Tootfinder

Totally Disconnected (non-metric) Gelfand Duality
Sebasti\'an Rodr\'iguez, Xavier Caicedo
https://arxiv.org/abs/2508.11188 https://arxiv.org/pdf/25…

Totally Disconnected (non-metric) Gelfand Duality
We characterize those algebras over a disconnected uniformly complete topological field which are represenable as algebras of continuous functions on compact topologicla spaces, generalizing thus Gelfand duality for non-archimedean normed fields (Van der Put theorem). More generally, we establish for any topological field F a (dual) adjunction between the category of compact F-Tychonoff spaces and a natural category of commutative F-algebras, which becomes a duality for fields satisfying the St…

Topological Data Analysis in Finsler Spaces
Rafael Cavalcanti, Nelson Leal, Serafim Rodrigues, Danilo Barros, Mathieu Desroches
https://arxiv.org/abs/2509.12660 https://

Topological Data Analysis in Finsler Spaces
We introduce a new approach to Topological Data Analysis (TDA) based on Finsler metrics and we also generalize the classical concepts of Vietoris-Rips and Cech complexes within this framework. In particular, we propose a class of directionally dependent Finsler metrics and establish key results demonstrating their relevance to TDA. Moreover, we show that several Information Theoretic perspectives on TDA can often be recovered through the lens of Finsler geometry.

Motivic Mirror Symmetry for Character Stacks
Lucas de Amorin
https://arxiv.org/abs/2508.12995 https://arxiv.org/pdf/2508.12995

Motivic Mirror Symmetry for Character Stacks
We propose a motivic version of T. Hausel and M. Thaddeus' Topological Mirror Symmetry for character stacks associated with arbitrary semisimple groups, which is an analogue of F. Loeser and D. Wyss' result for Chow motives of moduli spaces of Higgs bundles. As first steps towards it, we generalize A. Mellit's cell decomposition to arbitrary connected and reductive groups. We use it to describe all automorphisms on the associated character stacks. Then we show that the Weyl pairing induces a du…

Relation-Theoretic Banach Contraction Principle in Topological Spaces with an Application
Md Hasanuzzaman, Abhishikta Das, Sumit Som
https://arxiv.org/abs/2509.10497 https://

Relation-Theoretic Banach Contraction Principle in Topological Spaces with an Application
In this article, we extend several relation-theoretic notions to topological spaces. We introduce relation preserving contraction mapping into topological spaces and utilize the same to extend Banach contraction principle in topological spaces employing a binary relation. To illustrate the validity of our main result, we provide a concrete example along with a MATLAB-based visualization of the convergence behavior. Furthermore, we demonstrated the applicability of our main result by finding a s…

Toposes with enough points as categories of \'etale spaces
Sam van Gool, J\'er\'emie Marqu\`es, Umberto Tarantino
https://arxiv.org/abs/2508.09604 https://

Toposes with enough points as categories of étale spaces
We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be seen as an extension to ionads of Barr's equivalence between topological spaces and relational modules for the ultrafilter monad. In view of the correspondence between toposes and geometric theories, we obtain a strong conceptual completeness theorem, in the sen…

Some dynamical properties of weighted shifts on sequence spaces
Michal Hevessy, Tom\'a\v{s} Raunig
https://arxiv.org/abs/2509.11852 https://arxiv.org/p…

Some dynamical properties of weighted shifts on sequence spaces
Motivated by three recent open questions in the study of linear dynamics, we study weighted shifts on sequence spaces. First, we provide an~example of a~weighted shift on a~locally convex space whose topology is generated by a~sequence of complete seminorms which is generalized hyperbolic, but does not have the shadowing property. Next, we characterise uniform topological expansivity on Fréchet spaces satisfying some very natural conditions. Finally, we study the periodic shadowing property on…

The Realization of 3D Topological Spaces Branched Over Graphs
Christopher L Duston
https://arxiv.org/abs/2508.09026 https://arxiv.org/pdf/2508.09026…

The Realization of 3D Topological Spaces Branched Over Graphs
In this paper we present an implementation of a computer algorithm that automatically determines the topological structure of spacetime, using a branched covering space representation. This algorithm is applied to a few simple examples in dimension 3, and a complete set of the topological spaces branched over several graphs are found. We also include some new visualizations of the branched covering construction, in order to aid and clarify the understanding of how these structures can be used i…

Configuration spaces and peak representations
Marcelo Aguiar, Sarah Brauner, Victor Reiner
https://arxiv.org/abs/2508.09898 https://arxiv.org/pdf/2508.0989…

Configuration spaces and peak representations
Within the group algebras of the symmetric and hyperoctahedral groups, one has their descent algebras and families of Eulerian idempotents. These idempotents are known to generate group representations with topological interpretations, as the cohomology of configuration spaces of types A and B. We provide an analogous cohomological interpretation for the representations generated by idempotents in the peak algebra, called the peak representations. We describe the peak representations as sums of…

Isometry groups of Polish ultrametric spaces
Riccardo Camerlo, Alberto Marcone, Luca Motto Ros
https://arxiv.org/abs/2508.08480 https://arxiv.org/pdf/2508.…

Isometry groups of Polish ultrametric spaces
We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized wreath products of full permutation groups. Since our solution is developed in the finer context of topological (Polish) groups, it also solves a problem of Gao and Kechris from 2003. Furthermore, we provide an exact correspondence between the isometry groups …

Topological Sequence Analysis of Genomes: Delta Complex approaches
Jian Liu, Li Shen, Dong Chen, Guo-Wei Wei
https://arxiv.org/abs/2507.05452 https://

Topological Sequence Analysis of Genomes: Delta Complex approaches
Algebraic topology has been widely applied to point cloud data to capture geometric shapes and topological structures. However, its application to genome sequence analysis remains rare. In this work, we propose topological sequence analysis (TSA) techniques by constructing $Δ$-complexes and classifying spaces, leading to persistent homology, and persistent path homology on genome sequences. We also develop $Δ$-complex-based persistent Laplacians to facilitate the topological spectral analysis…

Supersymmetric Topological Sigma Models and Doubling Spaces
Daniel M. Halmrast
https://arxiv.org/abs/2507.08181 https://arxiv.org/pdf…

Supersymmetric Topological Sigma Models and Doubling Spaces
Witten's topological B-model on a Calabi-Yau background is known to reproduce, in the open string sector, the derived category of coherent sheaves. When the target space is a complex torus, the topological model enjoys a non-geometric symmetry known as T-duality, which relates the theories on the torus and dual torus backgrounds. By considering the "double field theory" of Hull and Reid-Edwards on the product of a torus and its dual, T-duality occurs as a geometric symmetry of the target space.…

Antiholomorphic correspondences and mating II: Shabat polynomial slices
Mikhail Lyubich, Jacob Mazor, Sabyasachi Mukherjee
https://arxiv.org/abs/2509.12357 https://

Antiholomorphic correspondences and mating II: Shabat polynomial slices
We study natural one-parameter families of antiholomorphic correspondences arising from univalent restrictions of Shabat polynomials, indexed by rooted dessin d'enfants. We prove that the parameter spaces are topological quadrilaterals, giving a partial description of the univalency loci for the uniformizing Shabat polynomials. We show that the escape loci of our parameter spaces are naturally (real-analytically) uniformized by disks. We proceed with designing a puzzle structure (dual to the in…

Moduli space of spin connections on three-dimensional homogeneous spaces
Matteo Bruno, Gabriele Peluso
https://arxiv.org/abs/2507.06104 https://

Moduli space of spin connections on three-dimensional homogeneous spaces
In this manuscript, we aim to classify and characterize the moduli space of homogeneous spin connections and homogeneous $SU(2)$ connections on three-dimensional Riemannian homogeneous spaces. An analysis of the topology of the associated moduli spaces reveals that they are finite-dimensional topological manifolds (possibly with boundary) possessing trivial homotopy groups. Owing to their deep connection with cosmological models in the Ashtekar-Barbero-Immirzi formulation of General Relativit…

Typed Topological Structures Of Datasets
Wanjun Hu
https://arxiv.org/abs/2508.14008 https://arxiv.org/pdf/2508.14008

Typed Topological Structures Of Datasets
A datatset $X$ on $R^2$ is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method \cite{carlsson}. In \cite{hu}, the concept of typed topological space was introduced and showed to have the potential for studying finite topological spaces, such as a dataset. It is a new method from the general topology perspective. A typed topological space is a topological space whose open sets are assigned types. Topological concepts and m…

A Preliminary Study on the Dimensional Stability Classification of Polynomial Spline Spaces over T-meshes
Bingru Huang, Falai Chen
https://arxiv.org/abs/2508.06217 https://

A Preliminary Study on the Dimensional Stability Classification of Polynomial Spline Spaces over T-meshes
This paper introduces the concept of dimensional stability for spline spaces over T-meshes, providing the first mathematical definition and a preliminary classification framework. We define dimensional stability as an invariant within the structurally isomorphic class, contingent on the rank stability of the conformality matrix. Absolute stability is proposed via structurally similar maps to address topological and order structures. Through the $k$-partition decomposition of T-connected compone…

Some Separation Axioms in Topological Spaces
Neeraj Kumar Tomar, M. C. Sharma, Amit Ujlayan, Fahed Zulfeqarr
https://arxiv.org/abs/2507.06843 https://

Some Separation Axioms in Topological Spaces
In this paper, we introduced the concepts of new separation axioms called $ SC^* $-separation axioms and $ H^* $-separation axioms by using $ SC^* $ and $ H^* $-open sets in topological spaces. The $ SC^* $-separation axioms include $ SC^* $-$ C_0 $, $ SC^* $-$ C_1 $, weakly $ SC^* $-$ C_0 $, and weakly $ SC^* $-$ C_1 $ spaces, while the $ H^* $-separation axioms include $ H^* $-$ T_{\frac{1}{2}} $, $ H^* $-$ T_b $, and $ H^* $-$ T_d $ spaces. Also, we obtained several properties of such spaces.

Topological components of surface group representations and the signature
Inkang Kim, Xueyuan Wan
https://arxiv.org/abs/2508.05000 https://arxiv.org/pdf/25…

Topological components of surface group representations and the signature
We study the topological components of the surface group representations into $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{PSL}(2,\mathbb{R})$. Utilizing the signature formula established in [10], we determine the number of connected components of the representation spaces with boundary elliptic, hyperbolic, and parabolic holonomies.

Matisse: Visualizing Measured Internet Latencies as Manifolds
Stephen Jasina, Loqman Salamatian, Joshua Mathews, Scott Anderson, Paul Barford, Mark Crovella, Walter Willinger
https://arxiv.org/abs/2509.08097

Matisse: Visualizing Measured Internet Latencies as Manifolds
Manifolds are complex topological spaces that can be used to represent datasets of real-world measurements. Visualizing such manifolds can help with illustrating their topological characteristics (e.g., curvature) and providing insights into important properties of the underlying data (e.g., anomalies in the measurements). In this paper, we describe a new methodology and system for generating and visualizing manifolds that are inferred from actual Internet latency measurements between different…

Modular functors from conformal blocks of rational vertex operator algebras
Chiara Damiolini, Lukas Woike
https://arxiv.org/abs/2507.05845 https://

Modular functors from conformal blocks of rational vertex operator algebras
For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel--Ben-Zvi generalized by Damiolini--Gibney--Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form vector bundles over a suitable moduli space of algebraic curves. In this article, we establish, under the same assumptions, the widely expected topological result that the spaces of conformal blocks produce a modular functor, i.e. a modular algebra over an …

A linear topological invariant for weighted spaces of holomorphic functions
Andreas Debrouwere, Quinten Van Boxstael
https://arxiv.org/abs/2506.23695 https…

A linear topological invariant for weighted spaces of holomorphic functions
We study the linear topological invariant $(Ω)$ for a class of Fréchet spaces of holomorphic functions of rapid decay on strip-like domains in the complex plane, defined via weight function systems. We obtain a complete characterization of the property $(Ω)$ for such spaces in terms of an explicit condition on the defining weight function systems. As an application, we investigate the surjectivity of the Cauchy-Riemann operator on certain weighted spaces of vector-valued smooth functions.

On distributional one-category, diagonal distributional complexity, and related invariants
Ekansh Jauhari, John Oprea
https://arxiv.org/abs/2508.06973 https://

On distributional one-category, diagonal distributional complexity, and related invariants
We develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants, these invariants are rigid on spaces with finite fundamental group. On Eilenberg-Mac Lane spaces, we identify these new invariants with distributional category and complexity, respectively, and use them to illuminate aspects of the behavior of the latter invari…

Dual spaces of lattices and semidistributive lattices
Andrew Craig, Miroslav Haviar, Jose Sao Joao
https://arxiv.org/abs/2509.04417 https://arxiv.org/pdf/2…

Dual spaces of lattices and semidistributive lattices
Birkhoff's 1937 dual representation of finite distributive lattices via finite posets was in 1970 extended to a dual representation of arbitrary distributive lattices via compact totally order-disconnected topological spaces by Priestley. This result enabled the development of natural duality theory in the 1980s by Davey and Werner, later on also in collaboration with Clark and Priestley. In 1978 Urquhart extended Priestley's representation to general lattices via compact doubly quasi-ordered…

Nonlinear Symmetry-Fragmentation of Nonabelian Anyons In Symmetry-Enriched Topological Phases: A String-Net Model Realization
Nianrui Fu, Siyuan Wang, Yu Zhao, Yidun Wan
https://arxiv.org/abs/2506.24115

Nonlinear Symmetry-Fragmentation of Nonabelian Anyons In Symmetry-Enriched Topological Phases: A String-Net Model Realization
Symmetry-enriched topological (SET) phases combine intrinsic topological order with global symmetries, giving rise to novel symmetry phenomena. While SET phases with Abelian anyons are relatively well understood, those involving non-Abelian anyons remain elusive. This obscurity stems from the multi-dimensional internal gauge spaces intrinsic to non-Abelian anyons -- a feature first made explicit in [1,2] and further explored and formalized in our recent works [3-8]. These internal spaces can tr…

Dynamically encircling an exceptional point through phase-tracked closed-loop control
Sen Zhang, Yangyu Huang, Lei Yu, Kaixuan He, Ning Zhou, Dingbang Xiao, Xuezhong Wu, Franco Nori, Hui Jing, Xin Zhou
https://arxiv.org/abs/2509.04940

Dynamically encircling an exceptional point through phase-tracked closed-loop control
The intricate complex eigenvalues of non-Hermitian Hamiltonians manifest as Riemann surfaces in control parameter spaces. At the exceptional points (EPs), the degeneracy of both eigenvalues and eigenvectors introduces noteworthy topological features, particularly during the encirclement of the EPs. Traditional methods for probing the state information on the Riemann surfaces involve static measurements; however, realizing continuous encircling remains a formidable challenge due to non-adiabatic…

G-kernels and Crossed Modules
Sergio Gir\'on Pacheco, Masaki Izumi, Ulrich Pennig
https://arxiv.org/abs/2509.04134 https://arxiv.org/pdf/2509.04134

G-kernels and Crossed Modules
We develop a unified framework based on topological crossed modules for various lifting obstructions for $Γ$-kernels. It allows us to identify actions, cocycle actions and $Γ$-kernels up to their natural equivalence relations with cohomology sets. The obstructions then appear as boundary maps in corresponding exact sequences. Since topological crossed modules are topological $2$-groups (in the categorical sense), they have classifying spaces, which come with a natural transformation from the …

Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles
Elsa Maneval
https://arxiv.org/abs/2507.02588 https://

Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles
The Hausel-Thaddeus conjectures concern topological mirror symmetry between moduli spaces of $SL_n$ and $PGL_n$ Higgs bundles on a curve. A non-archimedean approach was introduced by Groechenig, Wyss and Ziegler, proving the conjecture for coprime rank and degree. This article is concerned with its generalisation to the non-coprime case. We treat both the classical ($D=K$) and meromorphic ($D>K$) settings. We prove an equality of $p$-adic volumes twisted by gerbes between moduli spaces of $SL_n…

Nil-Equivariant Tropological Sigma Models on Filtered Geometries
Emil Albrychiewicz, Andr\'es Franco Valiente, Christopher Stites
https://arxiv.org/abs/2507.23072 https://…

Nil-Equivariant Tropological Sigma Models on Filtered Geometries
We investigate the behavior of tropological (tropical topological) sigma models on higher dimensional target spaces and show that higher dimensional spaces explicitly admit nested Maslov dequantizations which lead to nontrivial anisotropic filtration structures. We provide a classification of all inequivalent tropological sigma models that can be constructed for the case of 4D targets and show that, generically, the corresponding sigma-models are not defined on foliated geometries like in the 2…

Nonautonomous Dynamical Systems III: Symbolic and Expansive Systems
Zhuo Chen, Jun Jie Miao
https://arxiv.org/abs/2509.11130 https://arxiv.org/pdf/2509.111…

Nonautonomous Dynamical Systems III: Symbolic and Expansive Systems
A nonautonomous dynamical system $(\boldsymbol{X},\boldsymbol{T})=\{(X_{k},T_{k})\}_{k=0}^{\infty}$ is a sequence of continuous mappings $T_{k}:X_{k} \to X_{k+1}$ along a sequence of compact metric spaces $X_{k}$. In this paper, we study the nonautonomous symbolic systems $(\boldsymbolΣ(\boldsymbol{m}),\boldsymbolσ)$ and nonautonomous expansive dynamical systems. We prove homogeneous properties and provide the formulae for topological pressures $\underline{P},\overline{P},P^{\mathrm{B}},P^{\m…

Generalised ultracategories and conceptual completeness of geometric logic
Ali Hamad
https://arxiv.org/abs/2507.07922 https://arxiv.o…

Generalised ultracategories and conceptual completeness of geometric logic
We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in the theory of generalised ultracategories. Another example of these generalised ultracategories is points of toposes. In this paper, we show a conceptual completeness theorem for toposes with enough points, stating that any such topos can be reconstructed fr…

A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images
Erin W. Chambers, Guangyu Meng
https://arxiv.org/abs/2507.01171 …

A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images
Reeb graphs are a fundamental structure for analyzing the topological and geometric properties of scalar fields. Comparing Reeb graphs is crucial for advancing research in this domain, yet existing metrics are often computationally prohibitive or fail to capture essential topological features effectively. In this paper, we explore the application of the Gromov-Wasserstein distance, a versatile metric for comparing metric measure spaces, to Reeb graphs. We propose a framework integrating a symme…

Isomorphism of almost locally compact Polish metric structures
Maciej Malicki
https://arxiv.org/abs/2508.07765 https://arxiv.org/pdf/2508.07765

Isomorphism of almost locally compact Polish metric structures
A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always classifiable by countable structures. This allows to remove a gap in the proof presented in \cite{Ma} of a positive answer to a question of Gao and Kechris, who asked whether isometry of locally compact Polish metric spaces is classifiable by countable structure…

On numerical invariants of retraction map
Nursultan Kuanyshov
https://arxiv.org/abs/2509.05745 https://arxiv.org/pdf/2509.05745

On numerical invariants of retraction map
We introduce a notion of retraction between continuous maps of topological spaces and study the behavior of several numerical invariants under such retractions. These include (co)homological dimensions, the Lusternik-Schnirelmann category, the topological complexity, and the sequential topological complexity. We prove that, under the retraction map, the corresponding inequalities between invariants hold. Our results also apply to recent invariants are defined by Dranishnikov, Jauhari \cite{DJ} …

On the Context of Sequences of Intervals: Nonlinear Normed Complete Spaces of Uncertain Variables
Mehmet \c{S}eng\"on\"ul
https://arxiv.org/abs/2509.07189 https://

On the Context of Sequences of Intervals: Nonlinear Normed Complete Spaces of Uncertain Variables
In this paper it were investigated the algebraic and topological properties of the space \mathscr{C}_f, which consists of convergent sequences of uncertain variable intervals. It was established that \mathscr{C}_f is a normed space with a well-defined norm.

A groupoid approach to the study of fuzzy topological spaces
Anjeza Krakulli, Elton Pasku
https://arxiv.org/abs/2508.16549 https://arxiv.org/pdf/2508.16549…

A groupoid approach to the study of fuzzy topological spaces
The definition of the complement of a fuzzy subset is algebraic in nature and when it is used in the context of fuzzy topological spaces it does not share any similarity with the usual property of topological spaces that the complement of an open subset is closed. To tackle this inconsistency, we associate to any fuzzy topological space a topological space and use its fundamental groupoid equipped with the Lasso topology to give a topological characterization for the complementation of fuzzy su…

An extension of Priestley duality to fuzzy topologies and positive MV-algebras
Marby Zuley Bola\~nos Ortiz, Ciro Russo
https://arxiv.org/abs/2508.19423 https://

An extension of Priestley duality to fuzzy topologies and positive MV-algebras
We extend Priestley Duality to suitable categories of fuzzy topological spaces and ordered algebraic structures that generalize bounded distributive lattices. The duality we prove extends not only classical Priestley Duality between Priestley Spaces and bounded distributive lattices, but also the duality between limit cut complete MV-algebras and Stone MV-topological spaces (proved by the second author in a previous paper) which, on its turn, is an extension of classical Stone Duality.

Branching space of multipointed d-space
Philippe Gaucher
https://arxiv.org/abs/2507.08377 https://arxiv.org/pdf/2507.08377

Branching space of multipointed d-space
Using the notion of short directed path, we introduce the branching space of a multipointed $d$-space. We prove that for any q-cofibrant multipointed $d$-space, it is homeomorphic to the branching space of the q-cofibrant flow obtained by applying the categorization functor. As an application, we deduce a purely topological proof of the invariance of the branching space and of the branching homology of cellular multipointed $d$-spaces up to globular subdivision. By reversing the time direction,…

Replaced article(s) found for math.OA. https://arxiv.org/list/math.OA/new
[1/1]:
- Involutive quantales and quantale-enriched involutive topological spaces
Javier Guti\'errez Garc\'ia, Ulrich H\"ohle

Computability of Separation Axioms in Countable Second Countable Spaces
Andrew DeLapo, David Gonzalez
https://arxiv.org/abs/2507.18564 https://arxiv.org/pd…

Computability of Separation Axioms in Countable Second Countable Spaces
We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic completeness result for each of the Tychonoff separation axioms. Beyond this, we prove completeness results for various other topological properties, such as being Polish and having a particular Cantor-Bendixson rank, using tools from computable structure theory. Thi…

Nonautonomous Dynamical Systems I: Topological Pressures and Entropies
Zhuo Chen, Jun Jie Miao
https://arxiv.org/abs/2508.01363 https://arxiv.org/pdf/2508.…

Nonautonomous Dynamical Systems I: Topological Pressures and Entropies
Let $\boldsymbol{X}=\{X_{k}\}_{k=0}^{\infty}$ be a sequence of compact metric spaces $X_{k}$ and $\boldsymbol{T}=\{T_{k}\}_{k=0}^{\infty}$ a sequence of continuous mappings $T_{k}:X_{k} \to X_{k+1}$. The pair $(\boldsymbol{X},\boldsymbol{T})$ is called a nonautonomous dynamical system. Our main object is to study the variational principles of topological pressures and entropies on nonautonomous dynamical systems. In this paper, we introduce a variety of topological pressures ($\underline{Q}$,$\…

Empowering Graph-based Approximate Nearest Neighbor Search with Adaptive Awareness Capabilities
Jiancheng Ruan, Tingyang Chen, Renchi Yang, Xiangyu Ke, Yunjun Gao
https://arxiv.org/abs/2506.15986

Empowering Graph-based Approximate Nearest Neighbor Search with Adaptive Awareness Capabilities
Approximate Nearest Neighbor Search (ANNS) in high-dimensional spaces finds extensive applications in databases, information retrieval, recommender systems, etc. While graph-based methods have emerged as the leading solution for ANNS due to their superior query performance, they still face several challenges, such as struggling with local optima and redundant computations. These issues arise because existing methods (i) fail to fully exploit the topological information underlying the proximity …

Bootstrapping periodic quantum systems
Zhijian Huang, Wenliang Li
https://arxiv.org/abs/2507.02386 https://arxiv.org/pdf/2507.02386…

Bootstrapping periodic quantum systems
Periodic structures are ubiquitous in quantum many-body systems and quantum field theories, ranging from lattice models, compact spaces, to topological phenomena. However, previous bootstrap studies encountered technical challenges even for one-body periodic problems, such as a failure in determining the accurate dispersion relations for Bloch bands. In this work, we develop a new bootstrap procedure to resolve these issues, which does not make use of positivity constraints. We mainly consider …

Isoperiodic meromorphic forms with at least three simple poles
Liza Arzhakova, Gabriel Calsamiglia, Bertrand Deroin
https://arxiv.org/abs/2508.00099 https://

Isoperiodic meromorphic forms with at least three simple poles
In this paper we prove the connectedness of isoperiodic moduli spaces of meromorphic differentials with at least three simple poles on homologically marked smooth curves whose periods are either not contained in a real line, or not contained in the rational space generated by the peripheral periods. From this topological property we deduce dynamical properties of the underlying foliation in the moduli space meromorphic differentials, by describing leaf closures associated to those spaces.

A Completeness Theorem for Topological Doctrines
Silvio Ghilardi, J\'er\'emie Marqu\`es
https://arxiv.org/abs/2507.20768 https://arxiv.org/pdf/2507…

A Completeness Theorem for Topological Doctrines
We extend logical categories with fiberwise interior and closure operators so as to obtain an embedding theorem into powers of the category of topological spaces. The required axioms, besides the Kuratowski closure axioms, are a `product independence' and a `loop contraction' principle.

An Intrinsic Barrier for Resolving P = NP (2-SAT as Flat, 3-SAT as High-Dimensional Void-Rich)
M. Alasli
https://arxiv.org/abs/2508.13200 https://arxiv.org…

An Intrinsic Barrier for Resolving P = NP (2-SAT as Flat, 3-SAT as High-Dimensional Void-Rich)
We present a topological barrier to efficient computation, revealed by comparing the geometry of 2 SAT and 3 SAT solution spaces. Viewing the set of satisfying assignments as a cubical complex within the Boolean hypercube, we prove that every 2 SAT instance has a contractible solution space, topologically flat, with all higher Betti numbers bk equals 0 for k greater than or equal 1, while both random and explicit 3 SAT families can exhibit exponential second Betti numbers, corresponding to expo…

Matrix Invariants as Homotopy Invariants in Finite $T_0$-spaces
Pedro J. Chocano
https://arxiv.org/abs/2506.16861 https://arxiv.org/p…

Matrix Invariants as Homotopy Invariants in Finite $T_0$-spaces
We establish a bijection between the set of finite topological $T_0$-spaces (or partially ordered sets) and classes of square matrices. The absolute value of the determinant or the rank of these matrices are simple homotopy invariants for the corresponding topological spaces, and consequently, for finite simplicial complexes. We calculate these invariants for various families of topological spaces and explore other relationships and problems posed for finite posets within the context of these m…

$\mathbb R^{\omega_1}$-Factorizable Spaces and Groups
Anton Lipin, Evgenii Reznichenko, Ol'ga Sipacheva
https://arxiv.org/abs/2509.05105 https://arxiv.…

$\mathbb R^{ω_1}$-Factorizable Spaces and Groups
A topological space $X$ is $\mathbb R^{ω_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{ω_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{ω_1}$-factorizable if and only if $X\times D(ω_1)$, where $D(ω_1)$ is a discrete space of cardinality $ω_1$, is $z$-embedded in the product $βX\times βD(ω_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{ω_1}$-facto…

On Borel subsets of generalized Baire spaces
Tapani Hyttinen, Miguel Moreno, Jouko V\"a\"an\"anen
https://arxiv.org/abs/2507.15427 https://

On Borel subsets of generalized Baire spaces
We develop descriptive set theory in Generalized Baire Spaces without assuming $κ^{<κ}=κ$. We point out that without this assumption the basic topological concepts of Generalized Baire Spaces have to be slightly modified in order to obtain a meaningful theory. This modification has no effect if $κ^{<κ}=κ$. After developing the basic theory we apply it to the question whether the orbits of models of a fixed cardinality $κ$ in the space $2^κ$ (or $κ^κ$) are $κ$-Borel in our generalized…

Persistence and Topological Complexity
Facundo M\'emoli, Ling Zhou
https://arxiv.org/abs/2506.17888 https://arxiv.org/pdf/2506.17…

Persistence and Topological Complexity
Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris-Rips filtrations of compact metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice th…

Norms and Non-Equivalence in Infinite-Dimensional Banach Spaces
Renan J. S. Isneri, Josias V. Baca, Lucas M. Fernandes
https://arxiv.org/abs/2508.15686 https://

Norms and Non-Equivalence in Infinite-Dimensional Banach Spaces
This work explores the interaction between different norms in infinite-dimensional vector spaces, focusing on their impact on Banach space structures and topological properties. We examine norms induced by bijective linear maps, the existence of non-equivalent norms in Banach spaces, and the role of Hamel bases in normed spaces.

On ball expanding maps
Noriaki Kawaguchi
https://arxiv.org/abs/2507.04744 https://arxiv.org/pdf/2507.04744

On ball expanding maps
We study the dynamical properties of ball expanding maps, a class of continuous self-maps defined on compact metric spaces. For a ball expanding map, we show that: (1) the set of periodic points is dense in the chain recurrent set; (2) if the topological entropy of the map is zero, then the chain recurrent set is finite; (3) the map has only finitely many chain components; (4) if the space is perfect, then the topological entropy of the map is positive; and (5) if the space is connected, then t…

2-dimensional TFTs via modular $\infty$-operads
Jan Steinebrunner
https://arxiv.org/abs/2506.22104 https://arxiv.org/pdf/2506.22104…

2-dimensional TFTs via modular $\infty$-operads
This lecture series is based on joint work in progress with Shaul Barkan, as well as work in progress of the author. The five sections of these notes correspond to the five lectures, but more details have been added. $2$-dimensional topological field theories ($2$D TFTs) valued in vector spaces are commutative Frobenius algebras. The goal of this lecture series is to generalize from the $1$-category of vector spaces to any symmetric monoidal $\infty$-category $\mathcal{C}$, i.e. to study symm…

Weak countability axioms on the quotient spaces of topological gyrogroups
Ying-Ying Jin, Yi-Ting Wang, Li-Hong Xie
https://arxiv.org/abs/2507.19506 https://

Weak countability axioms on the quotient spaces of topological gyrogroups
In this paper, we mainly prove that if $H$ is a closed strong subgyrogroup of a strongly topological gyrogroup $G$ and $H$ is neutral, then (1) \( G/H \) is biradial if and only if \( G/H \) is nested; (2) \( G/H \) is metrizable if and only if \( G/H \) is a biradial space with countable pseudocharacter; (3) \( G/H \) is metrizable if and only if \( G/H \) has countable $cn$-character, given that \( G/H \) has the Baire property.

The Alpha Group Dynamic Mapping
Cleber Souza Corr\^ea, Thiago Braido Nogueira de Melo
https://arxiv.org/abs/2507.18303 https://arxiv.org/pdf/2507.18303

The Alpha Group Dynamic Mapping
This paper investigates the dynamical behavior of a system of ordinary differential equations (ODEs) governed by a matrix that represents the division in the algebra of the Alpha group. As the system evolves, the matrix induces topological transitions in geometric spaces, controlled by a rotational parameter. Numerical simulations are performed using a fourth-order Runge-Kutta method implemented in Python. The results reveal the emergence of topological nodes, the existence of critical points a…

The ball fixed point property in spaces of continuous functions
Antonio Avil\'es, Mar\'ia Jap\'on, Christopher Lennard, Gonzalo Mart\'inez Cervantes, Adam Stawski
https://arxiv.org/abs/2506.17995

The ball fixed point property in spaces of continuous functions
A Banach space $X$ has the ball fixed point property (BFPP) if for every closed ball $B$ and for every nonexpansive mapping $T\colon B\to B$, there is a fixed point. We study the BFPP for $C(K)$-spaces. Our goal is to determine topological properties over $K$ that may determine the failure or fulfillment of the BFPP for the space of continuous functions $C(K)$. We prove that the class of compact spaces $K$ for which the BFPP holds lies between the class of extremally disconnected compact spaces…

Intersection homotopy, refinements and coarsenings
Martintxo Saralegi-Aranguren, Daniel Tanr\'e
https://arxiv.org/abs/2508.03224 https://arxiv.org/pdf/…

Intersection homotopy, refinements and coarsenings
In previous works, we studied intersection homotopy groups associated to a Goresky and MacPherson perversity and a filtered space. They are defined as the homotopy groups of simplicial sets introduced by P. Gajer. We particularized to locally conical spaces of Siebenmann (called CS sets) and established a topological invariance for them when the regular part remains unchanged. Here, we consider coarsenings, made of two structures of CS sets on the same topological space, the strata of one bei…

Sober topologies on a set
Xiangrui Li, Qingguo Li, Dongsheng Zhao
https://arxiv.org/abs/2508.05419 https://arxiv.org/pdf/2508.05419

Sober topologies on a set
The collection of all topologies on a set X forms a complete lattice with respect to the inclusion order, which have been investigated by many researchers. Sobriety is one of the core and extensively studied properties in non-Hausdorff topology. This property plays a crucial role in characterizing the spectral spaces of commutative rings and topological spaces determined by their lattices of open sets. In this paper, we investigate the statute of sober topologies in the complete lattice of all …

A subbase property for describing edge-end spaces
Lucas Real
https://arxiv.org/abs/2508.17424 https://arxiv.org/pdf/2508.17424

A subbase property for describing edge-end spaces
In a previous joint work with Aurichi and Magalhães Jr., we showed that the topological spaces arising from the edge-end structure of infinite graphs define a proper subfamily of those obtained through the well-known (vertex-)ends. This result was later recovered by a more general approach due to Pitz, who also stated the problem of finding a purely topological characterization for the class of edge-end spaces. His question reads as an edge-related version of a similar conjecture posed by Dies…

Generation of Grothendieck topologies, provability and operations on subtoposes
Olivia Caramello, Laurent Lafforgue
https://arxiv.org/abs/2508.21134 https://

Generation of Grothendieck topologies, provability and operations on subtoposes
After reviewing the multiple roles of toposes - as generalized topological spaces, as universal invariants, as categorical analogues of the set-theoretic universe, and as semantic environments for first-order theories - we recall the notion of subtopos and its dual expression: both in terms of Grothendieck topologies and in terms of first-order logic. We emphasize the significance of this duality, which enables the translation of provability problems in first-order logic into problems concernin…

A pair of monads in Topology
Ando Razafindrakoto
https://arxiv.org/abs/2507.23403 https://arxiv.org/pdf/2507.23403

A pair of monads in Topology
In the article \cite{Sim}, H. Simmons describes two monads of interests arising from the dual adjunction between the category of topological spaces and that of (bounded) distributive lattices. These are the open prime filter monad and the ideal lattice monad. It is known that the ideal lattice monad induces the ideal frame comonad on the category of frames. We show that this ideal frame comonad can be paired with the open prime filter monad via the open set-spectrum adjunction. From this, we gi…

A new spin on polynomial relations among kappa classes
Alexander Alexandrov, Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin
https://arxiv.org/abs/2508.17865

A new spin on polynomial relations among kappa classes
We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov-Witten theory of the projective line and is dictated by $\mathbb{Z}_2$-equivariant topological recursion.

Spaces of homomorphisms, formality and Hochschild homology
Simon Gritschacher
https://arxiv.org/abs/2507.17683 https://arxiv.org/pdf/2507.17683

Spaces of homomorphisms, formality and Hochschild homology
Let $G$ be a discrete group. The topological category of finite dimensional unitary representations of $G$ is symmetric monoidal under direct sum and has an associated $\mathbb{E}_\infty$-space $\mathcal{K}^{\mathrm{def}}(G)$. We show that if $G$ and $A$ are finitely generated groups and $A$ is abelian, then $\mathcal{K}^{\mathrm{def}}(G\times A)\simeq \mathcal{K}^{\mathrm{def}}(G)\otimes \widehat{A}$ as $\mathbb{E}_\infty$-spaces, where $\widehat{A}$ is the Pontryagin dual of $A$. We deduce a …

Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences
Ralf Meyer
https://arxiv.org/abs/2506.19569 https://a…

Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences
A groupoid correspondence on an etale, locally compact groupoid induces a C*-correspondence on its groupoid C*-algebra. We show that the Cuntz-Pimsner algebra for this C*-correspondence relative to an ideal associated to an open invariant subset of the groupoid is again a groupoid C*-algebra for a certain groupoid. We describe this groupoid explicitly and characterise it by a universal property that specifies its actions on topological spaces. Our construction unifies the construction of groupo…

Turbulent Closed Relations
Judy Kennedy, Christopher Mouron, Van Nall
https://arxiv.org/abs/2507.02102 https://arxiv.org/pdf/2507.021…

Turbulent Closed Relations
This paper generalizes the classical notion of turbulence from dynamical systems generated by continuous functions to those defined by closed relations on compact metric spaces. Using the Mahavier product and the associated shift map, we introduce and explore CR-turbulence and reverse CR-turbulence, analyzing their relationship to topological entropy. A key focus is understanding when turbulence implies entropy and vice versa, with results showing that for finite closed relations, these propert…

Sites and Grothendieck toposes: an introduction
Olivia Caramello, Laurent Lafforgue
https://arxiv.org/abs/2508.21609 https://arxiv.org/pdf/2508.21609

Sites and Grothendieck toposes: an introduction
Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction that moves from familiar ground - groups and their actions, topological spaces, categories, and sheaves - to Grothendieck topologies and sites, then to the axiomatic and categorical foundations of toposes (via Giraud's theorem), to their geometry (morphisms, p…

Replaced article(s) found for math.LO. https://arxiv.org/list/math.LO/new
[1/1]:
- Topological games in Ramsey spaces
Juli\'an C. Cano, Carlos A. Di Prisco

Equivariant homotopic distance
Navnath Daundkar, J. M. Garc\'ia-Calcines
https://arxiv.org/abs/2508.20485 https://arxiv.org/pdf/2508.20485

Equivariant homotopic distance
In this paper, we introduce and investigate the equivariant homotopic distance between G-maps, an equivariant homotopy invariant that extends the notion of homotopic distance to the setting of spaces equipped with a continuous action of a topological group. This new invariant provides a unifying framework that generalizes both the equivariant Lusternik-Schnirelmann category and the equivariant topological complexity. We establish fundamental properties of the equivariant homotopic distance, inc…

On a local property of fences and fans
David S. Lipham
https://arxiv.org/abs/2508.03530 https://arxiv.org/pdf/2508.03530

On a local property of fences and fans
Two closely related classes of topological spaces are fences and fans. A fence is a compact metric space whose components are either arcs or singletons. A fan is a continuum formed by joining arcs at a common vertex, in such a way that intersections of subcontinua are always connected. We prove that every fence can be embedded in the plane and that both fences and fans admit a basis consisting of pierced open sets. This resolves a question by Iztok Banič, Goran Erceg, Ivan Jelić, Judy Kennedy…

Open 2D TFTs admit initial open-closed extensions
Shaul Barkan, Jan Steinebrunner, Adela YiYu Zhang
https://arxiv.org/abs/2509.02553 https://arxiv.org/pdf/…

Open 2D TFTs admit initial open-closed extensions
We show that any open 2-dimensional topological field theory valued in a symmetric monoidal $\infty$-category (with suitable colimits) extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of its value at the disk. As a corollary, we obtain an action of the moduli spaces of surfaces on the Hochschild homology object of $E_1$-Calabi-Yau algebras. This provides a space level refinement of previous work of Costello over $\mathbb{Q}$ and Wahl…

Weird $\mathbb R$-Factorizable Groups
Evgenii Reznichenko, Ol'ga Sipacheva
https://arxiv.org/abs/2506.18733 https://arxiv.org/pdf…

Weird $\mathbb R$-Factorizable Groups
The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $ω_1$. Closely related results concerning the $\mathbb R$-factorizability of products of topological groups and spaces are also obtained (a product $X\times Y$ of topological spaces is said to be $\mathbb R$-factorizable if any continuous function $X\times Y\to \mathbb R$ factors through a product of maps from $X$…

Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory
Xuecai Ma, Yifei Zhu
https://arxiv.org/abs/2509.00690 https://

Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory
Given an E-infinity ring spectrum R, with motivation from chromatic homotopy theory, we define relative effective Cartier divisors for a spectral Deligne-Mumford stack over Spet(R) and prove that, as a functor from connective R-algebras to topological spaces, it is representable. This enables us to solve various moduli problems of level structures on spectral abelian varieties, overcoming difficulty at primes dividing the level. In particular, we obtain higher-homotopical refinement for finite …

A Further Generalization of the Gale-Nikaido-Debreu Market Equilibrium Theorem
Ranjit Vohra
https://arxiv.org/abs/2507.17669 https://arxiv.org/pdf/2507.176…

A Further Generalization of the Gale-Nikaido-Debreu Market Equilibrium Theorem
We extend the important generalizations by Yannelis [25] and Cornet et al [7] of the classical result of Gale, Nikaido and Debreu (the "GND theorem") regarding existence of market equilibrium, by broadening the applicability of their results, which apply only to economies with commodity space that can be modeled by a locally convex Hausdorff space, to the wider class of economies with commodity spaces describable by any Hausdorff topological vector space with algebraic dual that separates point…

Exponentiable locales, revisited
Xu Huang
https://arxiv.org/abs/2507.15579 https://arxiv.org/pdf/2507.15579

Exponentiable locales, revisited
We give a moderately motivated exposition of exponentiable locales and the construction of exponentials in $\textsf{Loc}$, without assuming prior knowledge of exponential topological spaces or continuous posets.