2015年10月24日
第42回変換群論シンポジウム アブストラクト( PDF )
- 川上智博 (和歌山大学)
タイトル : Definable C^r submanifolds in a definable C^r manifold
アブストラクト:
Let $X$ be a definable $C^r$ manifold, $Y_1, Y_2$ definably compact definable $C^r$ submanifolds of $X$
such that $\dim Y_1+\dim Y_2<\dim X$ and $Y_1$ has a trivial normal bundle.
We prove that there exists a definable isotopy $\{h_p\}_{p \in J}$ such that
$h_0=id_X$ and $h_1(Y_1) \cap Y_2=\emptyset$.
( PDF )
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大橋耕(東京大学)
タイトル : Equivariant maps between representation spheres of cyclic p-groups
アブストラクト:
We will consider necessary conditions for
the existence of equivariant maps between the unit spheres of unitary
representations of a cyclic $p$-group $G$.
Bartsch gave a necessary condition
for some unitary representations of $G$ by using equivariant $K$-theory.
In this talk, we will give two necessary conditions following Bartsch's
approach.
One is a generalization of Bartsch's result for all unitary
representations of $G$ with fixed point free actions.
The other gives a stronger necessary condition under stronger
assumptions.
( PDF )
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松下尚弘(東京大学)
タイトル : Hom 複体が与えるグラフの彩色数の下界について
アブストラクト:
単純グラフ$G$に対し,辺で結ばれている頂点では異なる
ように,$G$の頂点に色を与えることを$G$の彩色という.$G$の彩色に必要な色
の個数を$G$の彩色数といい,$\chi(G)$で表す.
Hom複体とは,二つのグラフ$T,G$に対して定義されるCW複体であり,${\rm
Hom}(T,G)$で表す.任意のグラフ$G$に対し,
$$\chi(G) > {\rm conn}({\rm Hom}(T,G)) + \chi(T)$$
なる不等式が成り立つとき,$T$をホモトピーテストグラフであるという.ここ
で,${\rm conn}(X)$は,位相空間$X$が$n$-連結となる最大の$(-1)$以上の整数
(ただし$X=\emptyset$のときは$-\infty$とする)である.ホモトピーテストグ
ラフの例としては,$n\geq 2$に対する完全グラフ$K_n$ (Lov\'asz, Babson-
Kozlov)や,奇数次のサイクル$C_{2r+1}$ (Babson-Kozlov)などが知られている.
しかし$T=K_2$のときは,ホム複体の与える彩色数の下限と,実際の彩色数と
が一致しない例が知られている.特に Walker は1983年の論文において,「任意
の正の整数$n$に対し,上記の下界と,$G$の彩色数が$n$以上差がある$G$の例」
や「${\rm Hom}(K_2,G)$-複体がホモトピー同値だが,彩色数が$1$異なる例」を
発見している.
本講演では,上の Walker の結果を,以下のように一般化することを考える.
任意の有限グラフ$T$と,彩色数が3以上のグラフ$G$,および任意の整数$n$に対
して,$G$を部分グラフとして含む$H$であって,以下の二つの性質を満たすもの
が存在する.一つの性質は包含${\rm Hom}(T,G) \rightarrow Hom(T,H)$がホモ
トピー同値であること,もう一つは $H$ の彩色数が $n$ より大きいことである.
特に任意の有限グラフ$T$に対して,${\rm Hom}(T,G)$のホモトピー不変量は$G$
の彩色数の上界を与えないことがこのことからわかる.
( PDF )
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酒井高司(首都大学東京)
タイトル : The intersection of two real flag manifolds in a complex flag manifold
アブストラクト:
Tanaka and Tasaki studied the antipodal structure of the intersection of two real
forms in Hermitian symmetric spaces of compact type. An orbit of the adjoint
representation of a compact connected Lie group $G$ admits a $G$-invariant K\"ahler
structure, and called a complex flag manifold. Furthermore, any simply-connected
compact homogeneous K\"ahler manifold is a complex flag manifold. Using a torus
action, we can define (generalized) antipodal sets of a complex flag manifold.
An orbit of the linear isotropy representation of the compact symmetric space $G/K$
is called a real flag manifold, and is embedded in a complex flag manifold as
a real form. In this talk, we will give a necessary and sufficient condition for
two real flag manifolds, which are not necessarily congruent with each other,
in a complex flag manifold to intersect transversally in terms of symmetric triads.
Moreover we will show that the intersection is an orbit of a certain Weyl group
and an antipodal set, if the intersection is discrete.
This talk is based on a join work with Osamu Ikawa, Hiroshi Iriyeh, Takayuki Okuda
and Hiroyuki Tasaki.
( PDF )
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奥田隆幸(広島大学)
タイトル : Totally geodesic surfaces in Riemannian symmetric spaces
and nilpotent orbits
アブストラクト:
Let X = G/K be a Riemannian symmetric space of non-compact type with connected G.
We are interested in classifications of totally geodesic (complete) submanifolds in X = G/K.
It is known that any totally geodesic submanifold of X is homogeneous,
more precisely, any such submanifold can be realized as an orbit of a reductive subgroup L of G acting on X.
In this talk, we give a one-to-one correspondence between
the set of G-conjugate classes of non-flat totally geodesic oriented surfaces in the Riemannian symmetric space X = G/K
and the set of nilpotent orbits of the real semisimple Lie algebra Lie G, which were already classified in Representation theory.
This is joint work with Akira Kubo (Hiroshima Shudo Univ.), Katsuya Mashimo (Hosei Univ.) and Hiroshi Tamaru (Hiroshima Univ.).
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増田高行(大阪大学)
タイトル : Combination of Lorentzian transformation groups
アブストラクト:
We will consider classification of affine Lorentzian transformation groups
acting on $(2+1)$-Minkowski spacetime properly discontinuously. All
noncocompact affine Lorentzian transformations acting properly
discontinuously are obtained by affine deformations of noncompact
hyperbolic surfaces. Let a hyperbolic surface fixed. We introduce a new
parameter, the affine twist parameter. Then we show that the affine
deformation space can be parametrized by Margulis invariants and affine
twist parameters. We will also talk about some topics associated with this
theory.
( PDF )
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小林竜馬(石川工業高等専門学校)
タイトル : A normal generating set for the Torelli group of a compact non-orientable surface
アブストラクト:
The mapping class group M(N) of a compact non-orientable surface N is defined as the group consisting of isotopy classes of diffeomorphisms over N fixing the boundary pointwise.
The Torelli group I(N) of a compact non-orientable surface N is defined as the subgroup of M(N) consisting of mapping classes acting trivially on the integral first homology group of N.
Hirose and the speaker have obtained a normal generating set for I(N), where N is a genus g(>3) closed non-orientable surface. In this work, we obtain a normal generating set for I(N),
where N is a genus g(>3) compact non-orientable surface with b(>0) boundary components.
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曽昊智(大阪市立大学)
タイトル : Cohomology of non-orientable toric origami manifolds
アブストラクト:
Toric origami manifolds, introduced by A. Canas da Silva, V. Guillemin and A. R. Pires, are generalization of symplectic toric manifolds.
In this talk we will discuss cohomology groups of some kinds of toric origami manifolds.
This talk is based on the joint work with Anton Ayzenberg, Mikiya Masuda and Seonjeong Park.
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南範彦(名古屋工業大学)
タイトル : 有理点へのホモトピー論的手法
アブストラクト:
近年ホモトピー論が有理点研究に応用されている.
本講演ではこの状況ををホモトピー論的観点か概観する.
可能で有れば,あるホモトピー論手法の有理点への応用
の可能性についても,言及したい.
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杉村匡郁(岡山大学)
タイトル : A_5 の Burnside 環について
アブストラクト:
Let $G$ be a finite nontrivial group and
$\cF$ a set of subgroups of $G$
which is closed under conjugations
by elements in $G$ and under taking subgroups.
Let $\mathfrak{F}$ denote the category whose objects are elements in $\cF$
and whose morphisms are triples $(H,g,K)$
such that $H$, $K$ $\in \cF$ and $g \in G$
with $gHg^{-1} \subset K$.
We denote by $A(G)$ the Burnside ring of $G$.
For each morphism $(H,g,K)$, we have the associated
homomorphism $(H, g, K)^{*} : A(K) \to A(H)$.
In particular, if $H \leq K$ then
$(H, e, K)^{*}$ agrees with ${\rm{res}}^K_H : A(K) \to A(H)$.
We denote by $A(\mathfrak{F})$ the inverse limit
$$
{\text{inv-lim}}_{\mathfrak{F}}
A(\thinspace \bullet \thinspace ) \ \ \left(
\subset \prod_{H \in \cF} A(H) \right)
$$
associated with the category $\mathfrak{F}$.
We denote by ${\rm{res}}_{\cF}$
the restriction homomorphism $A(G) \to A(\mathfrak{F})$ and
by $A(G)|_{\cF}$ the image of the map ${\rm{res}}_{\cF}$.
It is interesting to ask whether $A(G)|_\cF$ coinsides with
$A(\mathfrak{F})$.
Y.Hara and M.Morimoto showed that in the case of $G = A_4$,
alternating group on four letters,
the answer is affirmative.
We consider that issue in the case of $G = A_5$.
( PDF )
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森本雅治 (岡山大学)
タイトル : 閉多様体のA_5-不動点集合としての実現
アブストラクト:
Let $G$ be the alternating group on $5$ letters and
let $¥fM$ denote the family of closed smooth manifolds
which can be obtained as the $G$-fixed point sets
of smooth $G$-actions on disks.
Let $S^n$ denote the sphere of dimension $n$, let
$P_{¥bC}^n$ and $P_{¥bR}^n$ denote the complex and
real projective space of dimension $n$, respectively,
and let $L^{2n-1}_m$ denote the lens space
$S(¥bC^n)/C_m$, where $m$ is an integer $¥geq 3$ and
$$C_m = ¥{ z ¥in¥bC ¥ | ¥ z^m = 1 ¥}.$$
Let $M ¥in ¥fM$.
We will discuss whether $M$
can be realized as the $G$-fixed point sets of
smooth $G$-actions on $S^n$, $P_{¥bC}^n$,
$P_{¥bR}^n$ and $L^{2n-1}_m$.
( PDF )
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内藤貴仁(東京大学)
タイトル : String topology of the Borel constructions
アブストラクト:
The theory of string topology is a study of algebraic structures
on the homology of the free loop space (called the loop homology).
It is known that the loop homology of a manifold has many algebraic
structures, for example graded commutative algebra, Batalin-Vilkovisky
algebra and 2-dimensional TQFT.
In this talk, we will study string topology of the Borel constructions,
especially TQFT structure on the loop homology. Moreover, we will
introduce some properties and give some computational examples of
the loop homology.
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中川征樹(岡山大学)
タイトル : Gysin formulas for the universal Hall-Littlwood functions
アブストラクト:
For certain kinds of maps (e.g., smooth maps between compact, oriented manifolds, or projections of fiber bundles),
the Gysin maps (sometimes called push- forwards, Umkehr maps, integration over the fiber etc.) can be defined in ordinary cohomology.
In Schubert calculus, there are many formulas for Gysin maps for Grassmann and flag bundles which relate Schubert classes with Schur S- and P-functions (Damon, Fulton, Harris-Tu, Pragacz). Recently Pragacz generalized the above formulas to the Hall- Littlewood functions which interpolate Schur S-functions and P-functions.
Our main goal is to generalize the above formulas in ordinary cohomology to the complex cobordism theory which is universal among complex-oriented generalized cohomology theories.
More precisely, we introduce the universal analogue of the Hall-Littlewood functions,
which we call the universal Hall-Littlwood functions,
and give analogous Gysin formulas in complex cobordism theory. This is joint work with H. Naruse.
( PDF )