Tuesday Seminar on Topology (April -- July, 2021)
[Japanese] [Past Programs]
This is an online seminar on Zoom.
Last updated September 14, 2021
Information :@
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai
April 13, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Tetsuya Ito (Kyoto University)
Quantitative Birman-Menasco theorem and applications to crossing number
Abstract: Birman-Menasco proved that there are finitely many knots having a given genus and braid index.
We give a quantitative version of Birman-Menasco finiteness theorem;
an estimate of the crossing number of knots in terms of genus and braid index.
As applications,
we give various supporting evidences of various conjectural properties of the crossing number of knots.
April 20, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Kenfichi Ohshika (Gakushuin University)
Realisation of measured laminations on boundaries of convex cores
Abstract: I shall present a generalisation of the theorem by Bonahon-Otal concerning realisation of measured laminations as bending laminations of geometrically finite groups,
to general Kleinian surface groups which might be geometrically infinite.
Our proof is based on analysis of geometric limits,
and is independent of the technique of hyperbolic cone-manifolds employed by Bonahon-Otal.
This is joint work with Shinpei Baba (Osaka Univ.).
April 27, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Katsuhiko Kuribayashi (Shinshu University)
On a singular de Rham complex in diffeology
Abstract: Diffeology gives a complete, co-complete,
cartesian closed category into which the category of manifolds embeds.
In the framework of diffeology,
the de Rham complex in the sense of Souriau enables us to develop de Rham calculus.
Moreover, Iglesias-Zemmour has been introduced homotopical concepts such as homotopy groups,
cubic homology groups and fibrations in diffeology.
Thus one might expect `differential homotopy theory'.
However, the de Rham theorem does not hold for Souriau's cochain complex in general.
In this talk, I will introduce a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space.
May 11, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Mayuko Yamashita (RIMS, Kyoto University)
The classification problem of non-topological invertible QFTfs and a differential model for the Anderson duals
Abstract: Freed and Hopkins conjectured that the deformation classes of non-topological invertible quantum field theories are classified by a generalized cohomology theory called the Anderson dual of bordism theories.
Two of the main difficulty of this problem are the following.
First, we do not have the axioms for QFTfs.
Second, The Anderson dual is defined in an abstract way.
In this talk, I will explain the ongoing work to give a new approach to this conjecture, in particular to overcome the second difficulty above.
We construct a new, physically motivated model for the Anderson duals.
This model is constructed so that it abstracts a certain property of invertible QFTfs which physicists believe to hold in general.
Actually this construction turns out to be mathematically interesting because of its relation with differential cohomology theories.
I will start from basic motivations for the classification problem,
reportthe progress of our work and explain future directions.
This is the joint work with Yosuke Morita (Kyoto, math) and Kazuya Yonekura (Tohokku, physics).
May 18, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Geoffrey Powell (CNRS and University of Angers)
On derivations of free algebras over an operad and the generalized divergence
Abstract: This talk will first introduce the generalized divergence map from the Lie algebra of derivations of a free algebra over an operad to the trace space of the appropriate associative algebra.
This encompasses the Satoh trace (for Lie algebras) and the double divergence of Alekseev, Kawazumi, Kuno and Naef (for associative algebras).
The generalized divergence is a Lie 1-cocyle.
One restricts to considering the positive degree subalgebra with respect to the natural grading on the Lie algebra of derivations.
The relationship of the positive subalgebra with its subalgebra generated in degree one is of particular interest.
For example, this question arises in considering the Johnson morphism in the Lie case.
The talk will outline the structural results obtained by using the generalized divergence.
These were inspired by Satoh's study of the kernel of the trace map in the Lie case.
A new ingredient is the usage of naturality with respect to the category of free, finite-rank abelian groups and split monomorphisms.
This allows global results to be formulated using 'torsion' for functors on this category and extends the usage of naturality with respect to the general linear groups.
May 25, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Taro Asuke (The University of Tokyo)
On a characteristic class associated with deformations of foliations
Abstract: A characteristic class for deformations of foliations called the Fuks-Lodder-Kotschick class (FLK class for short) is discussed.
It seems unknown if there is a real foliation with non-trivial FLK class.
In this talk, we show some conditions to assure the triviality of the FLK class.
On the other hand, we show that the FLK class is easily to be non-trivial for transversely holomorphic foliations.
We present an example and give a construction which generalizes it.
June 1, 17:30-18:30 [Joint with Lie Groups and Representation Theory Seminar] -- Online on Zoom.
Masatoshi Kitagawa (Waseda University)
On the discrete decomposability and invariants of representations of real reductive Lie groups
Abstract: A problem to determine the behavior of the restriction of an irreducible group representation to a subgroup is called the branching problem.
The restriction of an irreducible representation is not irreducible in general,
and if the representation is unitary,
the restriction has an irreducible decomposition described by a direct integral.
The decomposition can be regarded as a generalization of spectral decomposition of unitary operators,
and has continuous spectrum and discrete spectrum in general.
If the decomposition has no continuous spectrum, the representation is said to be discretely decomposable.
Discretely decomposable branching laws are technically easy to deal with,
and in the setting, it is relatively easy to extract information about representations of a small subgroup from that of a large group.
The following applications are known.
It is known that the operators, called the Rankin--Cohen brackets which make a new automorphic form from a automorphic form,
can be obtained as intertwiner from discretely decomposable representations to irreducible representations.
Many generalizations of the operators are obtained recently.
The discrete decomposability is used to construct discrete spectrum of the space of L^2 functions on homogeneous spaces (T. Kobayashi, J. Funct. Anal. ('98)).
In this talk, I will give several criterion about the discrete decomposability and G'-admissibility based on the general theory and criterion given by T. Kobayashi (Invent. math. '94, Annals of Math. '98, Invent. math. '98).
The criterion are written by associated varieties (algebraic invariants),
wave front sets (analytic invariants) and topological structure of representations.
June 8, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Takahiro Matsusita (University of the Ryukyus)
Graphs whose Kronecker coverings are bipartite Kneser graphs
Abstract: Kronecker coverings are bipartite double coverings of graphs which are canonically determined.
If a graph G is non-bipartite and connected, then there is a unique bipartite double covering of G,
and the Kronecker covering of G coincides with it.
In general, there are non-isomorphic graphs although they have the same Kronecker coverings.
Therefore, for a given bipartite graph X, it is a natural problem to classify the graphs whose Kronecker coverings are isomorphic to X.
Such a classification problem was actually suggested by Imrich and Pisanski, and has been settled in some cases.
In this lecture, we classify the graphs whose Kronecker coverings are bipartite Kneser graphs H(n, k).
The Kneser graph K(n, k) is the graph whose vertex set is the family of k-subsets of the n-point set {1, c, n},
and two vertices are adjacent if and only if they are disjoint.
The bipartite Kneser graph H(n, k) is the Kronecker covering of K(n, k).
We show that there are exactly k graphs whose Kronecker coverings are H(n, k) when n is greater than 2k.
Moreover, we determine their automorphism groups and chromatic numbers.
June 15, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Takamichi Sato (Waseda University)
Direct decompositions of groups of piecewise linear homeomorphisms of the unit interval
Abstract: In this talk, we consider subgroups of the group PLo(I)
of piecewise linear orientation-preserving homeomorphisms of the unit interval I = [0, 1]
that are differentiable everywhere except at finitely many real numbers, under the operation of composition.
We provide a criterion for any two subgroups of PLo(I) which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic.
As its application we give a necessary and sufficient condition for any two subgroups of the
R. Thompson group F that are stabilizers of finite sets of numbers in the interval (0, 1) to be isomorphic.
June 22, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Ryoma Kobayashi (National Institute of Technology, Ishikawa College)
On infinite presentations for the mapping class group of a compact non orientable surface and its twist subgroup
Abstract: An infinite presentation for the mapping class group of any compact orientable surface was given by Gervais,
and then a simpler one by Luo. Using these results,
an infinite presentation for the mapping class group of any compact non orientable surfaces with boundary less than or equal to one was given by Omori (Tokyo University of Science),
and then one with boundary more than or equal to two by Omori and the speaker.
In this talk, we first introduce an infinite presentation for the twisted subgroup of the mapping class group of any compact non orientable surface.
I will also present four simple infinite presentations for the mapping group of any compact non orientable surface,
which are an improvement of the one given by Omori and the speaker.
This work includes a joint work with Omori.
June 29, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Kenta Hayano (Keio University)
Stability of non-proper functions
Abstract: In this talk, we will give a sufficient condition for (strong) stability of non-proper functions (with respect to the Whitney topology).
As an application, we will give a strongly stable but not infinitesimally stable function.
We will further show that any Nash function on the Euclidean space becomes stable after a generic linear perturbation.
July 6, 17:30-18:30 -- Online on Zoom.
Pre-registration required.
Yosuke Kubota (Shinshu University)
Codimension 2 transfer map in higher index theory
Abstract: The Rosenberg index is a topological invariant taking value in the K-group of the C*-algebra of the fundamental group,
which is a strong obstruction for a closed spin manifold to admit a positive scalar curvature (psc) metric.
In 2015 Hanke-Pape-Schick proves that, for a nice codimension 2 submanifold N of M,
the Rosenberg index of N obstructs to a psc metric on M.
This is a far reaching generalization of a classical result of Gromov and Lawson.
In this talk I introduce a joint work with T. Schick and its continuation concerned with this `codimension 2 index' obstruction.
We construct a map between C*-algebra K-groups, which we call the codimension 2 transfer map, relating the Rosenberg index of M to that of N directly.
This shows that Hanke-Pape-Schick's obstruction is dominated by a standard one, the Rosenberg index of M.
We also extend our codimension 2 transfer map to secondary index invariants called the higher rho invariant.
As a consequence, we obtain some example of psc manifolds are not psc null-cobordant.
July 13, 17:00-18:00 -- Online on Zoom.
Pre-registration required.
Makoto Sakuma (Osaka City University Advanced Mathematical Institute)
Homotopy motions of surfaces in 3-manifolds
Abstract: We introduce the concept of a homotopy motion of a subset in a manifold,
and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds.
This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs,
homotopical behaviour of simple loops on a Heegaard surface, and
monodromies of virtual branched covering surface bundles associated to a Heegaard splitting.
This is a joint work with Yuya Koda (arXiv:2011.05766).