不変擬準同型とBavard双対

木村 満晃

擬準同型は群上の「準同型に近い」実数値函数であって、様々な側面から研究されている対象である。本講演ではG不変擬準同型という概念を導入し、Bavard双対の類似が成り立つことを観察する。応用として、シンプレクティック版のある種のNielsen実現問題についても考える。本講演は川崎盛通氏（京都大）との共同研究(arXiv:1911.10855)に基づく。

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.

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 \(X\).

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.

べき零群におけるBurago-Margulis問題とサブフィンスラー幾何学

田代賢志郎

\(\Gamma\)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.

逆は常に成り立ちます. この問題はアーベル群や\(3\)-ハイゼンベルグ群, 双曲群などで肯定的に解決されていましたが, べき零群では反例も確認されています. 今回ある広いクラスのべき零群についてBurago-Margulis問題が成立することを(サブリーマンに限らない)サブフィンスラー幾何の技術を用いて示したので, それについての話をさせていただきます.

有限生成群のなす位相空間

見村万佐人

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。

Erdös-Sierpinski 双対定理について

藤田 博司

大下達也さんとの共同研究の経過報告です。連続体仮説のもとで、ベールの第1類集合とルベーグの零集合を互いにうつしあう全単射が存在すると主張するのが Erdös-Sierpinski 双対定理です。その精密化や変種をいくつかご紹介します。

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).

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

藤田 玄

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.

Dugundji spaces and linear extension operators of small norms

山内 貴光

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.

本発表は, Dmitri Shakhmatov 氏(愛媛大)と Vesko Valov氏(Nipissing大)との共同研究に基づきます.

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

平出 耕一

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.

多面体に対する Gauss-Bonnet 定理と Alexander-Spanier コホモロジー

森吉 仁志

有名な Gauss-Bonnet 定理によれば，滑らかな閉曲面上でガウス曲率を積分すると， その値は閉曲面のオイラー数の\(2\pi\)倍となる．一方，閉多面体（多面体であって閉曲面に 同相な位相空間）を考えると，各頂点に対して角欠損（Angle defect)という実数が定まり， 角欠損の総和を取ると，同じくオイラー数の\(2\pi\)倍となることが知られている （この定理は1630年のデカルトまで遡る）．この２つの定理の類比から，多面体における 角欠損と，滑らかな曲面におけるガウス曲率との対応に関心が生じる．本講演では， Alexander-Spanierコホモロジーという概念を介して，角欠損とガウス曲率が幾何的に 結びつく様子について話をする．．時間があれば，高次元への一般化についても言及する．

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.

無限次元多様体の解析的指数とKK理論

高田土満

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:1701.06055，arXiv: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．

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

加藤本子

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。

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.

Decomposition spacesのホモロジカルな局所連結性について

小山 晃

PDFファイルをご覧ください: [日本語] [英語]

The Menger property of \(C_p(X, 2)\)

酒井 政美

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.

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.

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.

Jordan-like decompositions in differential dynamical systems

平出 耕一

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.

粗幾何版アダマール・カルタンの定理とその応用

尾國 新一

粗凸距離空間と、それらに対する粗幾何版アダマール・カルタンの定理を紹介する。また、定理の応用として、粗凸距離空間に対する粗バウム・コンヌ予想が得られることについても触れる。この講演は首都大学東京の深谷友宏氏との共同研究に基づく（arXiv:1705.05588）。

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.

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.

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.

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.

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.

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.

\(\omega_1\)の超コンパクト性

池上 大祐

集合論は無限について研究する数学の一分野である。 現代数学では通常選択公理を仮定して議論を進めるが、 集合論で重要な数学的対象である巨大基数の研究を進めるうえで、 “ZF+決定性公理” を初めとする選択公理の否定を導く集合論の公理系について考察するのが重要であることが ここ三十年ほどで明らかになってきた。 この講演では、集合論や巨大基数について紹介した後、 選択公理の否定を導く巨大基数の性質について考察する。 特に、\(\omega_1\) （可算でない最小の無限基数）が超コンパクト基数という巨大基数になっているとき、 実数直線にどのような影響があるかについて議論する。 この研究は Nam Trang 氏との共同研究である。

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.

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.

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.

非正曲率を持つ群の直積に対する粗Baum-Connes予想

深谷友宏

題目にある，「粗Baum-Connes予想」とは，「非可換幾何学」と「幾何学的群論」の交わるところに生まれた予想である． この講演ではまず，ある具体的な二つの作用素環を区別する問題を通して，非可換幾何学の主題について説明する．後半では，尾國新一氏（愛媛大学）との共同研究で得られた結果について紹介する．

Separately continuous weak selections on products

元岡耕一

位相空間\(X\)に対して，集合\([X]^2=\{\{x,y\}:x,y \in X, \, x \ne y\}\)から \(X\)への関数\(\sigma\)が条件「全ての\(\{x,y\}\in [X]^2\)について\(\sigma(\{x,y\})\in \{x,y\}\)」 を満たすとき，\(\sigma\)は弱選択関数（weak selection）といいます。 また，弱選択関数\(\sigma\)が条件「全ての\(x \in X\)について \(\{y\in X :\sigma(\{x,y\})=y\}\)と \(\{y\in X :\sigma(\{x,y\})=x\}\)が \(X\)の開集合である」を満たすとき，\(\sigma\)は各個連続弱選択関数（separately continuous weak selection）といいます。

本発表では，García-Ferreira, Miyazaki, Nogura （2013）と Gutev （2016）の結果を元に，積空間\(X\times Y\)において 各個連続弱選択関数の存在が因子空間\(X,Y\)に与える影響について得られた結果を紹介します。

分類空間への写像のなす空間のホモトピー型について

蔦谷充伸

位相空間の間の連続写像全体のなす空間（写像空間）は無限次元の空間としては基本的なものであり、写像空間自身への純粋な興味にとどまらず、直積の右随伴関手を与えることから応用上も重要である。写像空間は一般に弧状連結ではなく、また、弧状連結成分同士も一般にホモトピー同値ではない。本講演ではコンパクト単純Lie群の分類空間への写像のなす空間の弧状連結成分のホモトピー型について、講演者と岸本大祐氏（京都大）によって得られた結果を中心に、代数的位相幾何学の基礎事項に関する説明を適宜織り交ぜながら解説する。特に主結果の証明では、LannesのT関手やAdams-Wilkersonによる非安定代数のGalois理論といった代数的な道具が重要な働きをする。

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.

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.

Continuous (weak) selections on hyperspaces

野倉 嗣紀

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Toric origami多様体に対するDanilov型定理

藤田 玄

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。

The first return maps and invariant measures for random maps

井上 友喜

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。

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.

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

Salvador Garcia-Ferreira

(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.

非有界距離空間の大尺度幾何と無限遠

非コンパクト完備リーマン多様体や有限生成な無限群に語距離を入れたものは非有界な固有距離空間の例である. 大尺度幾何（あるいは粗幾何）はこのような空間を調べるのに適した一つの見方である. 今回の話では大尺度幾何的な不変量である粗ホモロジー群を考えている空間の無限遠を眺めることで捉えられるかということに関するいくつかの定理や最近考察したことを紹介する.

On subgroups of Polish Abelian group generated by Borel sets

前半では，ボレル集合と \(\bf\Sigma^1_1\)集合（解析集合）の理論の歴史に沿った導入をする．後半では，可換ポーランド位相群の \(\bf\Sigma^1_1\)部分群についての最近の研究について述べる．

ヒルベルト幾何のコロナ

固有な距離空間が良い無限遠境界を持つとき，その境界は空間の粗幾何学的情報を豊富に持っていることが知られている．一方ヒルベルト幾何はユークリッド空間の有界凸領域に定まる直線を測地線とするような距離空間で，その構成からユークリッド空間としての境界はこの空間の自然な無限遠境界になっていると考えられる．実際，ヒルベルト距離の幾何学性質は領域の形に大きく依存することが知られていて，一般に双曲的でもユークリッド的でもない．そこで素朴に「ヒルべルト幾何の自然な境界がいつ粗幾何学的に良いか」という疑問が浮かぶ．発表ではこの疑問について考察する．また，本講演は尾國新一 氏との共同研究に基づくものである．

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.

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).

空間の距離化定理とフラクタル次元

空間の距離化定理としては、開被覆の正規列を用いるAlexandroff-Urysohn metrization theoremがよく知られている。ここでは、この距離化定理を用いて距離を色々変化させ、その距離によって定まるフラクタル次元がどう変化するか考察する。特に、箱次元を取り扱う。

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.

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.

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.

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 \(\mathbf{R}\).

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.

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).

暗号分野への数学の応用に関するいくつかの話題

縫田光司

話者は数学を基盤とする暗号分野の研究に取り組んでいる。この発表では、数学と暗号分野の関わりについていくつかの話題を紹介するとともに、幾何学や集合論と関連のある（と話者が信じている）研究課題を紹介する。

On effective descriptive set theory and its applications

Vassilis Gregoriades (TU-Darmstat)

Effective descriptive set theory refines and extends results from the classical descriptive set theory using ideas from recursion theory. In this talk we present its basic ideas as well as some applications to problems in classical theory.

Infinite dimensional topology for computability theorists

Takayuki Kihara (JAIST)

V. Gregoriades and the speaker applied a theorem on the algebraic structure of the degrees of noncomputability (the Turing degrees) to obtain some results on the Borel structure of Polish spaces having the transfinite inductive dimension. In this talk, we discuss the relationship between the degree structure of represented spaces and their topological dimension. We will look at the degree structure of Roman Pol's counterexample space to Alexandrov's problem, and we will propose some problems, which may connect computability theory and infinite dimensional topology.