Accepted (13):
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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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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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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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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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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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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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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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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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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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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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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- 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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- 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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