1. TDS
3. Plenary

Dissipative Dynamics on the Disc — Sylvain Crovisier <sylvain.crovisier@universite-paris-saclay.fr>

The dynamics of continuous interval maps admit a remarkably rich topological description, including the structure of attractors, criteria for positive entropy, and the density of periodic points in the recurrent set. In recent years, several extensions of these results have been obtained for dissipative diffeomorphisms of the disc, including Hénon maps. In this talk, I will survey some of these developments and present a new closing lemma, proved in collaboration with Enrique Pujals.

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1. GSTT

$\mathbb H^*$ — Will Brian <wbrian.math@gmail.com>

Let $\mathbb H$ denote the half-line $[0,\infty)$, and let $\mathbb H^* = \beta \mathbb H \setminus \mathbb H$ denote its \v{C}ech-Stone remainder. We aim to discuss a recent theorem showing that the Continuum Hypothesis ($\mathsf{CH}$) implies $\mathbb H^*$ is the ``generic'' continuum of weight $\aleph_1$. What precisely this means is the main topic of the talk, but roughly it means that, in the appropriate generalized sense, $\mathsf{CH}$ implies $\mathbb H^*$ is the inverse Fra\"{i}ss\'e limit of the class of metrizable continua. This leads directly to a topological characterization of $\mathbb H^*$ under $\mathsf{CH}$: i.e., a topological property of $\mathbb H^*$ such that $\mathsf{CH}$ implies $\mathbb H^*$ is, up to homeomorphism, the only weight-$\aleph_1$ continuum with this property.

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1. CT

1/2-Indecomposability and positive entropy for inverse limits of Markov set-valued functions — James Kelly <james.kelly@cnu.edu>

We discuss a class of upper semi-continuous set-valued functions called Markov set-valued functions. Such set-valued functions have a corresponding symbolic dynamical system. Through examples, partial results, and open questions we explore the relationship between the topology of the set-valued function's inverse limit and the topological entropy of the symbolic system. In particular, we establish some conditions where positive topological entropy is equivalent to the inverse limit containing a 1/2-indecomposable subcontinuum.

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1. GGT

A Combinatorial Characterization of Sol 3-Manifolds — Leslie Mavrakis <l.mavrakis@utah.edu>

A family F of compact n-manifolds is locally combinatorially defined (LCD) if there is a finite collection of triangulated n-balls (called models) such that the set F is exactly the set of compact n-manifolds that have a triangulation which locally looks like one of these models. In previous work with Daryl Cooper and Priyam Patel, we show that being LCD is equivalent to the existence of a compact branched n-manifold W, such that F is precisely those manifolds that immerse into W. In this way, W can be thought of as a universal branched manifold for F. In this talk, I will explain why the set of Sol 3-manifolds is LCD by constructing a universal branched manifold for the family. The construction is based on regular languages that detect Anosov monodromies of torus bundles and the gluing maps of Sol semi-bundles.

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1. TDS

A Family of Wild Attractors for Unimodal Maps — Lori Alvin <lori.alvin@furman.edu>

It is well-known that a unimodal map $f$ has a unique metric attractor that is either an attracting periodic orbit, the union of $n$ open intervals that are cyclically permuted by $f$, or a Cantor set. A Cantor attractor that arises from a non-infinitely renormalizable map is called an _absorbing Cantor set_ or a _wild attractor_. We present a symbolic construction that can be used to generate the kneading sequences for a family of unimodal maps with embedded strange odometers and wild attractors. This is joint work with Jernej Činč.

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1. TDS

A Tits alternative for groups of surface homeomorphisms — Frédéric Le Roux <frederic.le-roux@imj-prg.fr>

I will present recent results using the natural action of Homeo(S), for a compact surface S, on the fine curve graph introduced by Bowden, Hansel and Webb, who also proved that this graph is Gromov hyperbolic.

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1. TMAA

A dynamical hierarchy of Banach algebras — Matthias Neufang <mneufang@math.carleton.ca>

The concept of stability in the sense of Krivine--Maurey has proven very useful in Banach space geometry. We introduce and study a corresponding notion in the setting of Banach algebras, which we call multiplicative stability. As we shall see, the algebra of Schatten $p$-class operators on a separable Hilbert space is multiplicatively stable, where $p \in [1, \infty)$, while no infinite-dimensional $C^*$-algebra is. We also explore stronger and weaker versions of this concept, using the rich structure of spaces of functions defined on topological semigroups, including almost periodic, weakly almost periodic, and tame functions. This leads us to a novel classification of Banach algebras, providing a dynamical hierarchy. In this context, we also investigate further important classes of Banach algebras, such as group algebras and Fourier algebras, algebras of compact operators on Banach spaces, and algebras of differentiable functions. The talk is based on joint work with (my former PhD student) Narjes Alabkary, (my former postdoctoral fellow) Reza Esmailvandi, and Stefano Ferri.

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1. GGT

A language-theoretic characterization of f.g. subgroups of Thompson V — Davide Perego <dperego9@gmail.com>

The intersection of formal language theory and group theory provides a fascinating lens for studying algebraic structures, most notably through the word problem. This connection has allowed mathematicians to classify groups based on the Chomsky hierarchy. While the landmark Muller-Schupp theorem completely characterized groups with context-free word problems, progressing beyond this boundary has remained a major challenge. In this talk, we shift toward a more combinatorial and geometric approach. We will introduce a new framework that yields a characterization of the f.g. subgroups of Thompson V. Notably, this group is central to a well-known 2008 conjecture regarding groups with co-context-free word problems. Joint works with Corentin Bodart, Daniele D'Angeli, Francesco Matucci and Emanuele Rodaro.

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1. GGT

A linearity criterion for automorphism groups of hyperbolic groups — Mark Pengitore <mpengito@gmail.com>

This talk will introduce various growth functions associated to a finitely generated group which measure the difficulty of separating an element from the identity using epimorphisms to a fixed family of nonabelian finite simple groups with characteristic kernels as a function of the word length. As an application of these functions, we provide a characterization of when the automorphism group of a hyperbolic group is linear.

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1. GSTT

A quantitative measure of non-Isbell convexity in T_0-quasi-metric spaces — Collins Amburo Agyingi <agyingic@gmail.com>

Isbell-convex (q-hyperconvex) T₀-quasi-metric spaces are known to be bicomplete, although the converse does not hold in general. This talk introduces a quantitative invariant that measures the extent to which a bicomplete T₀-quasi-metric space fails to be Isbell-convex. The invariant is defined as a two-component parameter h_q(X)=((h_q)₁(X),(h_q)₂(X)) capturing the inherently asymmetric forward and backward deviations from convexity. Using the framework of minimal function pairs arising in the Isbell hull, we show that h_q(X)=(0,0) characterizes Isbell-convexity. We further establish key properties of this invariant, including non-negativity, invariance under isometries, and monotonicity with respect to subspaces and nonexpansive maps, as well as boundedness in the finite case. Concrete examples demonstrate how bicomplete spaces may fail to be Isbell-convex and illustrate how the invariant quantifies this deviation. This provides a new quantitative tool for studying the interplay between convexity and completeness in asymmetric topology.

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1. GSTT

A soft version of Magill's theorem on n-point Hausdorff compactifications — Robert Gryczka <robert.jakub.gryczka@gmail.com>

Magill's theorem on necessary and sufficent conditions for topological space to have n point Hausdorff compactification is one of the classical tools in the theory of compactifications of topological spaces. In this talk, a soft version of this theorem will be presented in the context of soft topological spaces. Basic concepts of soft set theory and soft topology will be discussed, with particular emphasis on soft Hausdorff compactifications and their relationships with classical topological constructions. Subsequently, a generalization of Magill's theorem to the framework of soft topology will be presented, together with conditions characterizing the existence of soft n-point Hausdorff compactifications.

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1. GSTT

A topological Ramsey space for pseudotrees — David Chodounsky <chodounsky@math.cas.cz>

Topological Ramsey spaces introduced by Todorcevic are spaces equipped with further structure (an ordering, a notion of approximation) which satisfy certain axioms. These spaces can be seen as abstract variations of the classical Ellentuck space and provide a unified framework for proving combinatorial partition theorems. We introduce a new type of A topological Ramsey space consisting of certain strong (sub)trees, which can be can be used for coding rational countable pseudotrees.

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1. GSTT

A tree-based perfectly normal space whose square is not countably metacompact — Assaf Rinot <rinotas@math.biu.ac.il>

The study of the interval topology of trees traces back to the 1960's with Jones' work on the normal Moore space problem. There are various limitations on the kind of spaces that can be obtained this way, for instance, Nyikos proved that for every tree $T$, if $X_T$ is normal, then it is also countably metacompact (CMC), i.e., there are no Dowker trees. Throughout the years, many consistency results were proven concerning the topological characteristics of spaces of the form $X_T$, but we couldn't find similar results dealing with the spaces' square. Here, we present a consistent construction of an Aronszajn tree $T$ such that $X_T$ is perfectly normal but $(X_T)^2$ it not even CMC. A key component of the construction is the use of `elevators' -- a device that enables to construct the tree level-by-level while optimally controlling features of its powers.

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1. GGT

Acyclicity of homeomorphism groups of stable Stone spaces — Michael Kopreski <michaelkopreski@gmail.com>

A stable second-countable Stone space is a closed subspace of the Cantor space with nice local structure. Examples arise as the end spaces of stable infinite-type surfaces and graphs. We classify the acyclicity of the homeomorphism groups of such spaces and describe progress toward computing their homology when not acyclic. These results are joint work with Mladen Bestvina and Rachel Skipper.

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1. QTBD

Adjoining germs of exponentials to R_an, exp while preserving o-minimality — Salma Kuhlmann <salma.kuhlmann@uni-konstanz.de>

Let κ be a regular uncountable cardinal. We construct non-archimedean exponential logarithmic models (R, an, exp) of cardinality κ, for T_an, exp (the elementary theory of the reals with restricted analytic functions and exponentiation), which admit a family F of 2^κ exponentials of pairwise distinct growth rates. These model exhibits the following remarkable features: 1. For each exponential exp' in F, (R, an, exp') is a model of T_an, exp and is thus o- minimal. 2. All exponentials in F agree exactly on the convex valuation ring of R. In particular, the germs of these exponentials are incompatible, in the sense that the structure (R, an, exp, exp') is no longer o-minimal.

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1. GGT

Algebraic fibrations, hyperbolic Coxeter groups and a hidden icosahedron. — Giovanni Italiano <giovanni.italiano.math@gmail.com>

A group is said to algebraically fibre if it admits an epimorphism onto $\mathbb{Z}$ whose kernel satisfies suitable finiteness properties, such as finite generation or finite presentability. The study of algebraic fibrations of hyperbolic groups has been a major theme in geometric group theory over the past two decades, motivated in part by the virtual fibering conjecture for odd-dimensional hyperbolic manifolds. Recently, Lafont, Minemyer, Sorcar, Stover, and Wells constructed, for every $d \geq 2$, a $d$-dimensional hyperbolic group admitting an algebraic fibration with finitely generated kernel. We strengthen this result by showing that, for every $d \geq 3$, there exists a $d$-dimensional hyperbolic group admitting an algebraic fibration whose kernel is _finitely presented_. Our construction combines right-angled Coxeter groups and Bestvina--Brady theory with a new collar-coning procedure inspired by a family of polytopes introduced by Löbell in the 1930s. This is joint work with M. Migliorini and A. Ng.

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1. GGT

An alternative for subgroups of Thompson group V — Corentin Bodart <cobodart123@gmail.com>

In this talk, I will explain some applications of the characterization of finitely generated subgroups of Thompson group V given in Davide Perego's talk. I will recall some of the previously known obstructions, start building up by rephrasing them in our new framework, and then give some ideas towards the following stronger alternative: every finitely generated subgroup of V is either virtually abelian, or contains a free non-abelian semigroup.

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1. QTBD

An unbounded number of canard limit cycles in linear regularizations of piecewise linear systems — Renato HUZAK <renato.huzak@uhasselt.be>

It is known that the number of limit cycles of piecewise linear (PWL) systems is bounded. We show, using Hopf and jump-breaking mechanisms, that the number of (canard) limit cycles in linear regularizations of PWL systems is unbounded.

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1. GSTT

Analogues of Hindman's Theorem for Topological Groups — Serhii Bardyla <sbardyla@gmail.com>

We shall discuss the partition regular properties of topological groups, and obtain an extension of Hindman’s theorem where the monochromatic sets of finite sums are required to satisfy additional topological constraints. In particular, our results imply that for every nowhere dense subset $C\subseteq \mathbb R^n$ there exists an open set $P\supseteq C$ such that for every finite coloring of $\mathbb Q^n\setminus P$ there exists a family $\mathcal A$ of sequences in $\mathbb Q^n\setminus P$ satisfying the following conditions: (i) the set of finite sums $\operatorname{FS}(A)$ is a closed discrete subset of $\mathbb R^n$ for all $A\in\mathcal A$; (ii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is monochromatic; and (iii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is dense in an open unbounded subset of $\mathbb R^n$. The aforementioned result was obtained using a new characterization of spaces whose Stone-Čech compactifications possess remote points.

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1. TDS

Area-preserving surface homeomorphisms without zero entropy — Fabio Armando Tal <fabiotal@ime.usp.br>

The dynamics of area-preserving flows on closed orientable surfaces is a well-understood topic, and there exists a canonical invariant decomposition of the phase space into a region (a collection of topological annuli) where the dynamics is integrable, and a finite number of pieces of positive genus where the dynamics is quasi-minimal (closely resembling the dynamics of an irrational flow on a torus). In this work, we show that a very similar canonical decomposition remains valid when dealing with conservative homeomorphisms with zero topological entropy. We present this decomposition while also exhibiting examples of different phenomena that may arise, as well as several properties of the “quasi-minimal” regions. Part of the work involves proving a Thurston–Nielsen–type reduction result, showing that maps homotopic to Dehn twists and with zero entropy actually possess invariant “Dehn-like” annuli. Time permitting, we will also discuss some applications to Reeb flows on three-dimensional manifolds.

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1. TDS

Birkhoff-like attractors — Martin Sambarino <martinsambarino@gmail.com>

Birkhoff attractors arise from the study of dissipative annulus maps that twist the vertical direction. When their rotation set is nontrivial, these attractors exhibit a complicated topological structure, namely that of an indecomposable continuum. In this talk, we introduce a class of Birkhoff-like attractors for dissipative annulus maps and study the continuity properties of these attractors and their rotation sets under perturbations of the dynamics.

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1. GGT

Book decompositions and Morse complexity — Bena Tshishiku <bena_tshishiku@brown.edu>

A book decomposition of a manifold is a way of decomposing it as a union of codimension-1 submanifolds (the pages) that are glued along a codimension-2 submanifold (the binding). For example, each fibered knot gives a book decomposition of the 3-sphere; more generally, every odd dimensional manifold has a book decomposition by work of Alexander, Lawson, and Quinn. In this talk I’ll explain why even-dimensional hyperbolic manifolds do not have book decompositions. The main tool is Morse complexity, a norm on singular homology introduced by Gromov for which we give some new computations for locally symmetric manifolds. This is joint work with Fedor Manin and Shmuel Weinberger.

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1. CT

Building Spaces with Non-trivial Self Covers — Mathew Timm <mtimm@bradley.edu>

We consider a dynamical systems approach for building spaces which have non-trivial self covers and a connection to self similar groups.

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1. TC

Building confidence regions for Reeb graphs using the interleaving distance — Matteo Pegoraro <matteopegoraro.91@gmail.com>

Reeb graphs and Mapper graphs are widely used in topological data analysis to summarize the evolution of connected components of level sets of a scalar function. However, using these summaries in practice requires principled methods for parameter selection and uncertainty quantification under sampling assumptions. In this talk, I will present a framework for building confidence regions for Reeb graphs using the interleaving distance. The key advantage of this metric viewpoint is that zero distance corresponds to isomorphism of the underlying Reeb-type objects, so confidence balls provide object-level guarantees rather than guarantees only on persistence signatures. Starting from a finite sample, we define intrinsic and extrinsic Mapper-type cosheaf estimators and prove stability bounds comparing them to the target Reeb cosheaf. These bounds lead to confidence regions once the sampling scale is controlled, either through standard sampling assumptions or via subsampling-based estimates. I will also explain how these interleaving bounds relate to classical persistence-based guarantees: in particular, we prove that the extended-persistence pseudometric is controlled by the interleaving distance, with sharp constant 1 for the $H_0$-related components and global constant 2. This provides a direct bridge with previous Mapper confidence frameworks, while giving stronger, metric-level control of the underlying Reeb graph.

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1. TC

Chromatic topological data analysis and the stability of the six-pack via constrained Gromov–Hausdorff distances — Nicolò Zava <nicolo.zava@ist.ac.at>

Topological Data Analysis (TDA) utilise topology-inspired invariants, most notably persistent homology, to extract structural features from complex datasets. A fundamental requirement for these invariants in computational applications is stability under spatial perturbations. Within the standard setting of TDA, the Gromov–Hausdorff distance serves as a rigorous metric framework for comparing underlying datasets and establishing stability guarantees, ensuring that small metric deformations result in bounded changes in the corresponding persistence diagrams. While classical TDA focuses primarily on the geometric arrangement of unlabelled point clouds, modern applications frequently require integrating qualitative features, usually represented by different colours, directly onto the points of a dataset. A prominent example is bioimages of tissues, where different cell types are represented in different colours. This necessity has driven the emergence of chromatic TDA. To capture the homological interactions between distinct coloured subsets, recent techniques utilise the "six-pack", a collection of six interlinked persistence diagrams. In this talk, we recall the standard framework of TDA and the role of the Gromov–Hausdorff distance. We then present some of the techniques utilised to study and compute features from these coloured datasets. Finally, we introduce the $C$-constrained Gromov–Hausdorff distance, a suitable variation of the classical metric adapted for chromatic frameworks, and demonstrate its application in evaluating invariants in chromatic TDA—specifically by establishing the stability of the six-pack.

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1. TDS

Classification of Henon maps with strange attractors via the topology of a stable manifold — Sonja Stimac <sonja@math.hr>

In an earlier work with Boronski, we classified (up to conjugacy) the Henon maps with strange attractors in terms of three invariants that we introduced for them: (a) kneading sequences, (b) pruned trees, and (c) folding patterns of the unstable manifold of the hyperbolic fixed point $X$ in the attractor. In my talk, I will introduce yet another way to determine conjugacy classes of these maps, this time purely from the topology of the stable manifold $W^s$ of $X$. We consider a region of dissipation $D$ for the Henon map and study the connected components of $D \cap W^s$. To each such component, we assign a separation type and prove that two Henon maps are conjugate if and only if their corresponding components share the same separation type. This is joint work with Jan Boronski.

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1. GSTT
3. Plenary

Classification of countably tight groups. — Alex Shibakov <ashibakov@tntech.edu>

We provide a classification of convergence structures in countably tight groups using the Invariant Ideal Axiom and the related Definable Ideal Axiom. We then show some applications to countable groups and the boolean groups with the bounded topology, answering a few published questions. A number of open problems will also be mentioned.

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1. GGT
3. Plenary

Cohomology of arithmetic lattices and link complements — Jean Raimbault <jean.raimbault@univ-amu.fr>

I will present a proof of the following conjecture of Baker--Reid: given any rational homology 3-sphere N, there are at most finitely many congruence arithmetic quotients of hyperbolic space which are homeomorphic to the complement of a link in N. There are many ingredients to the proof but the final step is an asymptotic lower bound on the cuspidal homology of certain congruence subgroups of Bianchi groups. I will therefore use the conjecture as an excuse to talk about various ways to give such bounds, and finally present the somewhat new method we used to get to the result we needed. (Joint work with Steffen Kionke).

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1. QTBD
3. Plenary

Composition of transseries, monotonicity, and analyticity — Vincenzo Mantova <v.l.mantova@leeds.ac.uk>

Transseries generalise power series by including exponential and logarithmic terms, if not more, and can be interpreted as germs of a non-standard Hardy field by composition (for instance, on surreal numbers). I'll discuss a few results that must 'obviously' be true, yet their proofs are not obvious: that composition is monotonic in both arguments, only proved by Edgar for LE-series, that it satisfies a suitable Taylor theorem and that in fact composition is 'analytic with large radius of convergence' (joint with V. Bagayoko), something which appeared before in various special forms, but not in full generality. I'll discuss briefly what I cannot prove yet (convexity!). I'll show how monotonicity and Taylor can be used to prove some fairly general normalisation results for hyperbolic transseries (joint with D. Peran, J.-P. Rolin, T. Servi).

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1. TC

Computability of Common Fixed Points of Isometries — Lucija Validžić <luc.validzic@gmail.com>

For a metric space $X$, the fixed points of $X$ are points that are fixed by all isometries of $X$. We will study computable properties of fixed points of compact subsets of $\mathbb{R}^n$. A natural question is: If $X$ is a computable subset of $\mathbb{R}^n$ whose set of fixed points is non-empty, is there a computable fixed point of $X$? The answer is negative and we will show an example of such $X$, but if the set of fixed points is finite, then the answer is positive. More generally, we will prove that the set of fixed points of a computable set is necessarily semicomputable. Additionally, we will show that the convexity of a computable set $X$ implies that the set of its fixed points is computable, so in that case $X$ contains computable fixed points.

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1. TC

Computable categoricity in Euclidean spaces — Patrik Vasung <patrik.vasung@grad.unizg.hr>

A computable metric space $(X, d)$ is computably categorical if every two effective separating sequences in $(X, d)$ are equivalent up to isometry. We investigate computable categoricity of effectively compact metric spaces. We prove that every effectively compact metric space whose isometry group has computable type is computably categorical. Using this result, we prove that every effectively compact subspace of $\mathbb{R}^n$ is computably categorical.

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1. TC

Computable inner approximation of topological graphs — Matea Čelar <matea.celar@math.hr>

In this talk, we will discuss conditions under which a metric space is inner approximated by its computable subspaces. We focus on (generalised) topological graphs, which are spaces obtained by gluing arcs and rays together at their endpoints. First, we show that every non-vertex point in a semicomputable topological graph has a neighbourhood which is a computable arc with computable endpoints. Using this, we show that every semicomputable topological graph is inner approximated by computable topological graphs with computable endpoints. This talk is based on joint work with Vedran Čačić, Marko Horvat and Zvonko Iljazović.

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1. GSTT

Concentrated sets and the Hurewicz property — Valentin Haberl <valentin.haberl.math@gmail.com>

A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint from $D$ has size smaller than $\mathfrak{b}$. $\kappa$-concentrated sets play a crucial role in the investigation of combinatorial covering properties. It is independent of ZFC if all $\mathfrak{b}$-concentrated sets are Hurewicz. We present ZFC results about how structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property can be characterized with the notion of being meager-unbounded, which we then use to get that such structures are productively Hurewicz. We obtain that assuming that the semifilter trichotomy holds each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. In particular, in the Miller model $\mathfrak{b}$-concentrated sets are also productively Rothberger. We analyze the Laver model, where the behavior of Hurewicz $\mathfrak{b}$-concentrated sets differs from the one under the semifilter trichotomy. This is joint work with Piotr Szewczak (University of Warsaw) and Lyubomyr Zdomskyy (TU Vienna).

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1. GGT

Conjugacy separability in free-by-cyclic groups — Monika Kudlinska <mak74@cam.ac.uk>

A group G is conjugacy separable if any pair of non-conjugate elements remain non-conjugate in a finite quotient of G. While the original motivation for studying conjugacy separability stems from applications to algorithmic problems in group theory, more recently it has been successfully leveraged to exhibit certain rigidity properties of manifolds. In my talk, I will briefly discuss the applications of conjugacy separability to topology, before discussing the new result that all free-by-cyclic groups - a family closely related to 3-manifold groups - are conjugacy separable. This is joint work with Francois Dahmani, Sam Hughes and Nicholas Touikan.

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1. GSTT

Constructions of crowded zero-dimensional Hausdorff P-spaces without the Axiom of Choice — Eliza Wajch <eliza.wajch@gmail.com>

#Constructions of crowded zero-dimensional Hausdorff _P_-spaces without the Axiom of Choice **Eliza Wajch** ***Institute of Mathematics, University of Siedlce, 3 Maja 54, 08-110 Siedlce, Poland*** The results presented here form part of the author's joint work with Eleftherios Tachtsis [2]. They are motivated by the question posed in [1]: is the existence of a non-discrete Tychonoff _P_-space provable in **ZF**? A topological space whose every _G__&delta;-set is open is called a _P_-space. Throughout, our set-theoretic framework is **ZF** or **ZFA**. Among several related results, we show that non-empty zero-dimensional crowded Hausdorff _P_-spaces exist in every permutation model of **ZFA** and in every model of **ZF** having an aleph of uncountable cofinality. A key ingredient in our analysis is the following construction of zero-dimensional Hausdorff spaces which, under additional hypotheses, yields _P_-spaces. Let _X_ be an infinite set, and let &Zscr; be a family of subsets of _X_ closed under finite unions and containing [_X_]^{&lt;*&omega;*}. For _x_ ∈ [X]^{&lt;*&omega;*} and _z_ ∈ &Zscr; with _x_ &cap; _z_ = &empty;, define _B_(_x_,_z_) = \{_y_ ∈ [_X_]^&lt;&omega; : _x_ ⊆ _y_ ⊆ _X_ \ _z_}. Let &Tscr; be the topology on [_X_]^{&lt;*&omega;*} such that, for each _x_ ∈ [_X_]^{&lt;*&omega;*}, the family {_B_(_x_,_z_): _z_∈ &Zscr; and _x_ &cap; _z_ = &empty;} forms a neighborhood base at _x_. The resulting space **S**(_X_, &Zscr;) = ([_X_]^{&lt;*&omega;*} , &Tscr;) is Hausdorff and zero-dimensional, and it is crowded whenever _X_ is not a member of &Zscr;. Assuming that &Zscr; is a bornology on _X_, we obtain that the space **S**(_X_, &Zscr;) is homogeneous, and if it is a _P_-space, then &Zscr; is a &sigma;-ideal. In particular, **S**(_X_, [_X_]^{&lt;*&omega;*}) is a _P_-space if and only if the set _X_ is quasi Dedekind-finite. The space **S**(*&omega;*₁, [*&omega;*₁]^{&le;*&omega;*}) is a _P_-space if and only if *&omega;*₁ has uncountable cofinality. Every denumerable family of non-empty finite sets admits a choice function if and only if, for every infinite set _X_, either _X_ is Dedekind-infinite or **S**(_X_, [_X_]^{&le;*&omega;*}) is a _P_-space. If every denumerable family of non-empty subsets of the real line &#8477; has a choice function, then there exists a topology &Tscr; on &#8477; such that (&#8477; &Tscr;) is a zero-dimensional, crowded Hausdorff _P_-space. The converse implication is false in the Basic Cohen Model of **ZF**. In fact, in the Basic Cohen Model, for every infinite set _X_, the space **S**(_X_, [_X_]^{&le;*&omega;*}) is a _P_-space. **References** - [1] K. Keremedis, A. R. Olfati, and E. Wajch, *On P-spaces and G_&delta;-sets in the absence of the axiom of choice*, Bull. Belg. Math. Soc. Simon Stevin 30 (2), 194–236 (2023). - [2] E. Tachtsis and E. Wajch, *Constructing crowded Hausdorff P-spaces in set theory without the axiom of choice*, submitted manuscript (2025), https://arxiv.org/abs/2510.11935

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1. CT

Continua that admit an inscribed polygon: the Euclidean and hyperbolic settings — Ulises Morales-Fuentes <ulises.morales@uaem.mx>

A plane continuum $X$ is said to admit an inscribed polygon, $P$, if every embedding of $X$ into $\mathbb{R}^2$ (the Euclidean plane) contains the vertices of a polygon similar to $P$. In this talk we adapt this definition to the hyperbolic geometry setting: A plane continuum quasi-inscribes a polygon $Q$ in the hyperbolic plane $\mathbb{H}$, if given any embedding $\gamma:X\hookrightarrow\mathbb{H}$, we have that for all $\varepsilon>0$, $\gamma(X)$ admits a polygon whose inner angular sum is $\varepsilon$-close to the sum in $Q$; and both polygons share geometric structure. In particular, we show that there is a wide class of continua that quasi-inscribe rectangles. We will also include some results, obtained in the Euclidean setting, regarding the inscription of squares and rectangles in continua, in this case we will focus on continua that are ray compactifications.

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1. TDS

Continuation of attractors for discrete semidynamical systems and applications to generalized Hopf bifurcations — Héctor Barge <h.barge@upm.es>

In this talk we shall study continuations of attractors of embeddings in manifolds. We shall introduce the abstract basin of attraction of such attractors and we shall see that if two attractors are related by continuation, their abstract basins are homeomorphic. Moreover, if this continuation is achieved by a small perturbation, then, the homeomorphism can be chosen to be the identity close to the original attractor. We shall make use of this rigidity property in order to characterize the Cech cohomology of the attractors expelled after a generalized Hopf bifurcation of an attractor. These results have been obtained in collaboration with J.J. Sánchez-Gabites

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1. GSTT

Convergence with respect to a semitopogenous order on a complete lattice — Josef Slapal <slapal@fme.vutbr.cz>

A. Czaszar introduced the concept of a semitopogenous order as a binary relation on the power sets of a given set. We extend semitopogenous orders from power sets to arbitrary complete lattices and investigate their behaviour. In particular, we study convergence of generalized nets (upwards closed and centered subsets) with respect to a semitopogenous order and give conditions under which the convergence behaves analogously to the filter convergence in topological spaces. Separation and compactness with respect to a semitopogenous order are discussed, too.

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1. GGT

Convex cocompact groups with three-dimensional limit sets — Lorenzo Ruffoni <lorenzo.ruffoni2@gmail.com>

Given a discrete group of isometries of a real hyperbolic space, we can look at its limit set, i.e., the set of accumulation points of its orbits on the sphere at infinity. This is a compact metric space on which the group acts, and which enjoys many interesting geometric, topological, and dynamical features. For example, classical Kleinian groups provide many examples with limit sets that are quasicircles, Cantor sets, and Sierpinski carpets. In this talk, I will discuss how to construct examples whose limit sets are various low-dimensional trees of manifolds. In particular, we answer a question of M. Kapovich, by constructing convex cocompact groups of isometries of real hyperbolic spaces, whose limit sets are Čech cohomology 3-spheres not homeomorphic to S^3. These groups are right-angled Coxeter groups and our construction is flexible enough to produce infinitely many quasi-isometry classes. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.

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1. QTBD

Cyclicity of piecewise linear centers — Rafel J. Prohens <rafel.prohens@uib.cat>

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1. CT

Decomposability of inverse limits of positive entropy systems on the Gehman dendrite — Jakub Tomaszewski <tomaszew@agh.edu.pl>

The problem of the decomposability of inverse limits of dynamical systems on continua has been of long-standing interest to many researchers in topological dynamics. A cornerstone result by Barge and Martin ([1](https://doi.org/10.1090/S0002-9947-1985-0779069-7)) shows that if a topological dynamical system on a unit interval has positive entropy, then the inverse limit of this system must contain an indecomposable continuum. Since then, Ingram ([2](https://doi.org/10.1090/S0002-9939-1989-0984796-1)), Ye ([3](https://doi.org/10.1016/0166-8641(94)00035-0)), Mouron ([4](https://doi.org/10.1090/S0002-9939-2010-10783-9)), and others have carried out a number of studies, e.g., investigating maps exhibiting a local periodic behavior of a special kind on arc-like continua, and especially homeomorphisms of such spaces with positive entropy. A result by Darji and Kato ([5](https://doi.org/10.1016/j.aim.2016.09.012)) states that the inverse limit of a topological dynamical system on a graph-like continuum with positive entropy must contain an indecomposable continuum. In this talk, we will show that if we consider the Gehman dendrite, then it is possible to construct a system with arbitrarily large entropy whose inverse limit is hereditarily decomposable.

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1. QTBD

Determining the local cycle locus of a vector field with a Hopf singularity — María Martín Vega <martinvega@imj-prg.fr>

Let $\xi$ be an analytic vector field at $0\in \mathbb{R}^3$ with a Hopf singularity, i.e. with eigenvalues $\pm i, c$ with $c\in \mathbb R$. We describe the germ of the subanalytic set $\mathcal{C}(\xi)$ defined by the union of the cycles on small neighborhoods of the singularity. We prove $\mathcal{C}(\xi)$ is the union of a finite number of surfaces or the complement of a curve of singularities of $\xi$. We also prove that the set $\mathcal{C}(\xi)$ is formally determined by any given formal normal form of $\xi$. This talk is based on a joint work with Nuria Corral and Fernando Sanz Sánchez.

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1. GSTT

Dimension under Dense Linear Mappings of Function Spaces — Krzysztof Zakrzewski <kzakrze@sgh.waw.pl>

### Dimension under Dense Linear Mappings of Function Spaces All topological spaces are assumed to be Tichonoff. For a space $X$, by $dim(X)$ we denote the covering dimension of the space $X$. For a topological space $X$, let $C_p(X)$ denote the space of real continuous functions on the space *X* endowed with the pointwise convergence topology. Let $\kappa$ be an infinite cardinal number. A normal space $X$ is called strongly $\kappa$-dimensional if it is a union of $\kappa$ many closed finite dimensional spaces. For $\kappa=\omega$, one obtains the well known class of strongly countable dimensional spaces. A space is called $\kappa$-compact if it is a union of $\kappa$ many its compact subspaces. We improve results concerning invariance of dimension-like properties under transformations of function spaces from [Za]. In particular we show that for a normal, strongly $\kappa$-dimensional space $X$ and a normal, metacompact, locally $\sigma$-compact space $Y$ if there exists a continuous linear operator $T:C_p(X)\xrightarrow[]{}C_p(Y)$ with dense image, then $Y$ is strongly $\kappa$-dimensional as well. The finite-dimensional case is examined as well. We also obtain a compactification theorem that may be of independent interest: every normal strongly $\kappa$-dimensional space admits a strongly $\kappa$-dimensional $\kappa$-compactification. Recall that for a space $X$, we have $dim\,(X)=dim\,(\beta X)$ and that there exist a normal, strongly countable dimensional space without strongly countable dimensional compactification [EP,Example 5.5]. 1. [EP] R. Engelking, E. Pol, _Countable-dimensional spaces: a survey_, Dissertationes Math. 216, (1983). 1. [Za] K. Zakrzewski, _Function spaces on Corson-like compacta_, Results Math. 80, 75 (2025).

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1. QTBD

Discrete hyperbolic dynamical systems and surreal numbers — Dino Peran <dino.peran@pmfst.hr>

Determining the normal form of a map $f=\lambda z+\cdots$, where $\lambda>0$ and $\lambda\neq 1$, is a classical problem in dynamical systems. The goal is to "simplify" $f$ by finding a parabolic change of coordinates $\varphi=z+\cdots$ such that $\varphi^{-1}\circ f\circ\varphi=f_0$, where $f_0$ is a chosen normal form. This problem has been successfully solved in several settings, including analytic diffeomorphisms, various classes of real maps, Dulac maps, and logarithmic transseries. In this work, we investigate the normal form problem in the broader framework of surreal numbers. We review the main techniques used in existing normal form constructions and discuss how these methods can be extended to the surreal number setting. The study of normal forms in this context is connected with the dynamics of analytic planar vector fields because such objects arise as asymptotic expansions of Poincar\'e maps associated with hyperbolic and semi-hyperbolic polycycles of such fields. This is connected to the classical Dulac problem concerning the non-accumulation of limit cycles near polycycles of analytic planar vector fields. A deeper understanding of the formal dynamics of these asymptotic expansions may provide valuable insight into the dynamics of the corresponding Poincar\'e maps and could contribute to further progress on understanding of the Dulac problem. This is joint work in progress with V. Mantova, J.-P. Rolin, and T. Servi.

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1. GGT

Divergence in groups with microsupported action — Dominik Francoeur <dominik.francoeur@uam.es>

Divergence is a quasi-isometry invariant of groups that measures how difficult it is to connect two elements in the Cayley graph of a group by a path that does not pass close to the identity element. It is related to the existence of cut points in the asymptotic cones of the group. In this talk, we will explore divergence in groups with microsupported actions, a class of groups that include interesting examples such as Grigorchuk's groups and Thompson's groups. This is joint work with Letizia Issini and Tatiana Nagnibeda.

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1. TMAA
3. Plenary

Dynamics of unit groups of von Neumann's continuous rings — Friedrich Martin Schneider <martin.schneider@math.tu-freiberg.de>

In the 1930s, John von Neumann developed a continuous-dimensional analogue of finite-dimensional projective geometry. Inspired by conversations with Garrett Birkhoff as well as his collaboration with Francis Murray on rings of operators, von Neumann introduced and studied the notion of a continuous geometry, which is a complete complemented modular lattice possessing a certain continuity property. Among other remarkable results, von Neumann proved that every continuous geometry of order at least four can be coordinatized by some (up to isomorphism unique) ring, and that the continuous rings (i.e., rings corresponding to continuous geometries via this coordinatization theorem) are precisely those irreducible, regular rings which admit a complete rank function. The necessarily unique rank function of a continuous ring gives rise to a compatible metric and thus furnishes the ring with a natural topology. Unit groups of such continuous rings, equipped with the relative topology, constitute an interesting family of topological groups with many peculiar dynamical properties. The talk will provide an introduction to von Neumann's continuous geometry and discuss some of the latest advances concerning topological dynamics of unit groups of continuous rings.

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1. TC

Effective Decomposability of Continua — David Tarandek <david.tarandek@gmail.com>

**Abstract.** We study effective decomposability of continua in computable metric spaces. A continuum $X$ is called **decomposable** if there exist proper subcontinua $A,B$ of $X$ such that $X=A\cup B$. We investigate when such a classical decomposition can be replaced by a computable one. We say that a continuum is **effectively decomposable** if it can be written as the union of two proper computable subcontinua. If $(X,d,\alpha)$ is itself a continuum, then in order to ask whether decomposability can be made effective, one must first require $X$ to be effectively compact. Even under this assumption, it is not known whether decomposability implies effective decomposability. Our first result gives one sufficient condition for effective decomposability. Let $(X,d,\alpha)$ be an effectively compact computable metric space such that $(X,d)$ is a continuum. If $X$ contains an open subset homeomorphic to $\mathbb{R}$, then $X$ is effectively decomposable. Consequently, arcs, topological circles and, in general, topological graphs exhibit effective decomposability in this setting. We also prove a result for chainable continua. If a semicomputable chainable continuum $S$ is decomposable, say $S=K_1\cup K_2$, then we use the fact that $S$ can be inner approximated by a computable subcontinuum $H$. This construction yields computable proper subcontinua $K_1\cup H$ and $K_2\cup H$, and hence an effective decomposition of $S$. Classically, decomposability is also characterized by the following condition: $$ \textbf{(Ch)}\qquad X \text{ is decomposable if and only if } X \text{ contains a proper subcontinuum with nonempty interior.} $$ Motivated by $\textbf{(Ch)}$, we discuss desirable effective versions of this characterization.

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1. CT
3. Plenary

Embeddings of tree-like continua in the plane — Logan Hoehn <loganh@nipissingu.ca>

There are a number of interesting open problems in continuum theory that hinge on determining which tree-like continua can be embedded in the plane. Up to now, there are very few techniques available to show that a given tree-like continuum cannot be embedded in the plane. However, for a special class of tree-like continua, those which are inverse limits of simplicial inverse systems of trees, there is some hope that an algorithm may exist for checking planarity. I will describe this state of affairs, and pose some questions and computational challenges. At the same time, recent results are revealing that more tree-like continua can be embedded in the plane than perhaps were expected. I will discuss two such results: 1) Suppose $Y$ is any continuum of the form $Y = X \cup R$, where $X$ is an arc-like continuum, $R$ is a ray, $X \cap R = \emptyset$, and $\overline{R} \setminus R \subseteq X$. Then $Y$ can be embedded in the plane. 2) Suppose $Y$ is any continuum of the form $Y = K \cup \bigcup_{n=1}^\infty A_n$, where $K$ is a Knaster continuum and $\{A_n: n = 1,2,\ldots\}$ is a family of pairwise disjoint arcs, each intersecting $K$ in a single point, with $\mathrm{diam} A_n \to 0$. Then $Y$ can be embedded in the plane. This is joint work with Andrea Ammerlaan and Ana Anušić.

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1. GGT

Endomorphisms induced by self-maps in low dimensions — Christoforos Neofytidis <neofytidis.christoforos@ucy.ac.cy>

I will explain how residual finiteness and numerical invariants can be used to determine when all self-maps of non-zero degree induce an injective endomorphism or an automorphism of the fundamental group of a manifold in dimension three and in geometric settings in dimension four.

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1. TDS

Entropy Maximizing Measures for Coded Shifts: Beyond Uniqueness — Tamara Kucherenko <tkucherenko@ccny.cuny.edu>

For transitive subshifts of finite type and sofic shifts, the measure of maximal entropy is unique. This fails for coded shifts, which form a natural generalization of these classes. While non-uniqueness is often viewed as pathological, it is still possible to obtain a detailed description of entropy maximizing measures in this setting. We discuss coded shifts that are not intrinsically ergodic and show that an ergodic measure of maximal entropy can be associated with a generator for which it is Bernoulli. This perspective provides a unified framework for understanding both uniqueness and non-uniqueness of entropy maximizing measures and yields explicit descriptions even in non-intrinsically ergodic settings.

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1. GSTT
3. Plenary

Equivariant Means and Extension Properties — Natalia Jonard Pérez <nat@ciencias.unam.mx>

An $n$-mean on a topological space $X$ is a symmetric continuous operation $p:X^n\to X$ satisfying $p(x,\dots,x)=x$ for every $x\in X$. The existence of continuous means is closely related to several classical questions in topology, particularly in connection with retract theory and extension properties. In this talk, we discuss equivariant means associated with group actions on topological spaces and their connections with equivariant absolute extensors. Particular attention will be given to involutions (that is, $\mathbb Z_2$-actions) acting on spaces equipped with compatible lattice structures. We will present some existence results and applications in this setting, and explain how they relate to a classical open problem of Anderson. This is a joint work with Ananda López Poo.

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1. TC

Escape problems for semigroup actions on effective topological spaces — Eike Neumann <e.f.neumann@swansea.ac.uk>

A wide range of fundamental systems verification tasks, such as liveness and safety verification for stochastic or quantum automata, can be modelled as instances of the general problem of deciding whether a point escapes a set under the action of a given semigroup. Theoretical computer scientists traditionally study such problems from a symbolic algebraic perspective: all data is assumed to be provided by exact symbolic means, for example in terms of exact algebraic numbers. In this framework, questions of the above kind become undecidable very quickly. For example, threshold problems for stochastic automata are undecidable in general, and threshold problems for quantum automata are decidable if and only if the inequality with the threshold is taken to be strict. Further, real-world systems are in general not known exactly, but only to some fixed finite accuracy. In this talk, I will advocate for the study of verification problems such as the above from the perspective of effective topology and second-order computability, where we model the input data as points in effective topological spaces. This allows us to naturally model systems that are known only to finite accuracy. Regarding decidability, we will have to make concessions: if an input lies on the boundary of a decision problem, it is trivially impossible for any second-order algorithm to make a correct decision in finite time. The natural question to ask is hence whether there exists a sound decision procedure that halts on the entire complement of the boundary. On the positive side, excluding the boundary instances will often naturally yield a large set of instances where problems of interest do become decidable. I will give a sound decision method for the problem of detecting whether a given point in an effectively locally compact space escapes given a set under a given action of a compactly generated topological semigroup. I will show that this method is complete (in the sense of halting on the complement of the boundary instances) when the space is either (weakly) locally contractible or totally disconnected. I will further give examples of effectively locally compact spaces where there exists a complete method, but my "generic" method fails to be complete, and examples where there is no complete decision method at all.

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1. GSTT

Examining Properties of Selective Divergence — Christopher Caruvana <chcaru@iu.edu>

We discuss the properties of being discretely selective and selectively highly divergent, as well as close variants. We give a variety of examples separating the notions and note their equivalence in rings of continuous real-valued functions. Some relations to hyperspaces of finite subsets are also considered.

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1. TMAA

Extension of maps into equivariant hulls of convex sets — Sergey Antonyan <antonyan@unam.mx>

We will establish the following equivariant extension theorem. Let $G$ be a compact Lie group, $L$ a locally convex metrizable linear $G$-space, and $V$ a closed convex subset of $L$. Denote $G(V):=\{gv\mid g\in G, v\in V\}$ -- the equivariant hull of $V$. Then any $G$-equivariant map $f:A\to G(V)$ defined on a closed invariant subset of a metrizable $G$-space $X$, extends to a $G$-equivariant map $F:U\to G(V)$ over some invariant neighborhood $U$ of $A$ in $X$. If, in addition, $V$ contains a $G$-fixed point, the extension can be taken over the whole space, i.e. $U=X$. In particular, any continuous map $f:A\to G(V)$ from a closed subset of a metrizable space $X$, extends to a continuous map $F:U\to G(V)$ over some neighborhood $U$ of $A$ in $X$. Several applications will be discussed.

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1. TMAA

Frechet derivative is of first Baire class — Eva Kopecka <eva.kopecka@uibk.ac.at>

Let $X$ and $Y$ be Banach spaces, $G\subset X$ an open set and $f:G \to Y$ a mapping. We show that the Fr\'echet derivative $f'$ of $f$ is of first Baire class on the (possibly empty) set $D\subset G$ where it is defined.

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1. TDS

From Dust to Fences — Udayan Darji <ubdarj01@gmail.com>

Homeomorphisms of the Cantor set (“dust”) play a fundamental role in topology, dynamical systems, and descriptive set theory, where they are studied from different perspectives. Recently, various properties of so-called fence-like objects have attracted attention. These include the Lelek fan (from topology), the hairy Cantor set and Cantor bouquet (from dynamical systems), and the Fraïssé fence (from model theory). Several recent works investigate both the structure of these spaces and the dynamics of homeomorphisms defined on them. In this work, we develop a general technique that allows one to transfer—or lift—the dynamics of a given homeomorphism of the Cantor set to a homeomorphism of a fence of the types described above.

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1. TMAA
3. Plenary

From group theory to topological data analysis: asymptotic dimension and the Gromov–Hausdorff distance — Nicolò Zava <nicolo.zava@ist.ac.at>

In his seminal work on finitely generated groups, Gromov established that such groups possess a well-defined large-scale metric structure induced by the word metric of a finite generating set. This perspective transformed geometric group theory by introducing quasi-isometric invariants, a prominent example of which is the asymptotic dimension—a large-scale analogue of the Lebesgue covering dimension. A parallel milestone in this geometric framework was the proof of Gromov's polynomial growth theorem, which characterises groups with polynomial growth, utilising the Gromov–Hausdorff distance to quantify dissimilarities between metric spaces. Almost a decade later, Topological Data Analysis (TDA), a field at the interplay of computational geometry, computer science, and algebraic topology, emerged to study the shape of data. The main tools are topology-inspired invariants, such as persistent homology, used to extract geometric features from datasets. Within this framework, both classic metric notions found new, independent utilities. The Gromov–Hausdorff distance became a standard tool for comparing datasets and evaluating the stability of invariants. The asymptotic dimension was used to analyse the spaces of these invariants, thereby bounding the unavoidable information loss incurred during their vectorisation, a necessary step to integrate them into statistical and machine learning pipelines. In this talk, we discuss how the asymptotic dimension and the Gromov–Hausdorff distance, originally introduced in the realm of topological methods to study algebraic structures, have gained a crucial role in TDA, and present recent results that bridge these notions by determining the asymptotic dimension of the Gromov–Hausdorff space.

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1. GSTT

Function spaces on separable compact lines — Kacper Kucharski <k.kucharski6@uw.edu.pl>

A compact line is any linearly ordered compact topological space. During the talk we will provide a complete isomorphism classification of the spaces of real-valued continuous functions endowed with the topology of pointwise convergence $C_p(K)$ for separable compact lines $K$ of weight $\omega_1$, under the assumption of the Baumgartner's axiom BA. In particular, we will show that, up to linear homeomorphism, there are exactly two function spaces $C_p(K)$, for such $K$. This result should be compared with the recent work by Korpalski, Koszmider and Marciszewski in which it was proved that under the assumption of BA, whenever $K$ and $L$ are separable compact lines of weight $\omega_1$, then the Banach spaces $C(K)$ and $C(L)$ are isomorphic. We will also go over a construction of a ZFC example of a separable compact line $K$ of weight $2^{\omega}$, whose spaces of continuous functions with the pointwise convergence topology $C_p(K)$ and the weak topology $C_w(K)$ are not homeomorphic to their squares.

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1. GSTT

Fundamental properties and characterizations of new classes of δ− β* continuous mappings in m-Polar Neutrosophic Topological Spaces — Lorenzo Affè <lorenzo.affe1@gmail.com>

We introduce and study two classes of neutrosophic continuous mappings: the neutrosophic irresolute $\delta$-$\beta^\*$-continuous mappings (NIr $\delta$-$\beta^\*$ CM) and the $\delta$-$\beta^\*$-neutrosophic contra $\delta$-$\beta^\*$-continuous mappings (NC $\delta$-$\beta^\*$ CM). We establish their fundamental properties and provide characterizations in terms of preimages of $\delta$-$\beta^\*$-open and $\delta$-$\beta^\*$-closed sets. The role of each notion related to the other is shown and analyzed through implication chains, (non-)equivalences under mild hypothesis and stability result under composition, subspaces, and products. Then an extended framework to the $m$-polar setting is shown; in particular, the definitions of the $m$-polar neutrosophic irresolute $\delta$-$\beta^\*$-continuous mappings (MPNIr $\delta$-$\beta^\*$ CM) and $m$-polar neutrosophic contra $\delta$-$\beta^\*$-continuous mappings (MPNC $\delta$-$\beta^\*$ CM) are given. Moreover, this framework shows how core properties lift to the $m$-polar case and where new phenomena arise. Also examples and counterexamples are provided in order to separate the classes and to justify and illustrate the sharpness of the obtained results.

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1. GSTT

Further Observations on Locally Antisymmetric Spaces — Filiz YILDIZ <yfiliz@hacettepe.edu.tr>

Within the framework of asymmetry of the $T_0$-quasi-metric spaces [1], antisymmetric functions are appeared [3] as in some sense opposite to metric functions and studied [4] in detail. Following that in a previous study [2], the locality status of the $T_0$-quasi-metric spaces constructed with antisymmetric functions is described under the name local antisymmetricness. Hence, we are now in a position to ask that how local antisymmetric spaces behaves for subspaces, finite products and intersections-unions. Accordingly, some theorems and counterexamples will be presented about these observations in the context of $T_0$-quasi-metric spaces. Specifically, the question whether the images of locally antisymmetric spaces under an isometry have the same property or not, will be discussed as another problem worth examining.

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1. GSTT

Generalizations of known cardinal inequalities for topological spaces — Ivan Gotchev <gotchevi@ccsu.edu>

In this talk we will present some new results about cardinal inequalities on topological spaces. We introduce the cardinal invariant $nu_s(X)$, the non-Urysohn number for singletons, to generalize the Urysohn separation axiom. Using this invariant, we generalize and extend some known cardinal inequalities for Urysohn spaces to all topological spaces, particularly such that involve variations of tightness and pseudocharcter. The main results pertain to upper bounds on the cardinalities of closures and $\theta$-closures of sets, and variations of the Arhangelskii-Sapirovskii inequality.

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1. GSTT

Generalized almost disjoint families and injective Banach spaces — Chris Lambie-Hanson <lambiehanson@math.cas.cz>

We generalize the notion of almost disjoint family to the setting of arbitrary totally disconnected Hausdorff spaces. We present some results about the existence of such families on the Čech-Stone remainder of the integers. As an application, we present some modest progress concerning the open question of the injective dimension of the Banach space $$c_0$$. This is joint work with David Schrittesser.

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1. TMAA

Generalized spine algebras and their homomorphisms — Ross Stokke <r.stokke@uwinnipeg.ca>

For a locally compact group, *G*, its Fourier and Fourier--Stieltjes algebras *A(G)* and *B(G)* are Banach algebras of continuous functions on *G* that uniquely determine *G* as a topological group; when *G* is abelian, *A(G)* and *B(G)* can be identified via the Fourier--Stieltjes transform with the group and measure convolution algebras on the dual group of *G*. An old problem, solved in the abelian case by Paul Cohen in 1960, asks for a description of all homomorphisms from *A(G)* into *B(H)*. For non-abelian groups, M. Ilie, N. Spronk, M. Daws and H.L. Pham have, among others, made significant contributions to this problem. The difficulty of the problem of describing homomorphisms from *A* into *B(H)* where *A* is some other closed translation-invariant subalgebra of *B(G)* is significantly impacted by the complexity of the Gelfand spectrum of *A*. While the Gelfand spectrum of *A(G)* is just *G* and the spectrum of *B(G)* is often inaccessible, the spine of *B(G)*, *A'(G)*, is a subalgebra of *B(G)* containing *A(G)* whose spectrum is of intermediate complexity between the spectra of *A(G)* and *B(G)*. The spine algebra was introduced by J. Inoue and J. Taylor for abelian groups and by M. Ilie and N. Spronk for nonabelian locally compact groups. For any upper semilattice *D* of locally precompact topologies on *G*, we will define an associated generalized spine subalgebra *AD'(G)* of *B(G)*; when *D* is the set of all locally precompact topologies, we obtain the full spine algebra *A'(G)*. We will discuss properties of generalized spine algebras and identify their spectra as certain semilattices of topological groups. Using almost periodic compactifications, we will introduce a collection of examples of generalized spine algebras over whose spectra we exhibit fine control. Notions of compatible fusions of homomorphisms and affine maps will be introduced and used to characterize all completely positive, completely contractive and, when *G* is amenable, completely bounded homomorphisms from a generalized spine algebra *AD'(G)* to a Fourier--Stieltjes algebra *B(H)*. These results are new, even when *AD'(G)* is the full spine algebra *A'(G)* and even when *G* and *H* are abelian. Examples illustrating the scope of these theorems will be discussed. This is joint work with Nico Spronk and Aasaimani Thamizhazhagan.

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1. TDS

Generalizing the Poincaré-Hopf index in the discrete case — Nelson Schuback <nelson.schuback@imj-prg.fr>

In this talk, we will present a generalization of the Poincaré-Hopf index between trajectories of a non-singular flow on the plane to the discrete case. The main ingredient of the proof is to show that the space of pairs of positively-acessible points of a planar foliation forms a Serre fibration.

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1. TDS

Generic distributional chaos — Lenka Rucká <lenka.rucka@math.slu.cz>

The main result of this talk states, that for a continuous interval map $f$, the set of all Li-Yorke chaotic pairs which are not distributionally chaotic (of any type) is always of the first category in $I \times I$. This result has several immediate applications. For example, the characterization of generic Li-Yorke chaos by Snoha in [1] is valid also for distributional chaos. Following Geschke et al. in [2] we can deduce, that the existence of an uncountable $DCi$ scrambled set implies the existence of a Cantor $DCi$ scrambled set for the interval map $f$, where $i=1,2,3$. [1] L. Snoha; Generic chaos, Comment. Math. Univ. Carol., Vol. 31 (1990), No. 4, 793-810. [2] S. Geschke, J. Grebík, B. D. Miller; Scrambled Cantor sets, Proceedings of AMS, Vol.149, 10 (2021).

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1. GSTT

Hausdorff reflection preserves shape — Diego Mondéjar <diego.mondejar@cunef.edu>

We study the interaction between topological reflections and shape theory. We give general conditions under which a reflection preserves shape, showing in particular that the Hausdorff reflection induces a shape equivalence. This provides a categorical interpretation of reflections as operations that do not alter the global structure of spaces at the level of shape. Applications to inverse limits of finite $T_0$ spaces are discussed, where non-Hausdorff models retain the same shape as their Hausdorff counterparts.

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1. CT

Hedgehogs or how to make a continuum rigid — Teja Kac <teja.kac1@um.si>

For any Peano continuum $X$, we construct uncountable families of rigid, $\frac{1}{n}$-rigid, and $0$-rigid continua, of which all spaces contain a homeomorphic copy of $X$. We also show that for any continuum from the mentioned uncountable families, it holds that for every sequence of continuous surjective functions from the continuum into itself, the inverse limits of such a continuum with the described sequence are homeomorphic to the continuum itself.

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1. GSTT

Higher Lindelöf trees — Pedro Marun <marun@math.cas.cz>

Given an infinite cardinal $\kappa$ and a $\kappa$-splitting $\kappa^+$-tree $T$, we topologize $T$ as follows: if $x\in T$, then sets of the form $\uparrow x \setminus \uparrow F$, for $F$ a set of immediate successors of $x$ with $|F|<\kappa$, form a basis of neighbourhoods of $x$. We then ask whether $T$ is $\kappa^+$-compact with respect to this topology and characterize this property in purely order-theoretic terms. Such trees are necessarily $\kappa^+$-Aronszajn, so they may (consistently) not exist when $\kappa\ge\aleph_1$. In this talk, discuss how to construct such trees using Proxy Principles, introduced by Brodsky and Rinot. We will also mention a further consitency result on the non-existence of such trees together with the failure of the tree property at $\aleph_2$. This is joint work with Ari Meir Brodksy.

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1. GGT

Hurewicz-type formula for asymptotic dimension of countable approximate groups — Vera Tonić <vera.tonic@gmail.com>

In their theorem from 2006, A. Dranishnikov and J. Smith proved that if $f:G\to H$ is a group homomorphism, then the following formula for asymptotic dimension is true: $\mathrm{asdim} G \leq \mathrm{asdim} H + \mathrm{asdim} (\mathrm{ker} f)$. This result is known as the Hurewicz-type formula, after a 1927 theorem from classical topological dimension theory by W. Hurewicz, which inspired it. In this talk we will establish a similar formula to the one by Dranishnikov and Smith, for the following setup: whenever $(\Xi, \Xi^\infty)$ and $(\Lambda,\Lambda^\infty)$ are countable approximate groups and $f:(\Xi, \Xi^\infty) \to (\Lambda,\Lambda^\infty)$ is a (general) quasimorphism, i.e., a quasimorphism which need not be symmetric nor unital, then the following formula is true: $$ \mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda + \mathrm{asdim} (f^{-1}(f(e_\Xi)D(f)^{-1}D(f))), $$ where $D(f)$ is the defect set of the quasimorphism $f$. It follows as a corollary that if $f:G\to H$ is a quasimorphism of countable groups, then $$ \mathrm{asdim} G\leq \mathrm{asdim} H + \mathrm{asdim} (f^{-1}(f(e_\Xi)D(f)^{-1}D(f))).$$ In particular, whenever the quasimorphism $f$ is symmetric and unital, we can replace $f^{-1}(f(e_\Xi)D(f)^{-1}D(f))$ in the formulas above by $f^{-1}(D(f))$.

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1. GSTT

Independent sets in Abelian topological groups of prime exponent — Olga Sipacheva <ovsipa@gmail.com>

A subset $X$ of an Abelian group $G$ with zero element $0$ is said to be *independent* if, given any $n\in \mathbb N$, any pairwise distinct $x_1,\dots, x_n\in A$, and any $k_1,\dots, k_n\in \mathbb Z$, we have $k_1\cdot x_1=\dots= k_n\cdot x_n= 0$ whenever $k_1\cdot x_1 +\dots +k_n\cdot x_n=0$. In other words, $X\subset G$ is independent if the natural homomorphism $\bigoplus_{x\in X}\langle x\rangle \to G$ is injective (here $\langle x\rangle$ denotes the subgroup of $G$ generated by $x$). We say that $X$ is a *basis* of $G$ if $X$ is independent and $\langle X\rangle =G$. We consider independent subsets of Hausdorff Abelian topological groups of prime exponent $p$. It is well known that any such group $G$ is a direct sum of copies of the cyclic group $\mathbb Z/p\mathbb Z$ of order $p$ and hence can be treated as a vector space over the field $\mathbb F_p$. Therefore, $G$ has a basis $E$. Thus, on any Abelian topological group $G$ of prime exponent $p$ with basis $E$, there exists the natural topology induced by the Tychonoff product topology of $\prod_{e\in E}\langle e\rangle$. We refer to this topology as the *product topology on* $G$ *associated with* $E$. A subset $X$ of $G$ is said to be *topologically independent* if, given any $n\in \mathbb N$, any pairwise distinct $x_1,\dots, x_n\in X$, any $k_1,\dots, k_n\in \mathbb Z$, and any neighborhood $U$ of $0$, there exists a neighborhood $V$ of $0$ such that $k_1\cdot x_1, \dots, k_n\cdot x_n \in U$ whenever $k_1\cdot x_1 +\dots +k_n\cdot x_n\in V$. Clearly, any topologically independent set is independent, but the converse is not true: it is known that if $X\subset G$ is topologically independent, then the topology of $H=\langle X\rangle $ is coarser than the product topology on $H$ associated with the basis $X$ of $H$. Recall that the intersection of the kernels of continuous characters of a topological group is called the *von Neumann kernel* of $G$ and denoted by $n(G)$; a group $G$ with $n(G)= G$ is said to be *minimally almost periodic* and a group $G$ with trivial $n(G)$ is said to be *maximally almost periodic*. It is easy to see that an Abelian topological group $G$ of prime exponent is maximally almost periodic if and only if there exists a basis $E$ of $G$ such that the product topology on $G$ associated with $E$ is coarser than the original topology of $G$, i.e., $E$ is topologically independent. There exist examples of minimally almost periodic Abelian groups of any prime exponent. However, any infinite topological Abelian group of prime exponent contains an infinite maximally periodic subgroup (in other words, any such group contains an infinite topologically independent set). This is one of the main results of the report. The second main result is that any countable topological Abelian group of prime exponent has a closed discrete basis. Moreover, any countable-dimensional topological vector space over a finite field or over a complete second-countable valued field (such as $\mathbb R$ or $\mathbb C$) has a closed discrete basis. For uncountable-dimensional spaces, this is not true.

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1. QTBD

Infinitesimal and tangential center problems for planar hamiltonian vector fields — Maria Jesus Alvarez <chus.alvarez@uib.es>

The infinitesimal center problem concerns the persistence of a center under perturbations of a planar Hamiltonian differential system. Its first-order approximation is known as the tangential center problem. In this talk, we study the relationship between these two problems for systems whose origin is a non-degenerate center. We introduce an algorithm that yields necessary conditions for the tangential center problem and explain how its solutions can be employed to investigate the infinitesimal center problem. As an illustration of the method, we present a family of cubic systems for which the tangential center problem admits a complete solution.

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1. CT

Inscription problems in plane continua — Cristina Villanueva-Segovia <cristina@im.unam.mx>

Given a plane continuum $X$ and an annulus $A\subseteq\mathbb{R}^2$, we say that $X$ $A$-inscribes a polygon $P$ if every essential embedding of $X$ into the annulus $A$ contains a similar copy of (the vertices of) $P$. In this talk we will present conditions on $X$ that guarantee that $X$ $A$-inscribes squares for some fixed annulus $A$. Moreover, we will analyze how ubiquitous this property is among continua that separate the plane.

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1. CT

Interruptions and Chaos on Non-smooth fans — Jimmy Zakeršnik <jimmy.zakersnik1@um.si>

In this talk, we present the construction and dynamical properties of a family of arcwise connected continua known as fans. Firstly, we present a construction that, for any smooth fan $X$ containing a top and at least one accumulating leg, produces an uncountable family of pairwise non-homeomorphic non-smooth fans, such that the set of endpoints of each of them is homeomorphic to the set of endpoints of $X$. Secondly, we show that this construction preserves certain dynamical properties such as topological mixing, and some types of chaos. Finally, we apply the results of the paper on a few well-known examples.

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1. QTBD

Length of iterated integrals in Melnikov functions — Jessie Diana Pontigo Herrera <pontigo@ciencias.unam.mx>

Let $H\in \mathbb{R}[x,y]$, and assume that the Hamiltonian foliation $dH=0$ in $\mathbb{R}^2$ has a continuous family of cycles $\gamma(t)\subset \{H=t\}$. We consider a deformation $$ dH+\varepsilon\eta=0, $$ where $\eta$ is a polynomial 1-form and $\varepsilon$ is a small parameter. The question is then what happens to the family of cycles $\gamma(t)$ under this deformation. To study this problem, we complexify the foliations and consider the displacement map $$ \Delta(t,\varepsilon) =\varepsilon M_1(t)+\varepsilon^2 M_2(t)+\cdots, $$ where the functions $M_j(t)$ are analytic in a neighborhood of a regular value $t_0$ of $H$ and are called Melnikov functions (or Poincaré--Pontryagin functions). Depending on whether $\Delta\equiv0$ or $\Delta\not\equiv0$, the family either persists as periodic orbits or gives rise to limit cycles. In this context, the Melnikov functions provide essential information. It follows from Françoise's algorithm that if $\Delta\not\equiv0$, then the first nonzero Melnikov function $M_\mu$ can be expressed in terms of iterated integrals of length at most $\mu$. However, this bound depends explicitly on the deformation $\eta$. On the other hand, in 2018 we showed that there exists a constant $\kappa$, depending only on $H$ and on the orbit under monodromy of $\gamma(t_0)$, that bounds the length of the iterated integrals appearing in $M_\mu$. This constant was called the orbit depth. Later, however, we exhibited an example showing that the orbit depth can be infinite. This motivated us to develop new approaches for obtaining bounds on the length of the iterated integrals appearing in Melnikov functions. In this talk, I will explain the problem of bounding the length of Melnikov functions. The talk by P. Mardesic will continue this discussion and present recent joint work in this direction.

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1. TMAA

Limits of abstraction for convergence theory — Szymon Dolecki <dolecki@u-bourgogne.fr>

Convergence theory studies relations between filters and points. Continuity assures existence of initial and final convergences. This framework enables us to define functors objectwise, like in topological constructs, but a slightly higher level of abstraction in the latter case makes the formalism much more complex. The extension of the concept of adherence to arbitrary families makes it possible to treat various reflective classes of convergence as special cases of types of compactness of families, and not only of sets. This approach enables one to see various classes of quotientness and perfection of maps as forms of compactness of corresponding relations. Our approach has the advantage to represent classical topological properties as solutions to functorial inequalities. Is there any point to consider relations between arbitrary isotone families, not only filters, and points? Greco’s theory of limitoids constitutes the affirmative answer to this question. A limitoid is a functional $T:L^X \rightarrow L$, where $L$ is a complete lattice and $X$ is a set, which is isotone, and commutes with lattice complete homomorphisms. If $L$ is completely distributive, then each limitoid can be represented as a lower limit along an isotone family, which, in general, is not a filter. As all the variational limits of De Giorgi are limitoids, Greco’s theory is a powerful tool for the latter. But as contours are, in fact, lower limits over families of sets, limitoids apply to diagonality and to regularity in convergence theory.

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1. TDS

Mean dimension and finite-to-one maps — Yonatan Gutman <gutman@impan.pl>

We prove that any dynamical system $(X,T)$ that admits the marker property and has mean dimension strictly less than $d$ admits a continuous, finite-to-one equivariant map into $(([0,1]^d)^\mathbb{Z},\operatorname{shift})$. Moreover, in the above situation a generic continuous equivariant map from $X$ to $([0,1]^d)^\mathbb{Z}$ is finite-to-one. In particular when $\operatorname{mdim}(X,T) < \frac{1}{2}d$, we show that such a a generic continuous equivariant map is an embedding and this strengthens the optimal embedding theorem of Gutman, Qiao, and Tsukamoto (2019), for $\mathbb{Z}$-actions. Unlike earlier works, our proof relies on classical topological techniques originating in the work of Ostrand (1965), Kolmogorov (1957), and Arnold (1957). Based on a joint work with Michael Levin and Tom Meyerovitch.

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1. GSTT

Measuring the length of Borel hierarchies — Nick Chapman <nick.steven.chapman@gmail.com>

The class of Borel sets is one of the most fundamental structures on a topological space. Its study lies at the intersection of several areas of mathematics; in this talk, we will investigate properties of the Borel algebra from the viewpoint of descriptive set theory and topology, focusing on the length of this hierarchy on a given second-countable space $X$. The length $ord(X)$ of the hierarchy is defined as the least ordinal $\alpha$ for which every Borel subset of $X$ is $\Sigma^0_\alpha$. The exact value of this ordinal turns out to be highly malleable, and a sophisticated forcing technique was developed by Arnold Miller to produce models of set theory in which it takes on arbitrary values. We will discuss the basic building blocks of this technique, as well as sketch the nature of rank arguments that yield consistency results about assignments of $ord(X)$ to several spaces $X$ simultaneously. Time permitting, we will also delve into the speaker's recent contributions to this area, such as an extension of the framework to the study of generalized Borel hierarchies on topological spaces of uncountable weight.

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1. TC

Metric Bases and Computability of 1-Manifolds — Konrad Burnik <kburnik@gmail.com>

# Metric Bases and Computability of 1-Manifolds **Konrad Burnik** (kburnik@gmail.com), Independent Researcher, The Netherlands This talk is based on joint work with Zvonko Iljazović and Lucija Validžić (University of Zagreb). In computable topology, semicomputability of a space together with computability of its boundary often implies computability of the whole space. It is known that connected 1-manifolds with or without boundary are each homeomorphic to exactly one of $\mathbb{S}^1$, $[0,1]$, $[0,\infty)$ and $\mathbb{R}$ [4]. It was proved in [2] that in a computable metric space $(X,d,\alpha)$ each semicomputable 1-manifold with finitely many connected components, possibly with boundary, whose boundary is computable must itself be computable. The relationship between the computability of an arc and that of its endpoints is well studied: Miller [3] constructed a computable arc in $\mathbb{R}^2$ with noncomputable endpoints, while a computable arc in $\mathbb{R}$ must be a segment $[a,b]$ with $a$ and $b$ computable. The following property makes the endpoints special: a point $x_0$ is a *metric basis* for a metric space $(X,d)$ if $d(x,x_0)=d(y,x_0)$ implies $x=y$. Generalizing this, let $S\subseteq X$, $S \neq \emptyset$, be such that for all $x,y \in X$ if $d(x,s) = d(y,s)$ for all $s \in S$, then $x=y$. Then we call $S$ a metric basis for $(X,d)$. We show that if a computable metric space $(X,d,\alpha)$ is effectively compact and $(X,d)$ has finitely many connected components, then every singleton metric basis is a computable point; the assumption of effective compactness cannot be omitted. We also go beyond the compact setting: if $(X,d,\alpha)$ has the effective covering property [1] and compact closed balls, and $(X,d)$ is a topological ray, then any singleton metric basis is again a computable point. We show that the existence of a computable metric basis in the case of an arc or a topological ray implies the existence of a computable homeomorphism between $(X,d,\alpha)$ and the model space $[0,1]$ or $[0,\infty)$ with its canonical computability structure, respectively. Finally, we will briefly comment on the cases of the topological circle and the topological line, where a metric basis of cardinality more than one, as well as additional computability assumptions on the space are required. ## References [1] V. Brattka and G. Presser, Computability on subsets of metric spaces, Theoretical Computer Science 305 (2003), 43–76. https://doi.org/10.1016/S0304-3975(02)00693-X [2] K. Burnik and Z. Iljazović, Computability of 1-manifolds, Logical Methods in Computer Science 10(2:8) (2014), 1–28. https://doi.org/10.2168/LMCS-10(2:8)2014 (arXiv:1404.6487) [3] J.S. Miller, Effectiveness for Embedded Spheres and Balls, Electronic Notes in Theoretical Computer Science 66 (2002), 127–138. https://doi.org/10.1016/S1571-0661(04)80384-0 [4] A.R. Shastri, Elements of Differential Topology, CRC Press, Taylor and Francis Group, 2011.

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1. GSTT

More ZFC Dowker spaces — Menachem Kojman <kojman@woobling.org>

A construction scheme of topological spaces, which generalizes M. E. Rudin's construction of a Dowker space in ZFCC, is given, and is shown to produce a proper class of Dowker spaces. A proper subclass of this class of spaces are provably collectionwise normal Dowker in ZFC alone. The theory ZFC+SSH, where SSH is Shelah's Strong Hypothesis, proves that the whole class consists of collectionwise normal Dowker spaces. Whether all members of this class are Dowker in ZFC is still open.

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1. QTBD
3. Plenary

Multisummability relative to certain quasianalytic classes realted to Dulac's Problem — Patrick Speissegger <speisse@mcmaster.ca>

Using Tougeron’s characterization of multisummable series (in the positive real direction), the latter can be viewed as infinite series of convergent power series with radii of convergence shrinking to 0. In joint work with Jean-Philippe Rolin and Tamara Servi we showed that, if we replace “convergent power series” with “convergent generalized power series”, we obtain a larger class of multisummable series (again in the positive real direction). This class is shown to generate an o-minimal expansion $\mathbb{R}_{\mathcal{G}^*}$, whose expansion by the exponential function then defines the restrictions to some unbounded interval of both the Gamma and zeta functions. More recently, with Ilgwon Seo, we have been further generalizing this construction by replacing “convergent power series” with “almost regular generalized power series”. The resulting Hardy field is a first step towards filling the remaining gap in Ilyashenko’s proof of Dulac’s problem.

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1. TMAA

Negation functions in fuzzy metric spaces: topological aspects and fixed point results — Juan-José Miñana <juamiapr@mat.upv.es>

The theory of fuzzy metric spaces, originating from the foundational work of Kramosil and Michalek, continues to be an active and relevant area of research. From a topological perspective, these spaces have been extensively investigated, while fixed point theory within this framework remains a topic of ongoing interest. A recent contribution has explored the incorporation of negation functions as a tool to develop a more general setting in fuzzy metric spaces. In particular, such functions have been proposed both to define alternative topological structures and to extend existing fixed point results. In this talk, we examine the role of negation functions from these two viewpoints. Our analysis shows that whenever an alternative way of deriving a topology can be obtained through negation functions, it coincides with the classical topology introduced by George and Veeramani. Furthermore, when negation functions are assumed to be strict or strong, the resulting classes of fuzzy contractions do not provide genuine extensions of previously known ones. Consequently, the fixed point results obtained in this context can be regarded as direct corollaries of earlier theorems.

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1. QTBD

Newton Diagram and Topological Invariance for $\mu$-constant Deformations of Generalized Curves — Jesus Alberto Palma Marquez <jpalma@im.unam.mx>

We prove that $\mu$-constant deformations of generalized curves; that is, non-dicritical plane holomorphic foliations with no saddle-nodes in their desingularization, are equisingular. Furthermore, under the classical convenience assumption on the Newton diagram, we show that there exists an analytic family of coordinates preserving the Newton diagram throughout the deformation. Thus, we extend both the L\^{e}--Ramanujam theorem and Oka's Newton stability to germs of plane holomorphic foliations.

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1. QTBD

Noetherianity and Length of Melnikov Functions — mardesic pavao <mardesic.pavao@gmail.com>

We study foliations in $\mathbb{C}^2$ given by polynomial deformations of the form $dH+\epsilon \eta=0$, with $\gamma(t)\subset H^{-1}(t)$ a family of cycles. The Poincaré first return map is of the form $P(t)=t+\sum_j \epsilon^j M_j^\gamma(t).$ The functions $M_j^\gamma$ are called Melnikov functions and are given by iterated integrals of orbit length at most $j$. This length is a measure of the complexity of Melnikov functions. We show that, for each $k\in\mathbb{N}$, there exists a universal Noetherianity index $n_{ H,\gamma}(k)$, independent of the deformation $\eta$, such that, if $M_j^\gamma=0$, for $j=1,\ldots,n_{ H,\gamma}(k)$, then $M_j^\gamma$ is of orbit length $j-k$, for any Melnikov function $M_j^\gamma$. In order to prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt-Raudenbush differential algebra theorem. We calculate the universal Noetherianity index $n_{H,\gamma}(k)$ in various nontrivial examples. The presented work is a recent work which is a continuation of the work to be presented here by J. Pontigo-Herrera.

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1. CT
3. Plenary

Non-existence of common models for certain classes of continua — Eiichi Matsuhashi <matsuhashi@riko.shimane-u.ac.jp>

A continuum $X$ is called a \emph{common model} for a class $\mathcal{C}$ of continua if every member of $\mathcal{C}$ is a continuous image of $X$. One of the natural questions in continuum theory is whether a given class of continua admits a common model, and if not, how the non-existence of common models can be established. In this talk, we will discuss several recent results concerning the non-existence of common models for classes of continua arising in hyperspace theory and the theory of indecomposable continua. The main tool is a recent theorem on meandering continua, which provides a general method for establishing non-existence results. As applications, we will present new classes of continua associated with Whitney properties and Whitney reversible properties, together with several classes related to indecomposable continua, and show that these classes do not admit common models.

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1. GSTT

Non-meager P-filters, Miller-measurability, and a question of Hrušák — Andrea Medini <andrea.medini@tuwien.ac.at>

We will discuss our recent partial answer to a question of Hrušák: if a product of filters on ω is countable dense homogeneous, then the number of factors is smaller than **p** and each factor is a non-meager P-filter. Furthermore, we will show that non-meager P-filters can be characterized as the "chunkiest" filters with respect to Miller-measurability. As a rather "quotable" corollary, we will see that the intersection of fewer that **add**(_m_⁰) non-meager P-filters is a non-meager P-filter, where _m_⁰ denotes the ideal of Miller-null sets. All of these results build on an old joint paper with Kunen and Zdomskyy.

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1. QTBD

Normal forms for planar homoclinic 1:1 saddle loops — Loïc TEYSSIER <teyssier@math.unistra.fr>

We solve the embedding problem for Poincaré maps appearing in foliations on abstract complex surfaces near 1-polycycles corresponding to homoclinic connections of a 1:1 saddle point. We particularly prove that every such foliation is biholomorphic to a foliated neighborhood of some unique model saddle-loop in $\mathbb{C}^{2}$, defined in a neighborhood of an explicit singular elliptic curve.

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1. QTBD

O-minimality of some almost regular multisummable germs — Ilgwon Seo <seoi@mcmaster.ca>

The main goal of this project is to establish the o-minimality of an algebra containing multisummable functions and almost regular germs. An o-minimal structure is a framework for studying sets and functions with tame geometric behavior: in particular, every one-dimensional definable set is a finite union of points and intervals. This finiteness property leads to various uniform boundedness results and is a central source of tameness. Roughly speaking, the proof of o-minimality proceeds in two steps. The first is to construct a quasianalytic algebra of generalized variables. The second is to identify a suitable class of power series with coefficients in this algebra that remains stable under the operations needed in the construction. In this talk, I will describe the current progress of the project and explain the main ideas behind these two steps.

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1. GGT
3. Plenary

Obstructing Riemannian smoothings on CAT(0) manifolds — Jean-François Lafont <jlafont@math.ohio-state.edu>

CAT(0) geometry is a metric generalization of Riemannian non-positive curvature. One could wonder, in the context of closed manifolds, if this is a genuine generalization? Up to dimension three, every closed manifold supporting a CAT(0) metric also supports a Riemannian non-positively curved metric. But this is no longer true when the dimension is >3. I will give an overview of the various known constructions of "exotic" CAT(0) manifolds in higher dimensions, culminating in a sketch of some new high dimensional examples (joint work w/ Bakul Sathaye).

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1. GSTT

On Set-Relatively Star-Menger Subspaces and Related Star Covering Properties — Sumit Singh <sumit@ramjas.du.ac.in>

In this paper, we study set-relatively star-Menger subspaces and their connections with classical and star covering properties. We provide characterizations of set-RSM spaces and show that the family of such subspaces forms an admissible $\sigma$-ideal. Several examples are constructed to clarify relationships with existing notions and to correct earlier claims in the literature. We also investigate preservation properties under mappings and products, and establish equivalences between relative versions of star-K-Menger, star-C-Menger, and star-K-Hurewicz properties with their corresponding set versions.

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1. GSTT

On a Special Convergence in Cap Spaces — Meryem ATEŞ <mbiten@ankara.edu.tr>

In a topological space, Kuratowski convergence of hypernets is defined by Beer[1] and in a convergence space, Kuratowski convergence of hyperfilters is defined by Dolecki and Mynard [5]. In this study, we introduce and study upper and lower Kuratowski convergences of hyperfilters in the category Cap of convergence approach spaces and contractions. Given a convergence approach space $(X,\lambda)$, let $C_{c(\lambda)}$ denote $c(\lambda)$-closed subsets of $X$. For a hyperfilter $\mathfrak{F}$ defined on $C_{c(\lambda)}$ and $A\in C_{c(\lambda)}$ we defined: $ \lambda_{uK}\mathfrak{F}(A)=\bigvee_{x\notin A}1\oslash adh_\lambda (rdc\mathfrak{F})(x)$, $ \lambda_{lK}\mathfrak{F}(A)=\bigvee_{x\in A}1\oslash adh_\lambda (rdc\mathfrak{F}^\textit{#})(x)$ and $ \lambda_{K}\mathfrak{F}(A)=\lambda_{uK}\mathfrak{F}(A) \bigvee \lambda_{lK}\mathfrak{F}(A). $ Given an $\epsilon\in[0,\infty]$, the filter $\mathfrak{F}$ is said to be $\epsilon-$upper Kuratowski convergent (respectively $\epsilon-$lower Kuratowski convergent, respectively $\epsilon-$ Kuratowski convergent) to $A$ if $\lambda_{uK}\mathfrak{F}(A)\leq\epsilon$ (respectively $\lambda_{lK}\mathfrak{F}(A)\leq\epsilon$, respectively $\lambda_{K}\mathfrak{F}(A)\leq\epsilon$). We investigate the properties of this convergences and then obtain relations with these new notion of convergence and Fell approach structure defined by Ateş and Sagıroglu in [4]. We show that the upper Fell convergence approach structure is a non-Archimedean approach structure coarser than the upper Kuratowski convergence approach structure, but finer than the upper Fell approach structure introduced in [4]. We also obtain that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.

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1. CT

On chains of compacta — Bryant Rosado Silva <bryantrs99@hotmail.com>

In this talk, we are going to discuss what the typical maximal chain of compacta in the Cantor space looks like and use it as inspiration to discuss chains on the pseudoarc. This is a work in progress with Benjamin Vejnar.

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1. TDS

On continuum-wise hyperbolic dynamics on surfaces — Piotr Oprocha <piotr.oprocha@osu.cz>

Hyperbolicity is a central notion in the study of chaotic dynamical systems. Unfortunately, expansivity which is one if its main ingredients is very uncommon in typical dynamics. Because of this limitation, over the years some generalizations appeared in the literature, trying to preserve main features of hyperbolic, yet present in much more generality. In 1993 Kato introduced the notion of continuum-wise expansive homeomorphisms, and in 2024 it was used by Artigue, Carvalho, Cordeiro and Vieitez to define continuum-wise hyperbolicity. This definition combines cw-expansive with kind of local product structure, also expressed in terms of evolution of continua. In this talk we will survey selected results for surface dynamics and present new results obtained by the author jointly with several collaborators.

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1. GSTT

On densely defined linear continuous operators between function spaces — Arkady Leiderman <arkady@bgu.ac.il>

For any Tychonoff space $X$, let $D(X)$ denote either the space $C(X)$ of all continuous real-valued functions on $X$ or the space $C^*(X)$ of all bounded continuous real-valued functions on $X$.$\,\,\,$ We write $D_p(X)$ when $D(X)$ is endowed with the topology of pointwise convergence. In our recently published paper, A. Eysen, A. Leiderman and V. Valov, _Linear and uniformly continuous surjections between $C_p$-spaces over metrizable spaces_, Math. Slovaca, vol. 75 (2025), pp. 669--678, we obtained the following result: **Theorem.** If $T: D_{p}(X) \to D_{p}(Y)$ is a linear continuous surjection, where $X$ is a metrizable space and $Y$ is a perfectly normal space, then $Y$ inherits a given topological property $\mathcal{P}$ from $X$. A linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$ is said to be **densely defined** if $E(X)$ and $E(Y)$ are dense linear subspaces of $D_{p}(X)$ and $D_{p}(Y)$, respectively. In our talk, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$. In particular, $\mathcal{P}$ can be zero-dimensionality, strong countable-dimensionality, or $\sigma$-compactness. Additionally, for arbitrary Tychonoff spaces $X$ and $Y$, assuming only that $T: E_p(X)\to E_p(Y)$ is a densely defined linear continuous operator, we show that $X\in\mathcal P$ implies $Y\in\mathcal P$ where $\mathcal P$ is the property $(\kappa)$, the strong $\sigma$-scatteredness, or the property of being a $\Delta_1$-space.

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1. GSTT

On graph-induced betweenness — Aisling McCluskey <aisling.mccluskey@universityofgalway.ie>

Metric spaces give rise naturally to betweenness relations through the associated lens of generalised triangle (in)equalities. Examples include the usual metric betweenness of Karl Menger [1] whereby a point $c$ is said to be between points $a$ and $b$ in a metric space $(X,\rho)$ if $\rho(a,b) = \rho(a,c) + \rho(c,b)$. Another ultrametric version, contrasting sharply with Menger betweenness but aligning strongly with subcontinuum betweenness amongst hereditarily indecomposable continua, is where we declare $c$ to be between $a$ and $b$ if $\rho(a, b) = \max \{\rho(a,c), \rho(c,b)\}$. Such betweenness relations induced by metrics with values in a finite set turn out to be of interest through a natural correlation with simple graphs. We exploit this to identify when a given betweenness relation is graph-induced; namely, that edges between vertices (points of $X$) can be labelled from the set $\{1,2\}$ in such a way that the associated Menger betweenness relation from this metric (with values in the set $\{0,1,2\}$) coincides with the original betweenness relation. This is joint work with Paul Bankston (Marquette University, Wisconsin) and Steve Watson, York University, Toronto. [1] Karl Menger, Untersuchungen \"{u}ber allgemeine Metrik, Math. Ann. 100 (1928), 75--163

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1. GSTT

On the Gromov-Hausdorff quasi-metric distance — Olivier Olela-Otafudu <olivier.olela-otafudu@ul.ac.za>

In this talk, we introduce the concept of the Gromov-Hausdorff quasi-metric distance between two quasi-metric spaces. We then use this concept to study the stability estimates of two Isbell-hulls quasi-metric spaces. Moreover, we obtain the asymmetric version of the following well-known result: the Gromov-Hausdorff distance of two hyperconvex metric spaces generated by certain subsets is less than or equal to the Gromov-Hausdorff distance of these sets.

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1. GGT

On the characteristic classes of hyperbolic manifolds — Stefano Riolo <stefano.riolo@unibo.it>

Every finite-volume hyperbolic manifold is finitely covered by a stably parallelizable manifold, which in particular has trivial Stiefel-Whitney and Pontryagin classes. On the other hand, it is difficult to produce hyperbolic manifolds with non-trivial characteristic classes. In two recent works, one with Rizzi and one with Bustamante and Reyes, we complement the previously existing results of Long-Reid, Martelli-R-Slavich and Chen on the theme, solving one of the K3 problems and answering questions of Charney-Davis and of Belegradek on strict hyperbolization, respectively. The talk will essentially consist of an overview on the subject.

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1. TDS

On the growth of the number of periodic points of smooth maps — Luis Hernández-Corbato <luiherna@ucm.es>

A conjecture by Shub states that the asymptotic exponential growth rate of the number of periodic points of a C¹ map f : M → M is bounded from below by an algebraic topological quantity: the exponential growth rate of Lefschetz numbers L(f ⁿ). In particular, if M is a sphere the conjecture states that # Fix(f ⁿ) grows asymptotically at least as d ⁿ, where d denotes the degree of f. The conjecture is wide open in general, even in S². In the talk, we will review some results in very particular cases: maps on S³ leaving invariant a circle and maps preserving a singular foliation on a closed surface. This is joint work with H. Barge, A. Moreno, J. Sanchez-Gabites (Madrid).

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1. CT

On the hyperspace of completely regular curves — Paweł Krupski <pawel.krupski@pwr.edu.pl>

A nondegenerate continuum $X$ is _completely regular_ if each nondegenerate subcontinuum of $X$ has nonempty interior. The class of completely regular continua contains all nontrivial connected finite topological graphs and is contained in the class of all regular curves. Let $CR(I^n)$ denote the hyperspace of completely regular subcontinua of the cube $I^n$, $2\le n\le\infty$, considered as a subspace of the Vietoris hyperspace $C(I^n)$ of all subcontinua of $I^n$. We will discuss the descriptive complexity of $CR(I^n)$: the hyperspace is a Borel subset of $C(I^n)$ which is not $F_{\sigma\delta\sigma}$. In fact, in the spirit of the theory of absorbing sets, one can show that $CR(I^n)$ is an absolute retract which is strongly $G_{\delta\sigma\delta}$-universal in the topological Hilbert cube $C(I^n)$.

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1. QTBD

On the number of normally hyperbolic limit Tori in 3D polynomial vector fields — Lucas Arakaki <lucas.queiroz@unesp.br>

The second part of Hilbert's 16th problem concerns determining the maximum number $H(m)$ of limit cycles that a planar polynomial vector field of degree $m$ can exhibit. A natural extension to the three-dimensional space is to study the maximum number $N(m)$ of limit tori that can occur in spatial polynomial vector fields of degree $m$. In this work, we focus on normally hyperbolic limit tori and show that the corresponding maximum number $N_h(m)$, if finite, increases strictly with $m$. More precisely, we prove that $N_h(m+1)\geq N_h(m)+1$. Our proof relies on the torus bifurcation phenomenon observed in spatial vector fields near Hopf-Zero equilibria. While conditions for such bifurcations are typically expressed in terms of higher-order normal form coefficients, we derive explicit and verifiable criteria for the occurrence of a torus bifurcation assuming only that the linear part of the unperturbed vector field is in Jordan normal form. This approach circumvents the need for intricate computations involving higher-order normal forms.

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1. TDS

On visit numbers to semi-circles and automatic sequences — Henk Bruin <henk.bruin@univie.ac.at>

Some sequences related to circle rotations over simple quadratic irrationals turn up in the online encyclopaedia of integers sequences (OEIS). In this joint work with Robbert Fokkink some conjectures about A120243 are solved, using either automata theory or renormalization of circle rotations.

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1. CT

Persistent Recurrence and Inverse Limits of Unimodal Maps — Lori Alvin <lori.alvin@furman.edu>

Given a unimodal map, the recurrent critical point $c$ is said to be _reluctantly recurrent_ if there exists a $\delta > 0$ such that for every $\ell\in \mathbb{N}$ there is a backward orbit $\mathbf{x} = (x_{-\ell},\cdots x_{-2},x_{-1},x_0)$ in $\omega(c)$ such that $B(x_0,\delta)$ has a monotonic pull-back along $\mathbf{x}$; otherwise we say $c$ is _persistently recurrent_. Given a unimodal map $f$ with an infinite kneading sequence, it is known that the collection of endpoints for the inverse limit space $\varprojlim${$[c_2,c_1],f$} is precisely the collection of folding points if and only if $c$ is persistently recurrent. We revisit this known result and also show that when $c$ is infinitely recurrent and longbranched, it is not possible for $c$ to be persistently recurrent. This is joint work with Jernej Činč.

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1. CT

Plane continua, canals and dead ends — Rene Gril Rogina <rene.gril@um.si>

Given a continuum X in the Euclidean plane, a canal of X is a way of “approaching” the continuum from outside of X or the bounded components of its complement. Often we search for simple dense canals, which are rays with X as their remainder. While some things are known about planar continua with embeddings that admit such canals, there are still open questions on this topic. In this talk, we first define canals and then “dead ends”, which are used in a construction to obtain new planar continua and new embeddings of these continua, all of which have canals with the desired properties. This is joint work with my PhD advisor Jernej Činč. This work was co-financed by the Slovenian Research and Innovation Agency (ARIS) under Contract No. SN-ZRD/22-27/0552.

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1. GGT

Profinite rigidity of Kähler groups — Claudio Llosa Isenrich <claudio.llosaisenrich@uni.lu>

A classical problem in complex algebraic geometry is understanding the topology of smooth complex projective varieties, and, more generally, of compact Kähler manifolds. Two natural topological invariants to consider are the fundamental group and its profinite completion; the latter is also known as the algebraic fundamental group. In this talk I will address the following questions: When is the fundamental group of a compact Kähler manifold uniquely determined by its profinite completion? And, when does the profinite completion even determine the homeomorphism type of the underlying manifold? In particular, I will explain positive answers to both questions in the case of a direct product of fundamental groups of closed hyperbolic Riemann surfaces. This talk is based on joint work with Hughes, Py, Stover and Vidussi.

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1. GSTT

Ramsey spaces on trees with the successor operation — Jan Hubička <honza.hubicka@gmail.com>

Several Ramsey theorems and Ramsey spaces, such as the Milliken tree theorem and the Carlson-Simpson theorem, are naturally viewed as results about trees and their subtrees. Recently, the study of big Ramsey degrees of universal structures has led to a need for additional variants of these theorems where the notion of a subtree is modified. We discuss a general framework for proving Ramsey-type theorems on trees with finite but possibly unbounded branching and the associated Ramsey spaces. These spaces are formed by collections of infinite subtrees equipped with a topology generalizing the Ellentuck space. By verifying that these structures satisfy the abstract Ramsey space axioms, we ensure that every subset with the Baire property is Ramsey. This framework specifically incorporates the successor operation to maintain structural integrity during embeddings. This is joint work with Martin Balko, David Chodounský, Natasha Dobrinen, Matěj Konečný, Jaroslav Nešetřil, and Andy Zucker.

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1. CT

Reciprocating Domains in Classes of Continua — Iztok Banic <iztok.banic@um.si>

In topology, universal objects often serve as models for an entire class of spaces. A space $X$ is called a universal domain in a class $\mathcal C$ if every member of $\mathcal C$ is a continuous image of $X$. Classical examples include the Cantor set among compact metrizable spaces and the arc among Peano continua. In this talk, we give a new notion, called a reciprocating domain. A space $X$ in a class $\mathcal C$ is a reciprocating domain if whenever another space $Y\in\mathcal C$ admits a continuous surjection onto $X$, then $X$ also admits a continuous surjection onto $Y$. Intuitively, such a space cannot be reached from a ``larger'' space in the surjective order without also being able to map back onto that space. We discuss general properties of reciprocating domains and explain their relationship with universal domains. The main part of the talk will focus on examples arising in continuum theory, including chainable continua, tree-like continua, solenoids, circle-like continua, fans, and several related classes. Along the way, we will see that some familiar universal continua are also reciprocating, while in other natural classes reciprocating domains do not exist at all.

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1. TMAA

Recurrence and rigidity of multipliers on commutative Banach algebras I — Enrique Jordá <ejorda@mat.upv.es>

**Definition** [Costakis, Manoussos, Parissis] Let $X$ be a Banach space. We say that a bounded linear operator $T\colon X\to X$ is 1. *Recurrent*: if for each $x\in X$ there exists an increasing sequence $(n_k)$ of natural numbers such that $T^{n_k}(x)$ converges to $x$. 2. *Rigid*: if there exists an increasing sequence of natural numbers $(n_k)$ such that $T^{n_k}(x)$ is convergent to $x$ for every $x\in X$. 3. *Uniformly rigid*: if there exists an increasing sequence $(n_k)$ of natural numbers such that $\lim_k \|T^{n_k}-I\|=0$. In these consecutive talks, we report on joint work by M.J. Beltrán, E. Jordá and J. Galindo where we study the recurrence and rigidity of multipliers on semisimple Banach algebras and analyze the case of the Fourier algebra $A(G)$ of a locally compact group. We will address the following problems: -- Let $T\colon \mathfrak{A}\to \mathfrak{A}$ be a multiplier of a semisimple Banach algebra $\mathfrak{A}$ with spectrum $\Delta(\mathfrak{A})$. Its Gelfand transform, $\widehat{T}$, then defines a multiplier $M_{_{\widehat{T}}}\,\colon C_0(\Delta(\mathfrak{A}))\to C_0(\Delta(\mathfrak{A}))$. Find the relations between the recurrence and rigidity of these multipliers and determine under which conditions they coincide. -- Describe the recurrent and (uniformly) rigid multipliers of the Fourier algebra and find examples separating these concepts. Our results show that for power-bounded multipliers on the most common algebras (including Fourier algebras), the recurrence and rigidity properties of a multiplier $T$ coincide exactly with those of $M_{_{\widehat{T}}}$; this will be the subject of Talk I. In the absence of power-boundedness this equivalence fails, and we provide counterexamples within the framework of Fourier algebras; these will be discussed in Talk II.

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1. TMAA

Recurrence and rigidity of multipliers on commutative Banach algebras II — Jorge Galindo <jgalindo@uji.es>

**Definition** [Costakis, Manoussos, Parissis] Let $X$ be a Banach space. We say that a bounded linear operator $T\colon X\to X$ is 1. *Recurrent*: if for each $x\in X$ there exists an increasing sequence $(n_k)$ of natural numbers such that $T^{n_k}(x)$ converges to $x$. 2. *Rigid*: if there exists an increasing sequence of natural numbers $(n_k)$ such that $T^{n_k}(x)$ is convergent to $x$ for every $x\in X$. 3. *Uniformly rigid*: if there exists an increasing sequence $(n_k)$ of natural numbers such that $\lim_k \|T^{n_k}-I\|=0$. In these consecutive talks, we report on joint work by M.J. Beltrán, E. Jordá and J. Galindo where we study the recurrence and rigidity of multipliers on semisimple Banach algebras and analyze the case of the Fourier algebra $A(G)$ of a locally compact group. We will address the following problems: -- Let $T\colon \mathfrak{A}\to \mathfrak{A}$ be a multiplier of a semisimple Banach algebra $\mathfrak{A}$ with spectrum $\Delta(\mathfrak{A})$. Its Gelfand transform, $\widehat{T}$, then defines a multiplier $M_{_{\widehat{T}}}\,\colon C_0(\Delta(\mathfrak{A}))\to C_0(\Delta(\mathfrak{A}))$. Find the relations between the recurrence and rigidity of these multipliers and determine under which conditions they coincide. -- Describe the recurrent and (uniformly) rigid multipliers of the Fourier algebra and find examples separating these concepts. Our results show that for power-bounded multipliers on the most common algebras (including Fourier algebras), the recurrence and rigidity properties of a multiplier $T$ coincide exactly with those of $M_{_{\widehat{T}}}$; this will be the subject of Talk I. In the absence of power-boundedness this equivalence fails, and we provide counterexamples within the framework of Fourier algebras; these will be discussed in Talk II.

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1. TDS

Renormalization of regularly critical diffeomorphisms of the disk — Jonguk Yang <jongukyang@gmail.com>

A diffeomorphism of the disk is called _mildly dissipative_ if, for every invariant measure, the stable manifold of almost every point disconnects the domain. Crovisier, Pujals and Tresser proved that every mildly dissipative diffeomorphism with zero topological entropy that is not generalized Morse–Smale is infinitely renormalizable. In this talk, we survey a various regularity conditions under which such systems converge, under renormalization, to the universal renormalization attractor in the space of unimodal maps. We also discuss several ongoing projects and open directions related to this program. This talk is based on joint work with Sylvain Crovisier, Mikhail Lyubich, and Enrique Pujals.

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1. GGT

Rigidity for hyperbolic groups with Pontryagin sphere boundary — Emily Stark <emilyrstark@gmail.com>

The Pontryagin sphere is a homogeneous, nowhere planar, compact 2-dimensional fractal constructed as an inverse limit of closed orientable surfaces. The Pontryagin sphere arises naturally as the boundary at infinity of the fundamental group of a 3-dimensional hyperbolic pseudo-manifold. We prove that if the conformal dimension of the boundary is less than four, then such a group is action rigid: if it acts geometrically on the same proper metric space as another group, then the groups are virtually isomorphic. A key component of the proof is a generalization of Yang's Theorem regarding the structure of p-adic actions on a tree of manifolds. This is joint work with Chris Cashen, Pallavi Dani, and Kevin Schreve.

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1. QTBD

Rigidity of saddle loops — Maja Resman <mresman@math.hr>

We define an abstract complex saddle loop in $\mathbb C^2$ as a pair $(F,R)$ of a hyperbolic normalized saddle foliation $F$ with a corner Dulac map D and a regular map $R\in\mathrm{Diff}(\mathbb C,0)$. Up to an appropriate equivalence relation that corresponds to different determinations of complex Dulac and to transversal changes, the first return map is given by $F=RD$ on the universal cover of the standard quadratic domain. We show that such Poincar\` e maps are rigid, in the sense that their non-ramified formal conjugacy implies the analytic conjugacy.

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1. TDS

Rotational Axiom A homeomorphisms for higher genus surfaces — Pierre-Antoine Guihéneuf <pierre-antoine.guiheneuf@imj-prg.fr>

Consider a homeomorphism of closed surface of genus $g \ge 2$. I will explain that in the case its homological rotation set (a compact subset of $\mathbf R^{2g}$ capturing the rotational behaviour of the dynamics) is big enough, the whole rotational behaviour is contained in a compact set that resembles a finite union of homoclinic classes with some heteroclinic connections. This is related to $C^0$ rotational versions of properties like Markov partitions, rotational density of periodic orbits, stability under perturbations... as well as purely rotational features such as the description of the rotation set's shape or some bounded deviations properties. The whole thing is based on Le Calvez-Tal forcing theory but I will mainly focus on some examples.

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1. GSTT

Selection Principles in Cosmic Spaces — Davide Giacopello <dagiacopello@unime.it>

We introduce and investigate new selection principles involving countable networks in cosmic tychonoff spaces, namely, M-nw-selective, R-nw-selective, and H-nw-selective. These spaces represent a strengthening of both M-separability, R-separability, and H-separability, as well as the Menger, Rothberger, and Hurewicz properties. We also define and investigate two new games: the R-nw-selective game and the M-nw-selective game, which arise naturally from their corresponding selection principles. We give consistent results, and we define trivial R-, H-, and M-nw-selective spaces the cosmic ones having cardinality and weight strictly less than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$, respectively. We establish that spaces with cardinalities greater than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$ fail to possess the R-, H-, and M-nw-selective properties, respectively. Non-trivial examples, therefore, should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-nw-selective and H-nw-selective spaces. Finally, we study relations between nw-selective properties and a strong version of the HFD property.

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1. TC

Simplicial LS Category bounds for iterated subdivisions of pure simplicial complexes — Manuel Arriaza-Rincón <marriaza@us.es>

The Lusternik-Schnirelmann (LS) category is a fundamental invariant in modern algebraic topology. Its discrete analogue, the simplicial LS category, provides similar topological insights for finite simplicial complexes; however, computing its exact value remains remarkably difficult in most cases. In this talk, we introduce a novel approach to find an upper-bound for the simplicial LS category of pure simplicial complexes by using the point-arboricity of their underlying graphs, and discuss explicit categorical coverings of such spaces based on this approach.

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1. GSTT

Some relations between two topologies on a given set — Athanasios Megaritis <acmegaritis@upatras.gr>

The study of relations between two topologies on the same set is a classical and quite old subject in General Topology with the partial order by inclusion being one of the most essential relations. Various relationships between topologies have been studied (cf. [A], [C], [D], [W]). In [M] we introduced the strongly finer relation "&#x22B4;" between two topologies on a given set X. This relation defines a new order on the family _T_(X) of all topologies on X, which is stronger than the usual subset relation. In this talk, we continue the investigation of the poset (_T_(X),&#x22B4;). In addition, we introduce some new relations between topologies on X. **References**<br> [A] J. M. Aldaz, Uniformly finer topologies, Rend. Circ. Mat. Palermo (2) 45 (1996), no. 3, 453--458.<br> [C] Vitalij A. Chatyrko, On &pi;-compatible topologies and their special cases, Topology Appl. 374 (2025), Paper No. 109243, 10 pp.<br> [D] B. P. Dvalishvili, Bitopological spaces: theory, relations with generalized algebraic structures, and applications, North-Holland Mathematics Studies, 199. Elsevier Science B.V., Amsterdam, 2005.<br> [M] A. C. Megaritis, A new poset of topologies, Mathematica Slovaca (2026), in press.<br> [W] J. D. Weston, On the comparison of topologies, J. London Math. Soc. 32 (1957), 342--354.

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1. MER

Some results about topological groups — Jonathan Cancino-Manríquez <mhacajoh@gmail.com>

It was an old problem of van Dowen the existence of a countably compact topological group without non-trivial convergent sequences, which was finally solved in the positive by Hrusak, van Mill, Ramos-García and Shelah, in 2021. In their paper, besides the aforementioned result, the authors introduced a contruction of a p-compact topological group without non-trivial convergent sequences by means of iterated ultrapowers of the countable boolean group, where p is a selective ultrafilter, and left several open questions related to this contruction. In the present talk we will review some of such questions.

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1. TMAA

Spectra of Beurling Algebras of locally compact abelian groups — Nico Spronk <nspronk@uwaterloo.ca>

Consider a locally compact abelian group. I will construct its universal real topological vector space and demonstrate a bijective correspondence between Gelfand spectra of Beurling algebra and weak*-closed compact convex sets of the dual of this vector space.

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1. GSTT

Star-Proximal Games — Jocelyn Bell <bell@hws.edu>

We introduce star variants of the proximal game in which the uniform structure is replaced by covers of a topological space and Point moves through iterated stars. The cover-star game framework decomposes the proximal game into a hierarchy of topological games that can be studied separately. These games retain several of the preservation and separation features of the proximal game while applying in settings where no uniformity is fixed or assumed.

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1. QTBD

Strongly linear algebra and topological methods for algebraic problems — Vincent Bagayoko <bagayoko@imj-prg.fr>

Strongly linear algebra is an enrichment of linear algebra that allows infinite sums of a formal flavor. I will give some applications of this approach, which can be seen as extensions of topological methods, to the algebra of generalised formal series (such as transseries or multivariate formal series in commuting or non-commuting variables), and operators on these structures. The first application is a formal version of the Lie correspondence that applies to objects that are "formally nilpotent" without being nilpotent or topologically nilpotent. The second (related) application is a general result for treating the problem of normalisation of formal vector fields using asymptotic differential algebra. _This will be based on joint work with Lothar Sebastian Krapp, Salma Kuhlmann, Daniel Panazzolo, Michele Serra, and Vincenzo Mantova._

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1. GGT

Subgroups of Coxeter groups and Stallings Foldings — Jake Murphy Murphy <jmurphy4@oberlin.edu>

Stallings introduced the concept of Stallings foldings to aid in the study of free groups, which creates a "folded graph" associated with a subgroup of a free group. Dani-Levcovitz adapted this concept to the setting of Right-Angled Coxeter Groups. In this talk, we will generalize this idea to certain finitely generated subgroups of Coxeter groups to determine their index, whether they are normal, and to find generating sets of their intersections.

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1. TDS

Tameness, nullness, and amorphic complexity of automatic systems — Maik Gröger <maik.groeger@im.uj.edu.pl>

In the study of low-complexity aperiodic behaviour, tame and null systems arise naturally, yet providing concrete and easily testable conditions to establish their existence in a canonical class of systems is often nontrivial. In the talk I will present a recent result completely characterising tameness and nullness for minimal automatic systems generated by primitive constant-length substitutions in terms of a single numerical invariant: amorphic complexity, a topological invariant tailor-made to study zero-entropy systems with discrete spectrum. We show that for infinite automatic systems, tameness and nullness are equivalent to its value being one.

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1. GSTT

The Average Jones Polynomial: An Ensemble Approach to Knot Shadows via Tensors — Beomgyu Kim <posfn0319@gmail.com>

This talk introduces the Average Jones Polynomial (AJP), defined as the uniform expectation $V_{avg}(S, A) = 2^{-n} \sum_{D \in \mathcal{R}(S)} V_D(A)$, aimed to isolate the structural properties of the underlying 4-valent planar graph. We model the shadow as an uncontracted Temperley-Lieb tensor network, $\mathcal{T}(S) = \prod_{i} (a\mathbf{1} + b e_i)$. This formulation reduces the computational complexity of AJP calculations to $O(n\alpha(n)2^n)$ and maps the AJP to a finite loop-model partition function $Z_S(\delta, a, b)$. This can be utilized to evaluate some macroscopic observables (e.g., the expected number of loops). We analyze the behavior of AJP under shadow Reidemeister (SR) moves. The AJP is invariant under SR1 move, and transforms predictably under SR2 & SR3 moves. When evaluating $\Delta \mathcal{T} = \mathcal{T}(S') - \mathcal{T}(S)$, SR2 and SR3 moves appear as $e_i$ and $(e_i - e_{i+1})$ defect terms, respectively. This suggests a lower bound of required SR moves to transform one shadow into another.

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1. TC

The Convex Matching Distance in Multiparameter Persistence — Sara Scaramuccia <sara.scaramuccia@gmail.com>

In the context of multiparameter persistent homology, we introduce the convex matching distance, a novel metric for comparing multivalued functions. This metric measures the maximal bottleneck distance between the persistence diagrams associated with the convex combinations of the two function components. In the bi-parameter case, similarly to the traditional matching distance, the convex matching distance aggregates the information provided by two real-valued components. However, whereas the matching distance depends on two parameters, the convex matching distance depends on only one, offering improved computational efficiency. We further show that the convex matching distance can be more discriminative than the traditional matching distance in certain cases, although the two metrics are generally not comparable. Moreover, we prove that the convex matching distance is stable and characterize the coefficients of the convex combination at which it is attained. Finally, we demonstrate that this new aggregation framework benefits from the computational advantages provided by the Pareto grid, a collection of curves in the plane whose points lie in the image of the Pareto critical set associated with functions assuming values on the real plane. Experimental validation on MNIST digits, synthetic shapes, and chaotic attractors suggests that the convex matching distance provides a reliable and efficient alternative to the matching distance, at a significantly lower computational cost.

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1. TC

The Core Bifiltration and Multipersistence — Lars Moberg Salbu <lars.salbu@uib.no>

In topological data analysis, one often builds a filtered space from data and uses its persistent homology to describe properties of the data. One-parameter filtrations like the Vietoris-Rips complex or the offset filtration work well in many situations, but they are overly sensitive to outliers. A more robust approach is to add an additional density parameter to the filtration, leading to _multiparameter persistence_. We introduce the _core bifiltration_ given as the union of balls centered at data points in sufficiently dense areas, namely we consider balls that contain at least k data points for some density parameter k. By intersecting the balls with Voronoi cells, we obtain the _Delaunay core bifiltration_, which is smaller and for which we have a computationally efficient implementation for lower dimensions. Both bifiltrations share similar (Prohorov) stability properties.

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1. QTBD

The First Example of a Completely Integrable System with an A _2 Singularity — Gabriela Gutierrez <gabrielajgg4@gmail.com>

Completely integrable systems are Hamiltonian systems with “enough” first integrals and were originally introduced to model the phase spaces of mechanical systems with symmetries. Although the local structure of these systems is well understood, their global structure is far from being understood, particularly in dimensions greater than or equal to six. In this talk, I will give a geometric introduction to completely integrable systems and present a six-dimensional system for which we have established the existence of an A_2 singularity, a type of singularity that had not been observed before in this setting. I will give a complete description of the topology of its singular fibers and explain how Hamiltonian monodromy can be studied for this system. I will conclude with some perspectives arising from this work, which is joint with K. Efstathiou, P. Mardešić, and D. Sugny.

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1. TDS

The Riemann-Hurwitz formula and indecomposable continua — Juliana Xavier <mariajules@gmail.com>

The Riemann-Hurwitz formula establishes a relation between the degree and the number of critical points of branched coverings $f:X\to Y$. This relation involves the Euler characteristic of the spaces $X$ and $Y$. A priori, it has no sense when the spaces are not locally connected.The formula holds for branched coverings between finite cell complexes and also between open connected subsets of the sphere. We use it to define the Euler characteristic of a continuum even if it is not locally connected. For example, $\chi (X)=0$ for a solenoid and $\chi(K)=1/2$, where $K$ is the Knaster continuum. We give applications to dynamics of sphere branched coverings and provide several examples illustrating the interest of the results obtained.

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1. CT

The Shift Map on Mahavier Products — Van Nall <vnall@richmond.edu>

Some interesting continua can be represented in several different ways as a shift invariant subset of the Hilbert cube. In fact, several different representations as Mahavier products can be found for some continua each displaying different dynamical properties. We will review recent results concerning dynamical properties such as transitivity, mixing, shadowing, and specification that are exhibited by the shift map on Mahavier product embeddings of the Cantor fan and the Lelek fan into the Hilbert cube.

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1. CT

The Specification property on cones and suspensions of the Cantor set. — Christopher Mouron <mouronc@rhodes.edu>

In this talk I will show that if a homeomorphism of the cone over the Cantor set (i.e. the Cantor fan) with the specification property exists, it would be complicated to describe. However, an example of a homeomorphism of the suspension over the Cantor set (i.e. the Cantor fan) with the specification property is given and is easy to describe.

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1. TDS
3. Plenary

Topological Criteria for Annular Chaos — Alejandro Passeggi <apasseggi@cmat.edu.uy>

Although paradigmatic models of chaotic dynamics in low-dimensional systems are well understood, proving that a given system exhibits chaotic behavior often remains a challenging task. Moreover, identifying the underlying mechanisms responsible for such dynamics is frequently beyond the scope of the classical literature on the subject. In recent years, several topological criteria have been established for systems whose Poincaré map is defined on the annulus. These criteria provide simple and robust conditions guaranteeing the existence of chaos in the form of a rotational horseshoe. Roughly speaking, it is enough to find two topological disks with different rotation behavior under one iteration and whose forward iterates visit each other. This approach yields rigorous proofs of chaotic dynamics while relying on elementary information about the system [1,2]. Furthermore, effective implementations of these criteria have led to several concrete applications [3,4]. In this talk, I will review these results and discuss recent progress toward a natural next step: obtaining explicit constructions of the rotational horseshoe once the above criteria (or related ones) have been verified. Such constructions not only yield a rigorous computation of the map's topological entropy, but also allow one to locate the rotational horseshoe and its associated essential instability region. [1] A. Passeggi and F. A. Tal, Conditions Implying Annular Chaos, accepted to Inventiones Mathematicae. [2] A. Passeggi and F. Pirán, Annular Chaos for Non-Wandering Homeomorphisms, arXiv. [3] M. J. Capiński, M. Gröger, A. Passeggi and F. A. Tal, Conditions Implying Annular Chaos: Qualitative Results and CAP, arXiv. [4] M. J. Capiński, S. Llavayol and A. Passeggi, Rotational Chaos in the Driven Pendulum (to appear).

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1. TDS

Topologically mildly dissipative homeomorphisms and Wang-Young Strange Attractors — Jan Boronski <jan.boronski@uj.edu.pl>

In this joint work with Sonja Štimac, we extend R.F. Williams' result on 1-dimensional hyperbolic attractors to the non-uniformly hyperbolic setting, by showing that each Wang-Young strange attractor in the plane is conjugate to the shift on the inverse limit of a baobab (Peano continuum that contains at most one Jordan curve), generalizing our earlier result on Hénon attractors. More generally, the result holds on the core of the maximal attractor of any mildly dissipative diffeomorphism (in the sense of Crovisier and Pujals). We also generalize these results to the C⁰ setting, by introducing the class of topologically mildly dissipative surface homeomorphisms, providing a unified approach that covers many classes of dissipative dynamical systems scattered in the literature. Our purely topological conditions lead to a one-to-one correlation between the sets of ergodic measures of the one-dimensional and two-dimensional systems, as well as equality between the corresponding measure-theoretic entropies.

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1. GSTT

Topologies on the ring of Baire-one functions — Atasi Debray <adrpm@caluniv.ac.in>

A Baire class-one (or simply Baire-one) function $f : X \rightarrow \mathbb{R}$ is a function that can be expressed as the pointwise limit of a sequence of continuous functions on a topological space $X$. It is well known that the collection $B_1(X)$ of all Baire-one functions forms an overring of the ring $C(X)$ of continuous functions. Although $B_1(X)$ has resemblances with $C(X)$ in its algebraic behaviour, it exhibits substantial differences, particularly when endowed with topologies analogous to those commonly considered on $C(X)$. The objective of this paper is to discuss $B_1(X)$ from topological perspective and observe the behaviour of $C(X)$ as its subspace.

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1. TC
3. Plenary

Topology in the topos of countable reals — Andrej Bauer <andrej.bauer@andrej.com>

One of the best-known results in mathematics is the uncountability of the real numbers, which Georg Cantor proved by the diagonalization method. His proof relies on the law of excluded middle or the axiom of choice. In joint work with James E. Hanson we showed that this is necessary by constructing a mathematical universe, the [topos of countable reals](https://arxiv.org/abs/2404.01256), in which the Dedekind reals are countable, and consequently both the axiom of choice and the law of excluded middle are invalid. The construction rests on a piece of classical topology, a generalization of Kakutani's fixed-point theorem. In this talk we shall explore how topology behaves in the topos of countable reals. In many respects the reals are still well behaved. They form a Dedekind-complete archimedean ordered field and are connected. The closed interval is totally bounded and Cauchy-complete, although it lacks the stronger Heine–Borel property, as it can be covered by intervals whose lengths sum up to any desired small positive real. Brouwer's fixed-point theorem holds, as a corollary of the countability of the Hilbert cube and Lawvere's fixed-point theorem. Whether every function on the reals is continuous remains open.

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1. TC
3. Plenary

Topology via abstract computation — Alexander Melnikov <alexander.g.melnikov@gmail.com>

My talk will cover a broad range of topics highlighting interactions between abstract Turing computability and classification problems in topology. The subject is wide-ranging and can be roughly divided into two interconnected directions: (1) computable (“constructive”) aspects of topology, and (2) applications of formal models of computation to problems in topology that may initially appear unrelated to computation. As will be seen, these two directions are closely linked at a technical level, and no clear dividing line can be drawn between them.

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1. QTBD

Trajectories of vector fields asymptotic to formal invariant curves — Fernando Sanz Sánchez <fsanz@uva.es>

In this talk we present the following result obtained jointly with O. Le Gal, that generalizes to any dimension a result by F. Dumortier and P. Bonckaert in 1986 concerning the realizability of formal invariant curves of three-dimensional vector fields: Let $\xi$ be a $C^{\infty}$ vector field at $(\mathbb R^{n},0)$ and suppose that it has a formal invariant curve $\Gamma$. Then, as soon as the Taylor expansion of $\xi$ is not identically zero along $\Gamma$ (a necessary condition), there is a trajectory $\gamma\subset \mathbb R^{n}$ of $\xi$ which has $\Gamma$ as an asymptotic expansion at the origin. In fact, we realize the family of all trajectories which are asymptotic to $\Gamma$: we construct an invariant $C^0$ manifold in some open horn around $\Gamma$, entirely composed of asymptotic trajectories, and containing the germ of any such trajectory. Furthermore, if $\xi$ is analytic, we prove that there exists a trajectory $\gamma$ asymptotic to $\Gamma$ which is, moreover, non-oscillating with respect to subanalytic sets.

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1. CT

Ultrafilter orders on chainable continua — Julia Ścisłowska <j.scislowska@uw.edu.pl>

My talk will be devoted to discuss families of ultrafilter orders on a given chainable continuum X (such as e.g. arc, the Warsaw sine curve, the Knaster continuum etc.). These orders depend on a fixed sequence of chains, covering X (obtained from chainability of X), and on a fixed nonprincipal ultrafilter on N. Alternatively ultrafilter orders may be defined using representation of X as an inverse limit of a sequence of arcs and a fixed nonprincipal ultrafilter on N. During the talk I will present some known results in this topic. In particular, I will mention some ideas how we can express the “level of complexity” of a given chainable continuum in the language of ultrafilter orders. This is a joint work with Witold Marciszewski and Benjamin Vejnar, preprint is available at: https://arxiv.org/abs/2510.14577.

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1. GGT

Unipotents and linearity of amalgams — Sami Douba <doubasami@gmail.com>

I will discuss joint work with Konstantinos Tsouvalas where we investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain “doubles” of linear groups, and obtain new examples of finitely generated residually finite groups that fail to be linear.

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1. TDS

Universal bounds on the entropy of toroidal attractors — Jaime Jorge Sánchez-Gabites <jajsanch@ucm.es>

A compact set $K \subseteq \mathbb{R}^3$ is called \emph{toroidal} if it has a neighbourhood basis of solid tori. This is a natural generalization of the well-known notion of a cellular set. To any toroidal set one can assign a finite set of prime numbers called its prime divisors. These reflect purely topological properties of $K$. Suppose $f$ is a diffeomorphism of $\mathbb{R}^3$ and $K$ is an attractor for $f$ which happens to be a toroidal set (solenoids are the canonical example of this). We prove the following: the entropy of $f$ on $K$ is bounded below by ${\rm log}(p_1 \cdot \ldots \cdot p_n)$, where the $p_i$ are the prime divisors of $K$. Since the latter depend only on $K$, this provides a universal lower bound on any $\mathcal{C}^{\infty}$ attracting dynamics on $K$. In the talk we will discuss the geometric techniques used to prove this and discuss its (plausible?) validity when $f$ is just a homeomorphism.

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1. GSTT

Universal minimal flows of totally disconnected locally compact Polish groups — Dana Bartosova <dbartosova@ufl.edu>

Every topological group $G$ admits a universal minimal flow, $M(G),$ that is, a minimal flow that factors onto any minimal flow, that is unique up to isomorphism. Thus understanding $M(G)$ sheds light on how complicated minimal dynamics of $G$ can be. In the case of infinitely countable discrete groups, the underlying space of $M(G)$ is always the Gleason space of the Cantor cube of weight continuum, $\text{Gl}(2^{\mathfrak{c}})$, that can also be seen as the Stone space of the free completion of the free Boolean algebra on continuum many generators. Totally disconnected locally compact (TDLC) Polish non-discrete groups, in other words locally compact automorphism groups of countable structures, are topologically homeomorphic to the product of a countable discrete set and the Cantor space. In case a group $G$ is also algebraically isomorphic to a product of an infinitely countable discrete group and the Cantor group, $D\times 2^{\omega}$, then the underlying space of $M(G)$ is homeomorphic to the product $\text{Gl}(2^{\mathfrak{c}})\times 2^{\omega}$. The question is whether that is always the case. We show that the answer is positive in various scenarios, covering for instance the example of automorphism groups of finitely-branching regular countable tree. This is a joint work in progress with Andy Zucker.

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1. GSTT

Universality theorems for mappings — DIMITRIOS GEORGIOU <georgiou@math.upatras.gr>

In this talk, we study the universality problem for the existence of universal elements in classes of continuous mappings. Especially, we present: classical results regarding universal continuous mappings and the existence of universal elements in the class of all continuous mappings from a normal space of which covering dimension is not larger than n to a fixed compact Hausdorff space Y.

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1. GGT

Vanishing of bounded cohomology beyond amenability — Caterina Campagnolo <caterina.campagnolo@uam.es>

Bounded cohomology is an invariant of groups and spaces developed by Gromov in the 80's. Despite its purely topological definition, it turns out to have deep relations with geometric properties of the spaces or algebraic properties of the groups under consideration. In particular, its vanishing for a large family of coefficients modules allows to characterize amenability. In joint work with Fournier-Facio, Lodha and Moraschini, we develop a new criterion for the vanishing of bounded cohomology for a subfamily of coefficients modules and apply it to a variety of examples of groups of topological, geometric and dynamical origin. We also remark an interesting relationship with homological stability of these families of groups.

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1. TDS

Zero Entropy Locus of Lozi Maps Revisited — Kristijan Kilassa Kvaternik <kkkvaternik@fer.hr>

We consider orientation-reversing Lozi maps $L_{a,b}$ in a parameter region $\mathfrak{R}$ where there are no homoclinic points for the fixed point $X$ in the first quadrant, and the period-two cycle $\{P,P'\}$ is attracting. We first analyze the boundary of that parameter region: we show that all homoclinic points for $X$ on that boundary are tangential or there is a segment of homoclinic points, and we classify them. Moreover, we study the topological entropy $h_{top}$ of $L_{a,b}$ when $(a,b)\in\mathfrak{R}$. Consider the set $\ell$ of accumulation points of the unstable manifold of $X$. Misiurewicz and Štimac have recently proven that $h_{top}(L_{a,b})=0$ in a certain open subset of $\mathfrak{R}$; in that case, $\ell=\{P,P'\}$. We extend this result by showing that the $L_{a,b}$, restricted to the complement of $\ell$ in the plane, has zero entropy. Finally, we discuss that $\ell=\{P,P'\}$ does not hold in general for all parameters in $\mathfrak{R}$.

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