2021年4月 -- 7月
[English] [過去のプログラム]
17:00 -- 18:00 Zoom でのオンライン開催
Last updated September 14, 2021
世話係
河澄 響矢
北山 貴裕
逆井 卓也
4月13日 -- Zoom でのオンライン開催, 17:00 -- 18:00
伊藤 哲也 (京都大学)
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.
4月20日 -- Zoom でのオンライン開催, 17:00 -- 18:00
大鹿 健一 (学習院大学)
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.).
4月27日 -- Zoom でのオンライン開催, 17:00 -- 18:00
栗林 勝彦 (信州大学)
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.
5月11日 -- Zoom でのオンライン開催, 17:00 -- 18:00
山下 真由子 (京都大学数理解析研究所)
トポロジカルとは限らない invertible QFT の分類問題と, Anderson dual の differential なモデル
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 QFT’s.
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 QFT’s 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).
5月18日 -- Zoom でのオンライン開催, 17:00 -- 18:00
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.
5月25日 -- Zoom でのオンライン開催, 17:00 -- 18:00
足助 太郎 (東京大学大学院数理科学研究科)
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.
6月1日 [Lie群論・表現論セミナーと合同]
-- Zoom でのオンライン開催, 17:30 -- 18:30
北川 宜稔 (早稲田大学)
On the discrete decomposability and invariants of representations of real reductive Lie groups
Abstract: 群の既約表現を部分群に制限したときにどのように振る舞うかを記述する問題を分岐則の問題という。
既約表現の制限は一般には既約ではなくなり、ユニタリな場合には直積分で記述される既約分解が存在する。
この分解は、ユニタリ作用素のスペクトル分解の一般化とみなすことができ、一般には連続的なスペクトルと離散的なスペクトルが現れる。
連続的なスペクトルが現れない場合、つまりユニタリ表現の離散的な直和になっている場合、その表現は離散分解するという。
離散分解する分岐則は技術的に扱いやすいというだけでなく、
大きな群の表現の情報から小さい部分群の表現の情報が取り出しやすい状況になっており、
以下のような応用が知られている。
保型形式から別の保型形式を作り出す Rankin--Cohen ブラケットという作用素は、
離散分解する表現から既約表現への絡作用素として得られることが知られており、
近年でも多くの一般化が得られている。
また、等質空間の L^2 関数の空間の離散スペクトルを別の等質空間のものから構成するという結果にも用いられている。(T. Kobayashi, J. Funct. Anal. ('98))
本講演では、実簡約リー群の既約表現の制限の離散分解性について、
小林俊行氏が提唱した離散分解性とG'-許容性の一般論と判定条件
(Invent. math. '94, Annals of Math. '98, Invent. math. '98)を踏まえつつ、
最近得られた結果を紹介したい。
表現の代数的な不変量である随伴多様体、解析的な不変量である wave front set、表現空間の位相、の三つを用いて離散分解性を記述する。
6月8日 -- Zoom でのオンライン開催, 17:00 -- 18:00
松下 尚弘 (琉球大学)
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, …, 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.
6月15日 -- Zoom でのオンライン開催, 17:00 -- 18:00
佐藤 尚倫 (早稲田大学)
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.
6月22日 -- Zoom でのオンライン開催, 17:00 -- 18:00
小林 竜馬 (石川工業高等専門学校)
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.
6月29日 -- Zoom でのオンライン開催, 17:00 -- 18:00
早野 健太 (慶應義塾大学)
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.
7月6日 -- Zoom でのオンライン開催, 17:30 -- 18:30
窪田 陽介 (信州大学)
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.
7月13日 -- Zoom でのオンライン開催, 17:00 -- 18:00
作間 誠 (大阪市立大学数学研究所)
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).