# Tuesday Seminar on Topology

[Japanese]
17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:30 -- 17:00 Common Room

Last updated November 22, 2017
Information :@
Toshitake Kohno
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

September 26 [Joint with Lie Groups and Representation Theory Seminar] -- Room 056, 17:00 -- 18:30

Hideko Sekiguchi (The University of Tokyo)

Representations of Semisimple Lie Groups and Penrose Transform

Abstract: The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.
I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.
To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.
Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds, namely, that of positive $k$-planes and that of negative $k$-planes.

October 3 -- Room 056, 17:00 -- 18:00

Athanase Papadopoulos (IRMA, Université de Strasbourg)

Transitional geometry

Abstract: I will describe transitions, that is, paths between hyperbolic and spherical geometry, passing through the Euclidean. This is based on joint work with Norbert AfCampo and recent joint work with AfCampo and Yi Huang.

October 10 -- Room 056, 17:30 -- 18:30

Shoji Yokura (Kagoshima University)

Poset-stratified spaces and some applications

Abstract: A poset-stratified space is a continuous map from a topological space to a poset with the Alexandroff topology. In this talk I will discuss some thoughts about poset-stratified spaces from a naive general-topological viewpoint, some applications such as hyperplane arrangements and poset-stratified space structures of hom-sets, and related topics such as characteristic classes of vector bundles, dependence of maps (by Borsuk) and dependence of cohomology classes (by Thom).

October 17 -- Room 056, 17:00 -- 18:30

Atsushi Ishii (University of Tsukuba)

Generalizations of twisted Alexander invariants and quandle cocycle invariants

Abstract: We introduce augmented Alexander matrices, and construct link invariants. An augmented Alexander matrix is defined with an augmented Alexander pair, which gives an extension of a quandle. This framework gives the twisted Alexander invariant and the quandle cocycle invariant. This is a joint work with Kanako Oshiro.

October 24 [Joint with Lie Groups and Representation Theory Seminar] -- Room 056, 17:30 -- 18:30

Reiko Miyaoka (Tohoku University)

Approach from the submanifold theory to the Floer homology of Lagrangian intersections

Abstract: The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).

October 31 -- Room 056, 17:00 -- 18:30

Yash Lodha (École Polytechnique Fédérale de Lausanne)

Nonamenable groups of piecewise projective homeomorphisms

Abstract: Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

November 7 -- Room 056, 17:00 -- 18:30

Shin Hayashi (AIST-TohokuU MathAM-OIL)

On an explicit example of topologically protected corner states

Abstract: In condensed matter physics, topologically protected (codimension-one) edge states are known to appear on the surface of some insulators reflecting some topology of its bulk. Such phenomena can be understood from the point of view of an index theory associated to the Toeplitz extension and are called the bulk-edge correspondence. In this talk, we consider instead the quarter-plane Toeplitz extension and index theory associated with it. As a result, we show that topologically protected (codimension-two) corner states appear reflecting some topology of the gapped bulk and two edges. Such new topological phases can be obtained by taking a productff of two classically known topological phases (2d type A and 1d type AIII topological phases). By using this construction, we obtain an example of a continuous family of bounded self-adjoint Fredholm quarter-plane Toeplitz operators whose spectral flow is nontrivial, which gives an explicit example of topologically protected corner states.

November 21 -- Room 056, 17:00 -- 18:30

Keiichi Sakai (Shinshu University)

The space of short ropes and the classifying space of the space of long knots

Abstract: We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R^3. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way. This is joint work with Syunji Moriya (Osaka Prefecture University).

November 28 -- Room 056, 17:00 -- 18:30

Sang-hyun Kim (Seoul National University)

Diffeomorphism Groups of One-Manifolds

Abstract: Let a>=2 be a real number and k = [a]. We denote by Diff^a(S^1) the group of C^k diffeomorphisms such that the k--th derivatives are Hölder--continuous of exponent (a - k). For each real number a>=2, we prove that there exists a finitely generated group G < Diff^a(S^1) such that G admits no injective homomorphisms into Diff^b(S^1) for any b>a. This is joint work with Thomas Koberda.

December 5 -- Room 056, 17:00 -- 18:30

Kazuhiro Kawamura (University of Tsukuba)

Derivations and cohomologies of Lipschitz algebras

Abstract: For a compact metric space M, Lip(M) denotes the Banach algebra of all complex-valued Lipschitz functions on M. Motivated by a classical work of de Leeuw, we define a compact, not necessarily metrizable, Hausdorff space \hat{M} so that each point of \hat{M} induces a derivation on Lip(M). To some extent, \hat{M} may be regarded as "the space of directions." We study, by an elementary method, the space of derivations and continuous Hochschild cohomologies (in the sense of B.E. Johnson and A.Y. Helemskii) of Lip(M) with coefficients C(\hat{M}) and C(M). The results so obtained show that the behavior of Lip(M) is (naturally) rather different than that of the algebra of smooth/C^1 functions on M.

December 12 -- Room 056, 17:00 -- 18:30

Tatsuro Shimizu (RIMS, Kyoto university)

On the self-intersection of singular sets of maps and the signature defect

Abstract: Let M be a closed oriented n-dimensional manifold. We give a geometric proof of that the k-times self-intersection of singular set of a Morin map from M to R^p coincides with the corank k singular set of any generic map from M to R^{p+k-1} as homology classes with Z/2 coefficient (n>p+k-2). As an application we give a description of the signature defect of framed 3-manifold from the point of view of singular sets of maps.