\(G\)-invariant quasimorphisms and Bavard duality

Mitsuaki Kimura

A quasimorphism is a real-valued function on a group which is “almost homomorphism”. This concept has been studied from many perspectives. In this talk, we introduce the notion of \(G\)-invariant quasimorphism and observe that a Bavard-type duality holds. As an application, we consider a kind of Nielsen realization problem by symplectomorphisms. This talk is based on a joint work (arXiv:1911.10855) with Morimichi Kawasaki (Kyoto Univ.).

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.

The Burago-Margulis problem for nilpotent groups and subFinsler geometry

Kenshiro Tashiro

Let \(\Gamma\) be a finitely generated group. The Burago-Margulis problem asks the following. If two given word metrics on \(\Gamma\), \(\rho_1\) and \(\rho_2\), are asymptotically isometric \(\left(\frac{\rho_1(\gamma_1,\gamma_2)}{\rho_2(\gamma_1,\gamma_2)}\to 1~~\left(\rho_2(\gamma_1,\gamma_2)\to\infty\right)\right)\), then is the difference uniformly bounded \(\left(|\rho_1(\gamma_1,\gamma_2)-\rho_2(\gamma_1,\gamma_2)|<C=C(\Gamma,\rho_1,\rho_2)\right)\)?

Notice that the converse assertion immediately follows. It was known that the question is true for abelian groups, the \(3\)-Heisenberg group, word hyperbolic groups and etc, however there is a counterexample in the class of nilpotent groups. We show that the Burago-Margulis problem is true for some class of nilpotent groups with a technique of subFinsler geometry, not necessarily subRiemannian.

The space of finitely generated groups

M. Mimura

Fix a positive integer k. A k-marked group is (an equivalence class of) the pair of a k-generated group and a k-generating (oredered) set. Grigorchuk and other researchers have equipped the space of all k-marked groups with a compact and metrizable topology, which is called the Cayley topology in some literature. We discuss convergence of sequences of finite marked groups in this topology, and overview applications to coarse geometry and constructions of residually finite groups with noteworthy properties.

Remarks on Erdös-Sierpinski Duality

Hiroshi Fujita

This is just a progress report of joint work with T.Ohshita. We will present some variations of the Erdös-Sierpinski Duality Theorem, which states that assuming the continuum hypothesis there exists an involution on the real line which transfers each meager set into a null set and vice versa.

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

Hajime Fujita

(Note: the talk will be given in Japanese language.)

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.

Representation with special functions via Borel-Laplace transform to invariant curves of certain complex 2D nonlinear dynamics and the concerned chaotic sets

Koichi Hiraide

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.

Connections between the Gauss-Bonnet Theorem for polygons and the Alexander-Spanier cohomologies

Hitoshi Moriyoshi

Keywords: the Gauss-Bonnet Theorem, a combinatorial Gauss-Bonnet Theorem, the Alexander-Spanier cohomologies

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).

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.

An analytic index theory for infinite-dimensional manifolds and KK-theory

Doman Takata

The Atiyah-Singer index theorem is one of the monumental works in geometry and topology, which states the coincidence between analytic index and topological index on closed manifolds. The overall goal of my research is to formulate and prove an infinite-dimensional version of this theorem. For this purpose, it is natural to begin with simple cases, and my current problem is the following: For infinite-dimensional manifolds equipped with a ``proper and cocompact'' action of the loop group of the circle, construct a loop group equivariant index theory, from the viewpoint of KK-theory. Although this project has not been completed, I have constructed several core objects for the analytic side of this problem, including a Hilbert space regarded as an ``\(L^2\)-space'', in arXiv:1701.06055 and arXiv:1709.06205. In this talk, I am going to report the progress so far.

Metric Spaces, PDEs and Gromov-Hausdorff convergence

Shouhei Honda

In this talk we will discuss metric measure spaces with Ricci bounds from below. In particular we focus relationships between them and geometric PDEs. This talk is based on joint works with Luigi Ambrosio (Scuola Normale Superiore).

Fixed points for group actions on finite-dimensional nonpositively curved metric spaces

Motoko Kato

Helly's theorem is a basic result on the intersection of finitely many convex sets in a Euclidean space. Farb introduced a way to use Helly's theorem to discuss the existence of global fixed points for isometric actions of groups on nonpositively curved metric spaces. In this talk, we apply Farb's method to Richard Thompson's group \(T\) and its relatives. We show fixed point theorems for isometric actions of these groups on finite-dimensional CAT(0) cube complexes.

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.

On local homological connectivity of decomposition spaces

Akira Koyama

See PDF: [ja] [en]

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

Masami Sakai

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

Dmitri 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

Koichi Hiraide

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.

A coarse Cartan-Hadamard theorem and its application

Shin-ichi Oguni

We show a coarse version of the Cartan-Hadamard theorem for proper coarsely convex spaces. As an application, we see that such spaces satisfy the coarse Baum-Connes conjecture. This talk is based on a joint work (arXiv:1705.05588) with Tomohiro Fukaya in Tokyo Metropolitan University.

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.

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.

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.

On supercompactness of \(\omega_1\)

Daisuke Ikegami

Set theory is a study of infinity and large cardinals are one of the main research topics in set theory. While we usually assume the Axiom of Choice, it has been verified in these 30 years that the study of set theory (or set theoretic axioms) refuting the Axiom of Choice (such as “ZF + the Axiom of Determinacy”) is important for the research of large cardinals. In this talk, after introducing the background of set theory and large cardinals, we discuss some large cardinal property that implies the negation of the Axiom of Choice. We focus on the large cardinal property that \(\omega_1\) (i.e., the least uncountable cardinal) is a supercompact cardinal, and investigate the influence of the property on the real line. This is joint work with Nam Trang.

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.

Coarse Baum-Connes Conjecture about products of groups with non-positive curvature.

Tomohiro Fukaya

(Abstract in English language is not available for this talk.)

Separately continuous weak selections on products

Koichi Motooka

For a topological space \(X\), let \([X]^2=\{\{x,y\}:x,y \in X, \, x \ne y\}\). A function \(\sigma:[X]^2\rightarrow X\) is called a weak selection on \(X\) if \(\sigma(\{x,y\})\in \{x,y\}\) for each \(\{x,y\}\in [X]^2\). A weak selection on \(X\) is said to be separately continuous if \(\{y\in X :\sigma(\{x,y\})=y\}\) and \(\{y\in X :\sigma(\{x,y\})=x\}\) are open in \(X\) for each \(x\in X\).

In this talk, continuing the work of García-Ferreira, Miyazaki, Nogura (2013) and Gutev (2016), we consider to what extent the existence of a separately continuous weak selection on a product space \(X\times Y\) determines the topology of factor spaces \(X\) and \(Y\).

On the homotopy types of the space of maps to classifying spaces

Mitsunobu Tsutaya

The space of continuous maps called mapping space is a fundamental object in topology, which is not only purely interesting but also is useful in various applications. In general, a mapping space is not path connected and has path components of different homotopy types. In this talk, I will explain about the results about the homotopy types of the path components of the space of maps to classifying spaces. In particular, the main result is based on the recent work of Daisuke Kishimoto (Kyoto Univ.) and the speaker. I will also explain about basics of algebraic topology. In the proof of the main result, the algebraic tools called Lannes' T-functor and the Galois theory of unstable algebras by Adams-Wilkerson will play an important role.

Infinite sheeted covering map on solenoids

(See PDF)

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

Tsugunori Nogura

(See PDF)

A Danilov-type Theorem for Toric origami manifolds

Hajime Fujita

Key Phrases: Origami manifolds, symplectic structures, toric action on \(\mathit{spin}^c\) Dirac operator

The first return maps and invariant measures for random maps

Tomoki Inoue

Key-phrases: Random dynamical systems, first return map, invariant measures

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.

Large-scale geometry and topology at infinity of unbounded metric spaces

The speaker will report recent progress on coarse-geometric aspects of unbounded proper metric spaces.

On subgroups of Polish Abelian group generated by Borel sets

The talk will be divided into two parts:

1. A historical survey of theory of Borel and \(\bf\Sigma^1_1\) (i.e., analytic) sets.
2. Recent results about \(\bf\Sigma^1_1\) subgroups of Polish Abelian groups.

Coarse geometric properties of the Hilbert geometry

(Abstract in English language is not available for this talk.)

Universality of the Higson Compactifications in Coarse Geometry

Although Higson compactifications are considered to be “universal” among compactifications occuring in coarse geometry, there has not been literature that establishes such universality. There has not been even a widely accepted definition of the notion of “compactifications in coarse geometry.” In this talk, we shall give a tentative definition of the notion and discuss the universality of Higson compactifications.

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).

On the fractal-dimensions determined by Alexandroff-Urysohn metrics

The Alexandroff-Urysohn metrization theorem tells how to construct a metric from a normal sequence of open coverings. The speaker will explain the way that fractal-dimensions, especially the box-dimension, depend on such open coverings.

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

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

Entropy and its connection to Number Theory and Geometric Group Theory

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).

Topics from applications of mathematics to cryptography

Koji Nuida (AIST)

The speaker of this talk is working on mathematical cryptology. In this talk, the speaker presents some research topics in collaboration of mathematics and cryptology, and discusses some research themes which (the speaker believes) are relevant to geometry or set theory.

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.