Publishing type：Research paper (international conference proceedings)   Kind of work：Single Work   International / domestic magazine：International journal  

In the paper ``The Steenrod algebra and its dual'', J.Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme G_p represented by the dual Hopf algebra of the mod p Steenrod algebra.
Then, G_p assigns a graded commutative algebra A_* over a prime field of finite characteristic p to a set of isomorphisms of the additive formal group law over A_*, whose group structure is given by the composition of formal power series.
The aim of this paper is to show some group theoretic properties of G_p by making use of this presentation of G_p(A_*). We give a decreasing filtration of subgroup schemes of G_p which we use for estimating the length of the lower central series of finite subgroup schemes of G_p.
We also give a successive quotient maps of affine group schemes over a prime field such that the kernel of each quotient map is a maximal abelian subgroup.

DOI： https://doi.org/10.1016/j.topol.2020.107541

Repository URL： http://hdl.handle.net/10466/00017482

Kind of work：Joint Work  

Kind of work：Single Work  

Kind of work：Single Work  

Kind of work：Single Work  

Presentation type：Oral presentation (general)  

Venue：Osaka Metropolitan University Nakamozu Campus Science Hall  

For a Grothendieck site (C,J) and a functor F from C to the category of sets, we define a notion of "plots" which is a straightforward generalization of plots in diffeology. We denote by P((C,J),F) the category of plots associated with (C,J) and F. In the case that C is a category of open sets of Euclidean spaces and smooth maps, J is a Grothendieck topology generated by open coverings of each objects of C and F is a forgetful functor from C to the category sets, P((C,J),F) is nothing but the category of diffeological spaces and smooth maps. It can be shown that P((C,J),F) is a quasi-topos, that is, P((C,J),F) is (finitely) complete and cocomplete, locally cartesian closed and has a strong subobject classifier. We also observe that P((C,J),F) is a bifibered category over the category of sets whose inverse image functor is defined from “induction” and direct image functor is defined from “subduction". By considering the fibered category of morphisms in P((C,J),F), we define a notion of representations of groupoids in P((C,J),F) and show the existence of induced representations.

Other Link： http://www.las.osakafu-u.ac.jp/%7Eyamaguti/archives/rogitcop.pdf

Presentation type：Oral presentation (general)  

Venue：Shinshu University, Science Building A, Room A-401  

The notion of plots in diffeology is introduced to define diffeological spaces which generalize differentiable manifolds. We observe that the notion of plots in diffeology has an easy generalization by replacing the site (O,E) of open sets of Euclidean spaces and open embeddings by a general Grothandieck site (C,J) and the forgetful functor U:O → Set by a set valued functor F:C → Set. In this talk, we show that the category of “generalized” plots is a quasi-topos, namely it is (finitely) compltete and cocommplete, locally cartesian closed and has strong subobject classifier. We also show that groupoids associated with epimorphisms can be defined as in the text book “Diffeology” by P.I-Zemmour so that we can develop the theory of fibration in the category of “generalized” plots. Moreover, we mention the notion of F-topology which generalizes the D-topology in diffeology.

Other Link： https://drive.google.com/file/d/1N_mwx975mM0GKWxU2dXyneMFpGZmdszK/view?usp=sharing

Presentation type：Oral presentation (general)  

Venue：Osaka Metropolitan University Sugimoto Campus Science Building E room 408  

Other Link： http://www.las.osakafu-u.ac.jp/%7Eyamaguti/archives/rgghi_slide.pdf

Presentation type：Oral presentation (general)  

Venue：Osaka Metropolitan University Sugimoto campus  

Let G_p be an affine group scheme represented by the dual of the Steenrod algebra over a prime field of characteristic p. We call an affine group scheme G "a Steenrod group" if G is a quotient group of a subgroup of G_p. The aim of this talk is to report t he current status of my attempt to provide a foundation of a representation theory of Steenrod groups as a generalization of the theory of unstable modules over the Steenrod algebra developed by J.Lannes and others.

Other Link： https://drive.google.com/file/d/1JtcPmNME6uzgDTUzaLKzWE8ziNv0Fk6H/view?usp=sharing

Presentation type：Oral presentation (general)  

For a category C with finite limits, let M(C) be the category of morphisms of C, that is, M(C) is the category of functors from a finite category D to C, where D has two objects 0 and 1 and one non-identity morphism from 0 to 1. Then, it can be easily shown that the evaluation functor p from D to C at 1 is a fibered category whose inverse image functor obtained from a morphism f of C from X to Y given by the pull-backs along f of morphisms whose targets are Y. It is also easy to see that each inverse image functor of the fibered category p has a left adjoint, in other words p is a bifibered category.
On the other hand, for a groupoid object G=(G_0,G_1) in C and an object X of M(C) over G_0, the notion of a right action of G on X is defined which generalizes a right action of a group and we can consider the category Act(G) of right actions of G. If a morphism (f_0,f_1) of groupoids from H=(H_0,H_1) to G=(G_0,G_1) is given, the inverse image functor obtained from f_0 defines a functor from Act(G) to Act(H) by pulling back the action by f_1 and the left adjoint of the inverse image functor obtained from f_0 defines a left adjoint of the above functor from Act(G) to Act(H) under some mild conditions.
In this talk, we show that the inverse image functors of the fibered category p of morphisms of C also have right adjoints if C is the category of diffeological spaces. We also show that a morphism (f_0,f_1) of groupoids from H to G gives a right adjoint of the functor from Act(G) to Act(H) mentioned above. This fact applies to construct a diffeological fiber bundle with structure groupoid H from adiffeological fiber bundle with structure groupoid G.

Other Link： http://www.las.osakafu-u.ac.jp/%7Eyamaguti/archives/ofcsmadg.pdf

Type：Lecture

Number of participants：50（人）

Audience： Teachers

Type：Seminar, workshop

Audience： High school students

Audience： Teachers

Type：Seminar, workshop

Audience： High school students

Type：Cooperation with government and educational institutions

Audience：High school students,　College students,　Teachers,　Researchers,　General,　Company,　Civic organization

Keyword：位相幾何学の初歩的 

集合を用いて図形（多面体）を表す方法について解説し，多面体の面や辺のつながり方を測る「オイラー数」と呼ばれる整数を定義する．時間があればペーパークラフトで多面体を実際に作り，そのオイラー数を求めてみる．さらに2つ以上の多面体をつなぎ合わせてできる多面体のオイラー数の変化について考える．

Audience：High school students,　College students,　Teachers,　Researchers,　General,　Company,　Civic organization

Keyword：非ユークリッド幾何学 

座標平面においてy座標が正である点全体からなる「上半平面」において，通常とは異なる距離の測り方をすることにより，その上半平面における直線が，x軸上に始点をもちy軸に並行な半直線か，x軸上に中心をもつ半円になることを解説する．この上半平面では「与えられた直線 L 上にない点を通り，L と交わらない直線がただ1つ存在する．」というユークリッド幾何学の「平行線公理」が成り立たないことから，ユークリッド幾何学とは異なる幾何学である「非ユークリッド幾何学」が展開されることを紹介する．実習では，コンパスと定規を用いて「非ユークリッド幾何学」にける三角形を作図し，その内角の和が180度より小さくなることを，分度器を用いて確認する．

Audience：High school students,　College students,　Teachers,　Researchers,　General,　Civic organization

Keyword：微分, 積分,ベクトル,幾何学 

高校では、まず数と数式についての基本を学んだ後、ベクトルや微積分を学びますが、本講習では、ベクトルと微積分を用いて空間の曲線や曲面の性質を調べることについてお話しします。

Audience：High school students,　College students,　Teachers,　Researchers,　General,　Civic organization

Keyword：微分, 積分, 極限, 実数 

大学に入って学ぶ数学と高校で学んできた数学の間に大きなギャップを感じる学生が少なくありません。微積分学を題材に、高校で学ぶ数学と大学で学ぶ数学の違いについてお話します。

Audience：High school students,　College students,　Teachers,　Researchers,　General,　Civic organization

Keyword：幾何学, 空間, 連続関数 

幾何学といえば、三角形の合同や相似などを考えていろんな定理を証明する「初等幾何学」を思い浮かべる人が多いと思いますが、空間（＝図形）を連続的に変形させても変らない性質を研究する幾何学について紹介します。

Type：Academic society, research group, etc.