# 松山TGSAセミナー

Matsuyama Seminar on Topology, Geometry, Set Theory and their Applications

• 井上友喜
• 尾國新一
• 加藤本子
• シャクマトフ ディミトリ
• 野倉嗣紀
• 平出耕一
• 平田浩一
• 藤田博司
• 山内貴光

# 第58回セミナー

## 講演者

Dikran Dikranjan (Udine University, Italy)

## 講演アブストラクト

### Amenable (semi-)groups and the entropy of their actions

Dikran Dikranjan

Amenable groups were introduced by John von Neumann in 1929 in connection with the Banach-Tarski paradox. These are the groups $$G$$ admitting an invariant finitely additive measure $$\mu$$ with $$\mu(G) = 1$$ (known also under the name Banach measure). Amenable semigroups are defined similarly. All　commutative semigroups are amenable.

The talk will discuss the entropy of actions of an amenable semigroup $$S$$ on a topological groups and spaces. The leading example is the semigroup $$(\mathbb N,+)$$ of naturals, the $$\mathbb N$$-actions are simply self-maps (endomorphisms) and their iterations with composition as semigroup operation. We briefly recall the notions of topological entropy of a single self-map of a topological space $$X$$ (inspired by the measure entropy defined by Kolmogorov and Sinai sixty years ago). It is nicely connected to the notion of algebraic entropy defined in case the space $$X$$ is a locally compact group and the self-map is a continuous endomorphism. Next we discuss the possibility to extend the definition of these entropy to the case when the acting semigroup $$S$$ is amenable (e.g., $$\mathbb N^2$$, which means a the semigroup generated by a pair of two commuting self-maps).

If time allows, we will point out a possibility to obtain all entropies (topological, algebraic, measure entropy, etc.) of $$S$$-actions through a general construction based on a simply defined entropy of actions of the amenable semigroup $$S$$ on normed commutative monoids.

# 第57回セミナー

## 講演者

Matthew de Brecht (京都大学)

## 講演アブストラクト

### Some results on quasi-Polish semi-lattices and lattices

Matthew de Brecht

Quasi-Polish spaces are a class of countably based sober spaces that are general enough to include all Polish spaces (which have applications in analysis and measure theory) and all $$\omega$$-continuous domains (which are typically non-Hausdorff and have applications in domain theory and algebra), and yet quasi-Polish spaces behave well enough that techniques from both descriptive set theory and domain theory can be used to study them.

In this talk we will present some characterizations of certain classes of quasi-Polish semi-lattices and lattices, which are algebraic structures equipped with a quasi-Polish topology in such a way that the algebraic operations are continuous functions. We will be particularly interested in quasi-Polish semi-lattices $$X$$ with infinitary operations (joins and/or meets) that are continuous as a function from the (lower and/or upper) Vietoris powerspace $$P(X)$$ to $$X$$.

# 第56回セミナー

## 講演者

Nicolo Zava (University of Udine, Italy)

## 講演アブストラクト

### Coarse structures on topological groups

Nicolo Zava

This talk is based on a joint work with D. Dikranjan (University of Udine).

Coarse geometry, also known as large-scale geometry, is the study of large-scale properties of spaces. Following the work of Gromov in geometric group theory, a lot of interest was brought to the study of large-scale properties of finitely generated groups endowed with their word metrics.

Later this metric approach was extended to all discrete countable groups. In order to go beyond metrisable groups, coarse structures, a notion defined by Roe to encode the large-scale properties of spaces, have to be considered.

In this talk, we present possible coarse structures on topological groups that agree with both the topological and the algebraic structures. In particular, we mainly focus on the compact-group coarse structure and the impact that, in the locally compact abelian case, Pontryagin functor has on it.

# 第55回セミナー

## 講演アブストラクト

### べき零群におけるBurago-Margulis問題とサブフィンスラー幾何学

$$\Gamma$$を有限生成群とします. Burago-Margulis問題は, $$2$$つの相異なる$$\Gamma$$上の語距離$$\rho_1,\rho_2$$を与えたとき, その比が無限遠で$$1$$に収束するならば$$\left(\frac{\rho_1(\gamma_1,\gamma_2)}{\rho_2(\gamma_1,\gamma_2)}\to 1~~\left(\rho_2(\gamma_1,\gamma_2)\to\infty\right)\right)$$, 距離の差は一様に有界となるか　$$\left(|\rho_1(\gamma_1,\gamma_2)-\rho(\gamma_1,\gamma_2)|<C=C\left(\Gamma,\rho_1,\rho_2\right)\right)$$, を問うています.

# 第54回セミナー

## 講演アブストラクト

### 有限生成群のなす位相空間

1以上の整数kを一つ固定します。k元生成群とそのk元生成集合（順番つき）の組の同型類を「k-マーク付き群（k-marked group）」と呼びます。Grigorchukらは、k-マーク付き群全体のなす空間にコンパクト・距離付け可能な位相を導入しました。この位相は「ケーリー位相（Cayley topology）」などと呼ばれます。本講演では、ケーリー位相でのマーク付き有限群の収束と、それを利用してできる興味深い距離空間や剰余有限群の構成について概説します。群の基本的な諸定義（群・準同型写像・核・正規部分群・商群・生成集合など）と（可算）直積位相の定義くらいをご存知であればお聴きになれるようにお話しいたします。

# 第52回セミナー

## 講演者

Victor Hugo Yanez (愛媛大学大学院後期課程１年)

## 講演アブストラクト

### Selectively pseudocompact groups without infinite separable pseudocompact subsets

Victor Hugo Yanez

Let $$X$$ be a topological space. Given a family of subsets $$\mathcal{U} = \{U_i: i \in \mathbb{N}\}$$ of $$X$$ we say that a sequence of points $$\{x_i: i \in \mathbb{N}\}$$ is a “selection” for $$\mathcal{U}$$ if and only if $$x_i \in U_i$$ for every $$i \in \mathbb{N}$$. A space is called selectively pseudocompact if every sequence of non-empty open sets admits a selection which has a “$$p$$-limit point”, for some ultrafilter $$p$$. In a similar manner, a space is called selectively sequentially pseudocompact if every sequence of non-empty open subsets of $$X$$ admits a selection with a convergent subsequence. In our main result, we construct in ZFC an example of a topological group which belongs to the former class, while at the same time not admitting any infinite subsets which are separable and pseudocompact. As a consequence, this space cannot contain either convergent sequences or compact subsets, showing that it cannot belong to the latter class of spaces.

In the realm of topological groups, the “selective” types of compactness as above have played a prominent role in giving partial answers to famous open problems. In this talk we shall discuss many of these selective types of compactness, and we also provide some context as to how they all relate to each other. In the end, we shall also discuss open questions and where the next step in the study of these properties lies.

This is joint work with Dmitri Shakhmatov (Ehime University).

# 第51回セミナー

## 講演アブストラクト

### Distance functions on the moduli space of convex polytopes and symplectic toric manifolds (講演は日本語です)

This talk is based on a joint work with K.Ohashi, Y.Kitabeppu and A.Mitsuishi.

We introduce several distance functions on the set of convex bodies. The first is based on the Lebesgue volume of symmetric difference. The second is the Hausdorff distance induced from the Euclidean distance. The last one is based on the Wasserstein distance of probability measures. After that we introduce the moduli space of convex polytopes with respect to the action of integral affine transformations. We will discuss the induced distance functions and metric topologies on the moduli space. This moduli space of convex polytopes contains an important subspace, the moduli space of Delzant polytopes. Delzant's famous theorem says that the moduli space of Delzant polytopes can be identified with the set of isomorphism classes of symplectic toric manifolds.

Our construction leads to convergence/collapsing phenomena of symplectic toric manifolds with respect to Gromov-Hausdorff distance or Sormani-Wenger intrinsic flat distance.

# 第50回セミナー

## 講演アブストラクト

### Dugundji spaces and linear extension operators of small norms

1968年, Pelczynskiは, Borsuk-Dugundjiによる連続関数の拡張定理に示唆され, Dugundji空間を導入しました. Dugundji空間は, 連続関数のなすBanach空間の間の正則な線形拡張作用素の存在を用いて定義されます. 本発表では, 連続関数のなすBanach空間の間の線形作用素に関するPelczynskiの問題を振り返ると共に, Dugundji空間とノルムが2より小さい線形拡張作用素の存在について考えます.

# 第49回セミナー

## 講演アブストラクト

### ある複素2次元非線形力学系の不変曲線のBorel-Laplace変換による特殊関数表現とカオス的集合

New analytic functions describing the stable and unstable manifolds at saddle fixed points of Hénon maps are discussed. These functions are obtained by using Borel-Laplace transform, and represented by asymptotic expansions that are convergent in common domains of some half plane and some neighborhood of infinity. Historically, there exist entire functions that linearize polynomial maps on the stable and unstable manifolds. These classical functions, which were proposed by Poincaré, have been adopted for various qualitative theory in dynamical systems. We state the differences between the two functions from the both sides of quantitative and qualitative viewpoints.

# 第47回セミナー

## 講演者

Dmitri Shakhmatov (愛媛大学)

## 講演アブストラクト

### Factorization theorems in dimension theory and factorization of winning strategies for Banach-Mazur games

Dmitri Shakhmatov

In the first part of the lecture, we shall survey the most prominent factorization theorems in dimension theory and explain their applications to the problem of preservation of the covering dimension by isomorphisms of function spaces with the pointwise convergence topology (Pestov's theorem).

In the second part of the lecture, we shall recall the Banch-Mazur game and shall prove a new factorization theorem for the winning strategies in this game. Finally, we shall apply this factorization theorem to obtain a new characterization of pseudocompact groups in terms of the winning strategies for the Banach-Mazur game.

All new results are joint with Alejandro Dorantes-Aldama (Mexico).

# 第46回セミナー

## 講演者

Nicolo Zava (University of Udine, Italy)

## 講演アブストラクト

### Large-scale geometry of asymmetric spaces: an introduction to quasi-coarse spaces

Nicolo Zava

Large-scale geometry, also called coarse geometry, is the study of global properties of spaces, and it was initially developed for metric spaces, but then in the literature some generalisations emerged, such as Roe's coarse spaces. However, coarse spaces are inherently symmetric structures and thus they are not suitable object to describe interesting asymmetric spaces, such as quasi-metric spaces, preordered sets, and directed graphs. Quasi-coarse spaces were recently introduced to fill that gap (https://arxiv.org/abs/1805.11034). The goal of this talk is providing a gentle introduction to those structures and the theory developed so far, also by focusing on some examples.

# 第45回セミナー

## 講演アブストラクト

### 無限次元多様体の解析的指数とKK理論

Atiyah-Singerの指数定理は，閉多様体上の解析的指数と位相的指数が一致することを主張する，微分トポロジーの金字塔の一つである．私の研究目標は，その指数理論の無限次元多様体版を与えることである．そのためには，できるだけ単純な場合から始めるのが自然であるため，次の問題を考えることにした：円周 $$T$$ のループ群 $$LT$$ が，「固有かつ余コンパクトに」作用している無限次元多様体に対する $$LT$$ 同変指数理論を，KK理論的な観点から構築せよ．いまだにこの問題の解決には至っていないが，arXiv:1701.06055arXiv:1709.06205 では，「関数空間」と見なせるHilbert空間を始めとする，解析的指数理論を構築するのに不可欠な対象をいくつか構成した．本講演では，この問題に対する現時点での結果を説明する．

# 第43回セミナー

(第42回と同日)

## 講演アブストラクト

### 有限次元非正曲率距離空間への群作用の固定点性質について

Hellyの定理は、ユークリッド空間の有限の凸集合族が共通部 分を持つ状況についての古典的な結果である。 Farbはこの定理の一般化を用いて、有限次元非正曲率距離空間への群作用につい て、大域的な固定点の存在を調べる手法を導入した。 この講演では、Farbの手法をRichard Thompsonの群 $$T$$ とその一般化について適用 し、これらの群の有限次元CAT(0)方体複体への等長作用が固定点を持つことを述 べる。

# 第42回セミナー

(第43回と同日)

## 講演アブストラクト

### On the reversibility of topological spaces and related notions

Vitalij Chatyrko

In 1965 Rajagopalan and Wilansky consider a topological property which was not new at that time but seemed not to have systematically investigated. They called a topological space reversible if each continuous bijection of the space onto itself is a homeomorphism. In 1976 and later Doyle and Hocking looked at the concept for connected metric manifolds. In 2017 Shakhmatov and the speaker observed a possibility to generalize the reversibility to categories. In particular, they considered an analogue of the notion for topological groups. In this talk we will recall some old results and mention new ones.

# 第41回セミナー

（第40回と同日）

## 講演アブストラクト

### Decomposition spacesのホモロジカルな局所連結性について

PDFファイルをご覧ください: [日本語] [英語]

# 第40回セミナー

(第41回と同日)

## 講演アブストラクト

### The Menger property of $$C_p(X, 2)$$

For a Tychonoff space $$X$$, let $$C_p(X)$$ be the space of all continuous functions with the topology of pointwise convergence. A space $$X$$ is said to be Menger if for each sequence $$\mathcal{U}_0,\mathcal{U}_1, \cdots$$ of open covers of $$X$$, there exist finite $$\mathcal{V}_n\subset \mathcal{U}_n$$ such that $$\bigcup(\bigcup\mathcal{V}_n)=X$$. A.V. Arhangel'skii proved that $$C_p(X)$$ is Menger if and only if $$X$$ is finite. For a zero-dimensional space $$X$$, let $$C_p(X, 2)=\{f\in C_p(X):f(X)\subset\{0, 1\}\}\subset C_p(X)$$. It is not clear when $$C_p(X, 2)$$ is Menger. We discuss some necessary or sufficient conditions for $$C_p(X, 2)$$ to be Menger.

# 第39回セミナー

## 講演者

Alejandro Dorantes-Aldama (愛媛大学)

## 講演アブストラクト

### Compactness-like properties defined by topological games

Alejandro Dorantes-Aldama

For a topological space $$X$$ the game $$Ssp(X)$$, invented jointly with D. Shakhmatov, is defined as follows. In Round 1 of this game, Player A chooses a non-empty open subset $$U_1$$ of $$X$$, and Player B responds by choosing a point $$x_1$$ in $$U_1$$. In Round 2, Player A chooses a non-empty open subset $$U_2$$ of $$X$$, and Player B responds by selecting a point $$x_2$$ in $$U_2$$. The game continues to infinity. Player B wins if the sequence $$(x_n)$$ of points of $$X$$ selected by Player B has a convergent subsequence in $$X$$; otherwise, Player A wins. The (non-)existence of (stationary) winning strategies for the $$Ssp(X)$$ game defines new compactness-like properties of the space $$X$$ that lay between sequential compactness and (selective sequential) pseudocompactness. We will study examples that distinguish most of the new properties.

# 第38回セミナー

## 講演者

Dmitri Shakhmatov (愛媛大学)

## 講演アブストラクト

### Games topologists play

Dmtri Shakhmatov

The Banach-Mazur game on a topological space $$X$$ is played by two players. In Round 1, Player A chooses a non-empty open subset $$U_1$$ of $$X$$, and Player B responds by choosing an non-empty open subset $$V_1$$ of $$U_1$$. In Round 2, Player A chooses a non-empty open subset $$U_2$$ of $$V_1$$, and Player B responds by selecting a non-empty open subset $$V_2$$ of $$U_2$$. The game continues to infinity, yielding a decreasing sequence $$(U_1,V_1,U_2,V_2,\ldots,U_n,V_n,\ldots)$$ of non-empty open subsets of $$X$$. Player B wins if this sequence has a non-empty intersection; otherwise, Player A wins.

In the first part of the lecture, we shall define the meaning of (stationary) winning strategies for both players in the Banach-Mazur game, and review their connections to the completeness-type properties of the space $$X$$.

In the second part of the lecture, we introduce a new game on a topological space $$X$$ invented jointly with Alejandro Dorantes-Aldama. In Round 1 of this game, Player A chooses a non-empty open subset $$U_1$$ of $$X$$, and Player B responds by choosing a point $$x_1$$ in $$U_1$$. In Round 2, Player A chooses a non-empty open subset $$U_2$$ of X, and Player B responds by selecting a point $$x_2$$ in $$U_2$$. The game continues to infinity. Player B wins if the sequence $$(x_n)$$ of points of $$X$$ selected by Player B has a convergent subsequence in $$X$$; otherwise, Player A wins. We review the existence of (stationary) winning strategies for this game, leading to new compactness-like properties of the space $$X$$ sandwiched between its sequential compactness and (selective sequential) pseudocompactness.

# 第37回セミナー

## 講演アブストラクト

### Jordan-like decompositions in differential dynamical systems

This talk gives a decomposition theorem for bundle maps of linear vector bundles over compact metric spaces, which is like the Jordan decomposition for linear maps on vector spaces in linear algebra.

# 第35回セミナー

## 講演者

Yasser Ortiz Castillo (University of Sao Paulo)

## 講演アブストラクト

### Questions on the Higson compactification and $$\mathbb{N}^{\omega_1}$$

Yasser Ortiz Castillo

In this talk I will present the results on my visit to Ehime University. The principal point of the talk is the advance of the question whether every Higson compactification is a Wallman type. Furthermore I will discuss the question: “Is $$\mathbb{N}^{\omega_1}$$ weakly pseudocompact?”, and some approaches to solve it.

# 第34回セミナー

## 講演者

Alexander Shibakov (Tennesee Technological University, Cookeville, U.S.A.)

## 講演アブストラクト

### Sequential groups: independence results and their effective counterparts

Alexander Shibakov

A topological space is sequential if its topology is fully described by convergent sequences. In this talk we survey a number of recent results about sequential topological groups. We show that the answer to P. Nyikos 1980 question on the existence of exotic sequential groups is independent of the axioms of ZFC, as well as establish the independence of a related question about the size of such groups. We also build a continuum size sequential group that does not ‘reflect’ its convergence properties to its countable subgroups or small quotients.

We then switch our attention to countable sequential groups whose topology is an analytic subset of the irrationals (what Todorčevic calls the effective topology). We provide a full topological classification of such groups with the help of an elegant but not widely known result of E. Zelenyuk. This classification answers several questions asked by Todorčevic and Uzcategui in 2001.

We conclude the talk by mentioning a number of open questions about sequential and Fréchet-Urysohn groups in both the traditional and the effective realms.

# 第33回セミナー

## 講演者

Xabier Domínguez (University of A Coruña, Spain)

## 講演アブストラクト

### A precompact self-dual strongly reflexive abelian group need not be compact

Xabier Domínguez

For an abelian topological group $$G$$, we denote by $$G^\wedge$$ the Pontryagin dual of $$G$$; that, is, the group of all continuous homomorphisms from $$G$$ to the circle group $$\mathbb{T}=\mathbb{R}/\mathbb{Z}$$ endowed with the compact-open topology. An abelian topological group $$G$$ is called reflexive if it is topologically isomorphic to its second Pontryagin dual $$G^{\wedge\wedge}=(G^{\wedge})^{\wedge}$$, and $$G$$ is said to be strongly reflexive if all closed subgroups and all quotient groups of both $$G$$ and its dual group $$G^\wedge$$ are reflexive. It follows from the Pontryagin duality that locally compact abelian groups are strongly reflexive.

A topological group is precompact if it is (topologically isomorphic to) a subgroup of some compact group. Chasco, Dikranjan and Martín-Peinador asked if a precompact strongly reflexive abelian group must be compact. We resolve this question in the negative by constructing an example of a precompact strongly reflexive abelian group $$G$$ which is not even pseudocompact. In addition, our group $$G$$ is topologically simple (contains no proper closed subgroups) and strongly self-dual; the latter property implies that $$G$$ is topologically isomorphic to its Pontryagin dual $$G^\wedge$$. The construction of $$G$$ relies on an example of a topologically simple, free dense subgroup of $$\mathbb{T}^{\boldsymbol{c}}$$ of cardinality $$\boldsymbol{c}$$ which does not contain infinite compact subsets, due to Fujita and Shakhmatov. (Here $$\boldsymbol{c}$$ denotes the cardinality of the continuum.)

In this talk, we shall explain ideas behind the construction of our example and we shall survey relevant results from the duality theory.

This is a joint work with María Jesús Chasco and Dmitri Shakhmatov.

# 第32回セミナー

## 講演者

Yasser Ortiz-Castillo (Universidade de Sao Paulo, Brasil)

## 講演アブストラクト

### The hyperspace of convergent sequences

Yasser Ortiz-Castillo

A hyperspace of some given space $$X$$ is a family of subsets of $$X$$ provided with a topology which depends on the original topology of $$X$$. Some of the main known hyperspaces are certain specific families of closed sets (all nonempty closed sets, compact sets, subcontinua, finite sets for example) with the Vietoris Topology what has as a subbase the sets of the form $V^{+}=\{A\in CL(X): A\subseteq V\}\quad\text{and}\quad V^{-}=\{A\in CL(X): A\cap V\neq \emptyset\}$ Since the topology of Frechet Urysohn spaces are determined by their non-trivial convergent sequences it is interesting to study the relation between the properties of Frechet Urysohn spaces (as metric spaces) and the properties of their respective hyperspace of non-trivial convergent sequences, who is the subspace of the nonempty closed sets with the Vietoris Topology. In this talk we will present the basic properties of this hyperspace and some recent advances and open problems.

# 第31回セミナー

## 講演者

Víctor Hugo Yañez (愛媛大学)

## 講演アブストラクト

### Metric SSGP topologies on abelian groups of positive finite divisible rank

Víctor Hugo Yañez

Let $$G$$ be an abelian group. For a subset $$A$$ of $$G$$, $$\mathrm{Cyc}(A)$$ denotes the set of all elements of $$G$$ which generate the cyclic subgroup contained in $$A$$, and $$G$$ is said to have the small subgroup generating property (abbreviated to SSGP) if the smallest subgroup of $$G$$ generated by $$\mathrm{Cyc}(U)$$ is dense in $$G$$, for every neighbourhood $$U$$ of zero of $$G$$. SSGP groups form a proper subclass of the class of minimally almost periodic groups. Comfort and Gould asked for a characterization of abelian groups $$G$$ which admit an SSGP group topology, and they solved this problem for bounded torsion groups (which have divisible rank zero). Dikranjan and Shakhmatov proved that an abelian group of infinite divisible rank admits an SSGP group topology. In the remaining case of positive finite divisible rank, the same authors found a necessary condition on $$G$$ in order to admit an SSGP group topology and asked if this condition is also sufficient. We answer this question positively, thereby completing the characterization of abelian groups which admit an SSGP group topology. This is a joint work with Dmitri Shakhmatov.

# 第30回セミナー

## 講演者

Alejandro Dorantes-Aldama (愛媛大学)

## 講演アブストラクト

### Selective sequential pseudcompactness

Alejandro Dorantes-Aldama

We say that a topological space $$X$$ is selectively sequentially pseudcompact if for every family $$\{U_n:n\in\mathbb{N}\}$$ of non-empty open subsets of $$X$$, one can choose a point $$x_n\in U_n$$ for each $$n\in\mathbb{N}$$ in such a way that the sequence $$\{x_n:n\in \mathbb{N}\}$$ has a convergent subsequence. We show that the class of selectively sequentially pseudcompact spaces is closed under arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by García-Ferreira and Ortiz-Castillo. We prove, under the Singular Cardinal Hypothesis SCH, that if $$G$$ is an Abelian group admitting a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudcompact group topology. Since selectively sequentially pseudcompact spaces are strongly pseudocompact, this provides a strong positive answer to a question of García-Ferreira and Tomita. This is a joint work with Dmitri Shakhmatov.

# 第29回セミナー

## 講演アブストラクト

### Open retractions of indecomposable continua

Eiichi Matsuhashi

In 1971 Bellamy proved that if $$X$$ is a continuum, then there exists an indecomposable continuum $$Y$$ which contains $$X$$ as a retract. In 1990, van Mill proved that for each homogeneous continuum $$X$$, there exists a non-metrizable indecomposable homogeneous continuum $$Y$$ such that $$X$$ is an open retract of $$Y$$. Recently, Fukaishi and I proved that for each continuum $$X$$ there exist an indecomposable continuum $$Y$$ which contains $$X$$ and an open retraction $$r\colon Y \to X$$ such that each fiber of $$r$$ is homeomorphic to the Cantor set. Furthermore, $$Y$$ is homeomorphic to the closure of the countable union of topological copies of $$X$$ in some continuum.

In this talk I will present a sketch of the proof of our result and some related topics.

# 第27回セミナー

## 講演者

Franklin D. Tall (University of Toronto, Canada)

## 講演アブストラクト

### Definable versions of Menger's conjecture

Franklin D. Tall

A topological space is Menger if, given a countable sequence of open covers, there is a finite selection from each of them, so that the union of the selections is a cover. The Menger property lies strictly between σ-compactness and Lindelöfness. In 1925 Hurewicz conjectured that Menger was equivalent to σ-compact in metrizable spaces. This was refuted by Chaber and Pol in 2002. However, for spaces which are in some sense “definable”, the situation is less clear. Our principal result (joint with S. Todorcevic and S. Tokgoz) is that Hurewicz' Conjecture for projective sets of reals is equiconsistent with an inaccessible cardinal. In general topological spaces, the situation is more complicated, but we have a variety of similar results for co-analytic spaces, i.e. spaces having Cech-Stone remainders which are continuous images of the space of irrationals.

# 第26回セミナー

## 講演者

Mikhail Tkachenko (Universidad Autonoma Metropolitana, Mexico City, Mexico)

## 講演アブストラクト

### Productivity of properties in spaces and topological groups

Mikhail Tkachenko

A topological property $$P$$ is said to be productive if the product space $$\prod_{i\in I} X_i$$ has $$P$$ provided each factor $$X_i$$ has $$P$$. Our aim is to present several absolute'' and relative'' topological properties which fail to be productive in Tychonoff topological spaces, but do become productive in topological groups. Among those are PSEUDOCOMPACTNESS, BOUNDEDNESS (also called functional boundedness), RELATIVE PSEUDOCOMPACTNESS, etc.

We also explain the main reason for the productivity of these properties in the class of topological groups. Somehow this phenomenon is related to the fact that every topological group has many quotient spaces that admit weaker metrizable topologies.

# 第25回セミナー

## 講演者

Maria Jesus Chasco (University of Navarra, Pamplona, Spain)

## 講演アブストラクト

### Pontryagin duality versus Comfort-Ross duality for precompact groups

Maria Jesus Chasco

Pontryagin duality is a powerful tool in the context of locally compact abelian groups with many applications in knowledge of the structure of such groups, harmonic analysis, etc. In the case of precompact abelian groups, Comfort-Ross' theorem provides another "natural" duality. Throughout this talk the two dualities will be presented and explored. In this context, the problem of strong reflexivity of precompact Abelian groups and its solution will also be addressed.

Katsuya Eda

(PDF)

# 第20回セミナー

## 講演アブストラクト

### Homological selection theorems

Vesko Valov

A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem are established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically $$UV^n$$ set-valued maps.

# 第19回セミナー

## 講演者

Alejandro Dorantes-Aldama (愛媛大学外国人客員研究員)

## 講演アブストラクト

### Reflection numbers under large continuum

Sakaé Fuchino

The reflection number $$\kappa$$ of the Rado Conjecture is defined as the smallest cardinal $$\kappa$$ such that any non special tree $$T$$ has a non special subtree $$T'$$ of cardinality $$<\kappa$$ (or $$\infty$$ if there is no such $$\kappa$$). We show the consistency of the statement that the continuum is fairly large (e.g. that it is a large cardinal like supercompact in an inner model with the same cardinals as in the universe) while the reflection number of the Rado Conjecture is less than or equal to the continuum. We also consider some other reflection numbers and the relationship between them.

### Almost irresolvable spaces

Alejandro Dorantes-Aldama

A topological space $$X$$ is almost resolvable if $$X$$ is the union of a countable collection of subsets each of them with empty interior. We prove that under the Continuum Hypothesis, the existence of a measurable cardinal is equivalent to the existence of a Baire crowded ccc almost irresolvable T1 space. We also prove that:
(1) Every crowded ccc space with cardinality less than the first weakly inaccessible cardinal is almost resolvable.
(2) If $$2^{\omega}$$ is less than the first weakly inaccessible cardinal, then every T2 crowded ccc space is almost resolvable.

（PDF）

# 第17回セミナー

## 講演アブストラクト

### Toric origami多様体に対するDanilov型定理

Origami多様体とは超曲面でのある種の退化を許容する 閉2次微分形式が付与された多様体で、2011年にCannas da Silva-Guillemin-Piresによって導入されたsymplectic多様体の一般化である。 Origami多様体に対してsymplectic幾何の種々の概念や結果が拡張されている。 この講演では、toric作用をもつorigami多様体に対するDanilov型定理を紹介する。 ここで、Danilov型定理とは、$$\mathit{spin}^c$$ Dirac作用素の(同変)指数と toric作用に付随する多面体の格子点の数え上げの一致を主張する定理である。 時間が許せば、Furuta-Yoshidaとの共同研究による指数の局所化の立場からの証明の概要も説明したい。

# 第16回セミナー

## 講演アブストラクト

### The first return maps and invariant measures for random maps

ランダム力学系について説明した上で、ランダム力学系の First return map はどのようなときに定義できるかを述べる。また、その First return map が不変測度をもつとき、もとのランダム力学系の不変測度がどのようになるかについても述べる。

# 第15回セミナー

## 講演者

Valentin Gutev (University of Malta, Malta)
Dikran Dikranjan (Udine University, Italy)

## 講演アブストラクト

### Selections and deleted symmetric products

Valentin Gutev

It will be discussed a problem of continuous selections for the hyperspace of unordered n-tuples of different points of a connected space. The problem will be related to strong cut points and noncut points of such spaces. These considerations lead to a complete characterisation of continuous selections for such hyperspaces. They also settle an open question posed by Michael Hrusak and Ivan Martinez-Ruiz.

### Coarse structure on groups

Dikran Dikranjan

We consider functorial coarse structure on topological and abstract groups and study the impact on the algebraic or topological structure by taking account of cardinal invariants like asymptotic dimension, rank, etc.

# 第14回セミナー

## 講演者

Yasser F. Ortiz-Castillo (Universidade de Sao Paulo, Brasil)

## 講演アブストラクト

### The Glicksberg Theorem for Tychonoff extension properties and powers of ultrafilters

Yasser F. Ortiz-Castillo

A well known result due by Glicksberg says that the Stone-Čech compactification of a product can be factorized when such product is pseudocompact. In this talk we provide versions of this result for different Tychonoff extension properties. Furthermore we will define a linear ordered set in the $$p$$-compact extension of the natural numbers by using a new notion of the power of a free ultrafilter.

### Families of continuous functions and function spaces

(This is a joint work with R. Rojas-Hernandez.)

In this article, we mainly study certain families of continuous retractions ($$r$$-skeletons) having rich properties. By using full $$r$$-skeletons we solve some questions posed by R. Buzyakova concerning the Alexandroff duplicate of a space. Certainly, it is shown that if the space $$X$$ has a full $$r$$-skeleton, then its Alexandroff duplicate also has a full full $$r$$-skeleton and, in a very similar way, it is proved that the Alexandroff duplicate of a monotonically retractable space is monotonically retractable. The notion of $$q$$-skeleton is introduced and it is shown that every compact subspace of $$C_{\mathrm p}(X)$$ is Corson when $$X$$ has a full $$q$$-skeleton. The notion of strong $$r$$-skeleton is also introduced to answer a question suggested by F. Casarubias and R. Rojas by establishing that a space $$X$$ is monotonically Sokolov iff it is monotonically $$\omega$$-monolithic and has a strong $$r$$-skeleton. The techniques used here allows to give a very topological proof of a result of I. Bandlow that used elementary submodels.

# 第10回セミナー

## 講演アブストラクト

### Higsonコンパクト化とその普遍性

Higsonコンパクト化は, coarse幾何学における無限遠境界を与えるコンパクト化の中で普遍的であると考えられている. しかしながら, その普遍性を主張する文献はなく, また, そもそも「coarse幾何学におけるコンパクト化」という概念に明確な定義が与えられているわけではない. これについて, かりそめの定義を与えた上で, Higsonコンパクト化の普遍性について論じる.

# 第9回セミナー

## 講演者

ディクラニアン ディクラン(JSPS外国人特別研究員, イタリア・ウディネ大学)

## 講演アブストラクト

### The von Neumann kernel of a topological group under the looking glass of Zariski topology

The talk will discuss the relation of the von Neumann kernel of a topological group to the Zariski topology of the underlying abstract group and the recent solution (jointly with D. Shakhmatov) of problems posed by Comfort-Protasov-Remus (on minimally almost periodic group topologies), by Gariyelyan (on the realization of von Neumann kernel of an abelian group) and by Franklin Gould (on the three-space property related to minimal almost periodicity).

# 第7回セミナー

## 講演者

Udayan Darji (アメリカ・ルイビル大学)

## 講演アブストラクト

### P-domination and Borel sets

What is the Borel hierarchy? What are analytic and coanalytic sets? Why does one study them? We will leisurely discuss some basic concepts of descriptive set theory. Then, we will discuss a characterization of analytic sets arising from work of Talagrand in functional analysis and give a new characterization of Borel sets in the same spirit.

# 第6回セミナー

## 講演者

Shakhmatov Dmitri (愛媛大学)

## 講演アブストラクト

### Final solution of Markov's problem on the existence of connected Hausdorff group topologies

This is a joint work with Dikran Dikranjan (Udine University, Italy).

It is easy to see that a non-trivial connected Hausdorff group must have cardinality at least continuum. Seventy years ago Markov asked if every group of cardinality at least continuum can be equipped with a connected Hausdorff group topology. Twenty five years ago a counter-example to Markov's conjecture was found by Pestov, and a bit later Remus showed that no permutation group admits a connected Hausdorff group topology. The question (explicitly asked by Remus) whether the answer to Markov's question is positive for abelian groups remained widely open. We prove that every abelian group of cardinality at least continuum has a connected Hausdorff group topology, Furthermore, we give a complete characterization of abelian groups which admit a connected Hausdorff group topology having compact completion.

# 第5回セミナー

## 講演アブストラクト

### Mathias-Prikry forcing and dominating reals

For a countable set X, we call $$\mathcal{I}$$ an ideal on $$X$$ if $$\mathcal{I}$$ is a family of subsets of $$X$$ closed under the taking subsets and unions. We assume all ideals on $$X$$ contains the family of finite subsets of $$X$$. The Mathias-Prikry forcing associated with an ideal $$\mathcal{I}$$ on a countable subset $$X$$, denoted by $$\mathbb{M}_{\mathcal{I}}$$, consist of pairs $$(s,A)$$ such that $$s$$ is a finite subsets of $$X$$, $$A$$ in $$\mathcal{I}$$ and $$s\cap A=\emptyset$$. The ordering is given by $$(s,A)\leq (t,B)$$ if $$t$$ is a subset of $$s$$ and $$B$$ is a subset of $$A$$ and $$(s\setminus t)\cap B=\emptyset$$.

The Mathias-Prikry forcing adds a new subset of $$X$$ which diagonalize ideal $$\mathcal{I}$$, that is, $$\mathbb{M}_{\mathcal{I}}$$ adds a new subset $$\dot{A}$$ of $$X$$ such that $$X\cap I$$ is finite for every $$I$$ in $$\mathcal{I}$$. So Mathias-Prikry forcing plays significant role when we investigate ultrafilter, ideal or mad family.

Some additional nice properties of the Mathias-Prikry forcing depends on $$\mathcal{I}$$. For example, $$\mathcal{U}$$ is a Ramsey ultrafilter if and only if $$\mathbb{M}_{\mathcal{U}^{*}}$$ does not add Cohen real.

The speaker and Michael Hruš\'{a}k give a characterization of ideals $$\mathcal{I}$$ such that $$\mathbb{M}_{\mathcal{I}}$$ adds no dominating real. We say a forcing notion $$\mathbb{P}$$ adds dominating reals if $$\mathbb{P}$$ adds a new function $$\dot{g}$$ from $$\omega$$ to $$\omega$$ such that for $$f\in\omega^{\omega}\cap V$$, $$f(n)<\dot{g}(n)$$ for all but finitely many $$n\in\omega$$.

We show that $$\mathbb{M}_{\mathcal{I}^{*}}$$ adds dominating reals if and only if $$\mathcal{I}^{<\omega}$$ is $$P^{+}$$-ideal.

Recently, David Chodounský, and Dušan Repovš and Lyubomyr Zdomskyy give another characterization of ideal $$\mathcal{I}$$ with covering property such that $$\mathbb{M}_{\mathcal{I}}$$ adds no dominating real. We will talk about recent development of these result and application.

# 第4回セミナー

## 講演アブストラクト

### Certain outer measures on the real line and CH

Given a two point selection $$f\colon \mathbf{R}^2 \to \mathbf{R}$$ on the real line $$\mathbf{R}$$ we can define, following the definition of Lebesgue outer measure, an outer measure $$\lambda_f$$. We show that CH is equivalent to the existence of a two point selection $$f$$ for which the null sets of $$\lambda_f$$ are precisely the countable subsets of $$\mathbf{R}$$.

# 第3回セミナー

## アブストラクト

### Jordan-like decompositions in dynamical systems and the associated invariant manifolds

Koichi Hiraide (Ehime University)

This talk gives a topological version of the smooth ergodic theory.

# 第2回セミナー

## 講演者

Dikran Dikranjan (JSPS外国人特別研究員/イタリア･ウディネ大学)

## 講演アブストラクト

### Entropy and its connection to Number Theory and Geometric Group Theory

Dikran Dikranjan (Udine University, Italy)

The notion of entropy was invented by Clausius in Thermodynamics 160 years ago and carried over to Information Theory (by Shannon), Ergodic Theory (by Kolmogorov and Sinai), Topology and Group Theory (by Adler, Konheim, McAndrew and Bowen), Algebraic Geometry (by Bellon and Viallet). Nowadays entropy is one of the most relevant invariants of discrete dynamical systems of ergodic, topological or algebraic nature. The talk aims to give a self-contained introduction to some of these entropies (including also some more recent counterparts in Set Theory and Commutative Algebra) as well as their connection to other topics in Mathematics, such as Mahler measure and Lehmer Problem (from Number Theory), and the growth rate of groups and Milnor Problems (from Geometric Group Theory).

# 第１回セミナー

## 講演者

• 縫田 光司 (産業技術総合研究所)
• 木原 貴行 (国立先端科学技術大学院大学)