describe the chain type level of the Borel construction
of a homogeneous space as a computational example.

I will also relate the chain type level of a space to algebraic
approximations of the L.-S. category due to Kahl and to
the original L.-S. category of a map.

2010/01/19

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

Localization via group action and its application to
the period condition of algebraic minimal surfaces
[ Abstract ]
The optimal estimate for the number of exceptional
values of the Gauss map of algebraic minimal surfaces is a long
standing problem. In this lecture, I will introduce new ideas
toward the solution of this problem. The collective Cohn-Vossen
inequality" is the key idea. From this we have effective
Nevanlinna's lemma on logarithmic derivative for a certain class
of meromorphic functions on the disk. On the other hand, we can
construct a family holomorphic functions on the disk from the
Weierstrass data of the algebraic minimal surface under
consideration, which encodes the period condition.
Applying effective Lemma on logarithmic derivative to these
functions, we can extract an intriguing inequality.

2010/01/12

16:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)

Index problem for generically-wild homoclinic classes in dimension three
[ Abstract ]
In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.

On a generalized suspension theorem for directed Fukaya categories
[ Abstract ]
The directed Fukaya category $\\mathrm{Fuk} W$ of exact Lefschetz
fibration $W : X \\to \\mathbb{C}$ proposed by Kontsevich is a
categorification of the Milnor lattice of $W$. This is defined as the
directed $A_\\infty$-category $\\mathrm{Fuk} W = \\mathrm{Fuk}^\\to \\mathbb{V}$ generated by a distinguished basis $\\mathbb{V}$ of
vanishing cycles.

Recently Seidel has proved that this is stable under the suspension $W + u^2$ as a consequence of his foundational work on the directed
Fukaya category. We generalize his suspension theorem to the $W + u^d$
case by considering partial tensor product $\\mathrm{Fuk} W \\otimes' \\mathcal{A}_{d-1}$, where $\\mathcal{A}_{d-1}$ is the category
corresponding to the $A_n$-type quiver. This also generalizes a recent
work by the author with Kazushi Ueda.

2010/01/05

16:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)

The volume growth of hyperkaehler manifolds of type $A_{\\infty}$
[ Abstract ]
Hyperkaehler manifolds of type $A_{\\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, 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 $r0$ 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 松尾 信一郎 (東京大学大学院数理科学研究科) 17:30-18:30 On the Runge theorem for instantons [ Abstract ] A classical theorem of Runge in complex analysis asserts that a meromorphic function on a domain in the Riemann sphere can be approximated, over compact subsets, by rational functions, that is, meromorphic functions on the Riemann sphere. This theorem can be paraphrased by saying that any solution of the Cauchy-Riemann equations on a domain in the Riemann sphere can be approximated, over compact subsets, by global solutions. In this talk we will present an analogous result in which the Cauchy-Riemann equations on Riemann surfaces are replaced by the Yang-Mills instanton equations on oriented 4-manifolds. We will also mention that the Runge theorem for instantons can be applied to develop Yang-Mills gauge theory on open 4-manifolds. 2009/12/22 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) 寺杣 友秀 (東京大学大学院数理科学研究科) Relative DG-category, mixed elliptic motives and elliptic polylog [ Abstract ] We consider a full subcategory of mixed motives generated by an elliptic curve over a field, which is called the category of mixed elliptic motives. We introduce a DG Hopf algebra such that the categroy of mixed elliptic motives is equal to that of comodules over it. For the construction, we use the notion of relative DG-category with respect to GL(2). As an application, we construct an mixed elliptic motif associated to the elliptic polylog. It is a joint work with Kenichiro Kimura. 2009/12/15 17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) 砂田 利一 (明治大学) Open Problems in Discrete Geometric Analysis [ Abstract ] Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups. 2009/12/01 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) Andrei Pajitnov (Univ. de Nantes) Non-Abelian Novikov homology [ Abstract ] Classical construction of S.P. Novikov associates to each circle-valued Morse map a chain complex defined over a ring of Laurent power series in one variable. In this survey talk we shall explain several results related to the construction and properties of non-Abelian generalizations of the Novikov complex. 2009/11/24 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) Adam Clay (University of British Columbia) A topological approach to left orderable groups [ Abstract ] A group G is said to be left orderable if there is a strict total ordering of its elements such that gin G. Left orderable groups have been useful in solving many problems in topology, and now we find that topology is returning the favour: the set of all left orderings of a group is denoted by LO(G), and it admits a natural topology, under which LO(G) becomes a compact topological space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases. For example, the space of left orderings of the braid group B_n for n>2 contains isolated points (yet it is uncountable), while the space of left orderings of the fundamental group of the Klein bottle is finite. Twice in recent years, the space of left orderings has been used very successfully to solve difficult open problems from the field of left orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the newest understanding of this connection, and highlight some potential applications of further advances. 2009/11/17 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) 高田 敏恵 (新潟大学) On the$SO(N)$and$Sp(N)$free energy of a closed oriented 3-manifold [ Abstract ] We give an explicit formula of the$SO(N)$and$Sp(N)$free energy of a lens space and show that the genus$g$terms of it are analytic in a neighborhood at zero, where we can choose the neighborhood independently of$g$. Moreover, it is proved that for any closed oriented 3-manifold$M$and any$g$, the genus$g$terms of$SO(N)$and$Sp(N)$free energy of$M$coincide up to sign. 2009/11/10 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) Alexander Getmanenko (IPMU) Resurgent analysis of the Witten Laplacian in one dimension [ Abstract ] I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian. 2009/10/27 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) Alex Bene (IPMU) A new appearance of the Morita-Penner cocycle [ Abstract ] In this talk, I will recall the Morita-Penner cocycle on the dual fatgraph complex for a surface with one boundary component. This cocycle, when restricted to paths representing elements of the mapping class group, represents the extended first Johnson homomorphism \\tau_1, thus can be viewed as a (in some specific sense canonical) "groupoid extension" of \\tau_1. There are now several different contexts in which this cocycle can be constructed, and in this talk I will briefly review several of them, including one discovered in the context of finite type invariants of homology cylinders in joint work with J.E. Andersen, J-B. Meilhan, and R.C. Penner. 2009/10/20 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) 吉田 尚彦 (明治大学大学院理工学研究科) Torus fibrations and localization of index [ Abstract ] I will describe a localization of index of a Dirac type operator. We make use of a structure of torus fibration, but the mechanism of the localization does not rely on any group action. In the case of Lagrangian fibration, we show that the index is described as a sum of the contributions from Bohr-Sommerfeld fibers and singular fibers. To show the localization we introduce a deformation of a Dirac type operator for a manifold equipped with a fiber bundle structure which satisfies a kind of acyclic condition. The deformation allows an interpretation as an adiabatic limit or an infinite dimensional analogue of Witten deformation. Joint work with Hajime Fujita and Mikio Furuta. 2009/10/13 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) 笹平 裕史 (東京大学大学院数理科学研究科) Instanton Floer homology for lens spaces [ Abstract ] Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general. Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces. 2009/09/29 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) Sergei Duzhin (Steklov Mathematical Institute, Petersburg Division) Symbol of the Conway polynomial and Drinfeld associator [ Abstract ] The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We describe explicitly the resulting mapping from horizontal chord diagrams on 3 strands to univariante polynomials and evaluate it on the Drinfeld associator obtaining a beautiful generating function whose coefficients are integer combinations of multple zeta values. 2009/07/14 17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) 作間 誠 (広島大学) 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. 2009/06/30 16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.) 北山 貴裕 (東京大学大学院数理科学研究科) 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.

2009/06/23

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

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

2009/06/16

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

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.

2009/06/09

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

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.

2009/06/02

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
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.

2009/05/26

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

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.

2009/05/19

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Mark Hamilton (東京大学大学院数理科学研究科, 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.

2009/05/12

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

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.

2009/04/28

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)

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.

2009/04/21

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
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.