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A study of the class \Delta consisting of topological \Delta-spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes \Delta _2 \subset \Delta _1 properly containing \Delta. We observe that for every first-countable X the following equivalences hold: X\in \Delta _1 iff X\in \Delta _2 iff each countable subset of X is G_{\delta }. Thus, new proposed concepts provide a natural extension of the family of all \lambda-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class \Delta _1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class \Delta _2. We investigate whether the classes \Delta _1 and \Delta _2 are invariant under the basic topological operations. Similarly to \Delta, both classes \Delta _1 and \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to \Delta, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. C_p(X) admits a continuous linear map onto C_p(Y). We show that Y \in \Delta _1 whenever X \in \Delta _1. Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone–Čech compactification \beta Y is scattered. [ABSTRACT FROM AUTHOR]

In the paper [ The Teichmüller theory of harmonic maps , J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface S with the vector space of holomorphic quadratic differentials on a Riemann surface X homeomorphic to S. This coordinate system is via harmonic maps from the Riemann surface X to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from X in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification. [ABSTRACT FROM AUTHOR]

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces X and Y certifies that X and Y are approximately isomorphic, a homotopy interleaving between X and Y certifies that X and Y are approximately weakly equivalent. The main results of this paper are that homotopy interleavings induce an extended pseudometric d_{HI} on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that d_{HI} (or more generally, any pseudometric satisfying these two axioms and an additional "homology bounding" axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. Finally, we consider the problem of establishing a persistent Whitehead theorem in terms of homotopy interleavings. We provide a counterexample to a naive formulation of the result. [ABSTRACT FROM AUTHOR]

A subset A of a vector space X is called \alpha-lineable whenever A contains, except for the null vector, a subspace of dimension \alpha. If X has a topology, then A is \alpha-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of (\alpha,\beta)-lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of (\alpha,\beta)-lineability/spaceability and, as applications, we extend recent results of different authors. [ABSTRACT FROM AUTHOR]

The main result of this paper is that, under PFA, for every regular space X with F(X) = \omega we have |X| \le w(X)^\omega; in particular, w(X) \le \mathfrak {c} implies |X| \le \mathfrak {c}. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. In fact, this space X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X intersect, which is much stronger than F(X) = \omega. Moreover, any non-empty open set in X also has size 2^\mathfrak {c}, and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points. [ABSTRACT FROM AUTHOR]

For two Banach spaces X and Y, we denote by K(X,Y) (resp. B(X,Y)) the space of all compact (resp. bounded) linear operators from X to Y. In this paper, we show that for 1\leq p,q<\infty, K(\ell _{p},\ell _{q}) is an 8-Lipschitz retract of B(\ell _{p},\ell _{q}). [ABSTRACT FROM AUTHOR]

We establish a sharp reciprocity inequality for modulus in compact metric spaces X with finite Hausdorff measure. In particular, when X is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney [Ann. Acad. Sci. Fenn. Math. 44 (2019), pp. 681-692]. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua E and F in X and the modulus of the family of surfaces of finite Hausdorff measure that separate E and F. The paper also develops approximation techniques, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Suppose that K is a compact Hausdorff space, S is a locally compact Hausdorff space and X is a Banach space with Schäffer constant \mathrm {S}(X). In this paper, we prove that if there is a map T from C(K) to C_{0}(S, X) satisfying \begin{equation*} \frac {1}{M} \|f-g\| \leq \|T(f)-T(g)\|\leq M \|f-g\|,\ \forall f, g \in C(K), \end{equation*} with 1\leq M^{2}<\mathrm {S}(X), then there exists a compact subset S_{0} of S and a continuous function \varphi from S_{0} onto K. This theorem on Lipschitz embeddings of C(K) into C_{0}(S, X) is the first nonlinear vector extension of the classical 1966 Holsztyński Theorem. Our result is optimal for many Banach spaces X including the spaces \ell _p^n, \ell _p and L_{p}([0,1]), 1 < p <\infty, n \geq 2, even when T is linear. The motivation to prove this result comes from the fact that it immediately yields a nontrivial lower bound for the C_{0}(S, X)-distortion of the class of all separable Banach spaces whenever S is a scattered space and \mathrm {S}(X)>1, namely \mathrm {S}(X) itself. [ABSTRACT FROM AUTHOR]

Given a metric pair (X,A), i.e. a metric space X and a distinguished closed set A\subset X, one may construct in a functorial way a pointed pseudometric space \mathcal {D}_\infty (X,A) of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces \mathcal {D}_\infty (X,A) and obtain characterizations of their metrizability, completeness, separability, and geodesicity. [ABSTRACT FROM AUTHOR]

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if (X,d,\mu) is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space N^{1,p}(X). Here the measure \mu is Borel and is finite and positive on all metric balls. In particular, we don't assume properness of X, doubling of \mu or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to locally complete spaces X and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity. [ABSTRACT FROM AUTHOR]

A useful result of H. Rosenthal and J. Bourgain states that, given a Banach space X, an operator T:L_1[0,1]\to X is completely continuous if and only if its composition with the natural inclusion i_\infty :L_\infty [0,1] \to L_1[0,1] is compact. We extend this result to multilinear mappings on products of L_1[0,1] spaces, and consider also the composition with the natural inclusion i:C[0,1]\to L_1[0,1]. We show that a multilinear mapping on a product of L_1[0,1] spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range. [ABSTRACT FROM AUTHOR]

We prove that any region \Gamma in a homogeneous n-dimensional and locally compact separable metric space X, where n\geq 2, cannot be irreducibly separated by a closed (n-1)-dimensional subset C with the following property: C is acyclic in dimension n-1 and there is a point b\in C\cap \Gamma having a special local base \mathcal B_C^b in C such that the boundary of each U\in \mathcal B_C^b is acyclic in dimension n-2. In case X is strongly locally homogeneous, it suffices to have a point b\in C\cap \Gamma with an ordinary base \mathcal B_C^b satisfying the above condition. The acyclicity means triviality of the corresponding Čech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces. [ABSTRACT FROM AUTHOR]

We exhibit a Banach space Z failing the approximation property, for which there is an uncountable family \mathscr F of closed subideals contained in the Banach algebra \mathcal K(Z) of the compact operators on Z, such that the subideals in \mathscr F are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras \mathcal L(X) of bounded operators on X, where closed ideals \mathcal I \neq \mathcal J are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators \mathcal S(X) for classical spaces such as X = L^p with p \neq 2, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur. [ABSTRACT FROM AUTHOR]

We show that there exists a Banach space X which contains closed subspaces Y and Z with Y+Z=X such that the associated surjective summation operator \Sigma \colon Y\times Z\to X defined by \Sigma (y,z)= y+z for y\in Y and z\in Z has no Lipschitz right inverse. [ABSTRACT FROM AUTHOR]

The well-rounded retract for SL_n(\mathbb {Z}) is defined as the set of flat tori of unit volume and dimension n whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space X_n of metric graphs of rank n and the Teichmüller space \mathcal {T}_g of closed hyperbolic surfaces of genus g, we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the sets of graphs whose systoles fill either topologically or geometrically (two analogues of a set proposed as a spine for \mathcal {T}_g by Thurston) are spines for X_n but that their dimension is larger than the virtual cohomological dimension of Out(F_n) in general. [ABSTRACT FROM AUTHOR]

We prove that if every bounded linear operator (or N-homogeneous polynomials) on a Banach space X with the compact approximation property attains its numerical radius, then X is a finite dimensional space. Moreover, we present an improvement of the polynomial James' theorem for numerical radius proved by Acosta, Becerra Guerrero and Galán [Q. J. Math. 54 (2003), pp. 1–10]. Finally, the denseness of weakly (uniformly) continuous 2-homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained. [ABSTRACT FROM AUTHOR]

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X_n, the completion of c_{00} relatively to the norm \[ \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T[\theta, A_n] (0<\theta <1) associated with the regular family A_n, comprised of all subsets of \mathbb {N} with at most n elements. For this space, we characterize the extreme points and show that for \theta \leq \frac {1}{n}, T[\theta, A_n] supports standard isometries. We also derive the form for the surjective isometries of T[\theta, A_3]. [ABSTRACT FROM AUTHOR]

Let f: X \to Y be a regular covering of a surface Y of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group G. We give necessary and sufficient conditions for an integral homology class on X to admit a representative as a connected component of the preimage of a nonseparating simple closed curve on Y, possibly after passing to a "stabilization", i.e. a G-equivariant embedding of covering spaces X \hookrightarrow X^+. [ABSTRACT FROM AUTHOR]

Let (\Omega,\Sigma,\mu) be a finite measure space and \mathcal {N}(X,Y) stand for the Banach space of all nuclear operators between Banach spaces X and Y, equipped with the nuclear norm \|\cdot \|_{nuc}. It is shown that if X^* has the Radon-Nikodym property, then a bounded linear operator T:L^\infty (\mu,X)\rightarrow Y is a \sigma-order continuous nuclear operator between the Banach spaces L^\infty (\mu,X) and Y if and only if its representing measure m_T has a Radon-Nikodym derivative in L^1(\mu,\mathcal {N}(X,Y)) and if and only if T has a Bochner kernel in L^1(\mu,\mathcal {N}(X,Y)). [ABSTRACT FROM AUTHOR]

We prove that the Lipschitz-free space over a Banach space X of density \kappa, denoted by \mathcal {F}(X), is linearly isomorphic to its \ell _1-sum \left (\bigoplus _{\kappa }\mathcal {F}(X)\right)_{\ell _1}. This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 over a \mathcal {L}_p-space. More precisely, we establish that, for every 1\leq p\leq \infty, if X is a \mathcal {L}_p-space of density \kappa, then \mathrm {Lip}_0(X) is either isomorphic to \mathrm {Lip}_0(\ell _p(\kappa)) if p<\infty, or \mathrm {Lip}_0(c_0(\kappa)) if p=\infty. [ABSTRACT FROM AUTHOR]

Let X be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for X in terms of Lipschitz retractions onto generating compact subsets K of X. In one direction, if X admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set K. On the other hand, we prove that if X admits a "small" generating compact Lipschitz retract then X has the \pi-property. It is still unknown if the \pi-property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces X. We give an example of a small generating convex compact set which is not a Lipschitz retract of C[0,1], although it is contained in a small convex Lipschitz retract and contains another one. We characterize isomorphically Hilbertian spaces as those Banach spaces X for which every convex and compact subset is a Lipschitz retract of X. Finally, we prove that a convex and compact set K in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map. [ABSTRACT FROM AUTHOR]

It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces X under a linear group G (or under a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of X are connected. In this work we prove, under similar assumptions, that the étale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilizers. We also deal with some related questions, concerning strong approximation outside a finite set of valuations. Finally, we prove a compatibility result, suggested to be true by work of Cyril Demarche, between the Brauer-Manin obstruction pairing on quotients G/H, where G and H are connected algebraic groups and H is linear, and certain abelianization morphisms associated with these spaces. [ABSTRACT FROM AUTHOR]

Given a Banach space X, a w^*-compact subset of X^*, and 1

We show that for any separable reflexive Banach space X and a large class of Banach spaces E, including those with a subsymmetric shrinking basis but also all spaces L_p[0,1] for 1\le p \le \infty, every bounded linear map \mathcal {B}(E)\to \mathcal {B}(X) which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism \mathcal {B}(E)\to \mathcal {B}(X). That is, the pair (\mathcal {B}(E),\mathcal {B}(X)) has the AMNM property in the sense of Johnson [J. London Math. Soc. (2) 37 (1988), pp. 294–316]. Previously this was only known for E=X=\ell _p with 1

The lower dimension \dim _L is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu [Adv. Math. 329 (2018), pp. 273–328] introduced the modified lower dimension dim_\textit {{ML}} by making the lower dimension monotonic with the simple formula dim_\textit {{ML}}X=\sup \{\dim _L E: E\subset X\}. As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We prove a new, simple characterization for the modified lower dimension. For a metric space X let \mathcal {K}(X) denote the metric space of the non-empty compact subsets of X endowed with the Hausdorff metric. As an application of our characterization, we show that the map dim_\textit {{ML}}\colon \mathcal {K}(X)\to [0,\infty ] is Borel measurable. More precisely, it is of Baire class 2, but in general not of Baire class 1. This answers another question of Fraser and Yu. Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of \ell ^1 endowed with the Effros Borel structure. [ABSTRACT FROM AUTHOR]

A space is called minimal if it admits a minimal continuous selfmap. We introduce and study the notion of product-minimality. We call a compact metrizable space Y product-minimal if, for every minimal system (X,T) given by a metrizable space X and a continuous selfmap T, there is a continuous map S\colon Y\to Y such that the product (X\times Y,T\times S) is minimal. If such a map S always exists in the class of homeomorphisms, we say that Y is a homeo-product-minimal space. We show that many classical examples of minimal spaces, including compact connected metrizable abelian groups, compact connected manifolds without boundary admitting a free action of a nontrivial compact connected Lie group, and many others, are in fact homeo-product-minimal. Then we give examples of metrizable continua X admitting both minimal homeomorphisms and minimal noninvertible maps, whose squares X\times X are not minimal (i.e., they admit neither minimal homeomorphisms nor minimal noninvertible maps). This shows that the product of two minimal spaces need not be minimal. [ABSTRACT FROM AUTHOR]

We consider in this article reaction-diffusion equations of the Fisher-KPP type with a nonlinearity depending on the space variable x, oscillating slowly and non-periodically. We are interested in the width of the interface between the unstable steady state 0 and the stable steady state 1 of the solutions of the Cauchy problem. We prove that, if the heterogeneity has large enough oscillations, then the width of this interface, that is, the diameter of some level sets, diverges linearly as t\rightarrow +\infty along some sequences of times, while it is sublinear along other sequences. As a corollary, under these conditions, generalized transition fronts do not exist for this equation. [ABSTRACT FROM AUTHOR]

It is proven that if X is a Banach space, K and S are locally compact Hausdorff spaces and there exists an (M, L)-quasi isometry T from C_{0}(K,X) onto C_{0}(S, X), then K and S are homeomorphic whenever 1 \leq M^{2}< S(X), where S(X) denotes the Schäffer constant of X, and L \geq 0. As a consequence, we show that the first nonlinear extension of Banach-Stone theorem for C_{0}(K, X) spaces obtained by Jarosz in 1989 can be extended to infinite-dimensional spaces X, thus reinforcing a 1991 conjecture of Jarosz himself on \epsilon-bi-Lipschitz surjective maps between Banach spaces. Our theorem is optimal when X is the classical space \ell _p, 2 \leq p< \infty. [ABSTRACT FROM AUTHOR]

Assume that X, Y are compact Hausdorff spaces. Let C(X)_+, C(Y)_+ be the sets of all positive real functions on X,Y respectively. Let T be a surjective map from a point separating multiplicative subgroup G\subset C(X)_+ onto a point separating multiplicative subgroup H\subset C(Y)_+ satisfying T1=1 and \begin{equation*} \frac {1}{M}\|f\cdot g^{-1}\|\leq \|Tf\cdot (Tg)^{-1}\|\leq M \|f\cdot g^{-1}\| \end{equation*} for all f, g\in G and for some constant M\geq 1. Then there exists a homeomorphism \tau from the Šilov boundary \partial H of H onto the Šilov boundary \partial G of G such that \begin{equation*} \frac {f(\tau (y))}{M^2}\leq (Tf)(y)\leq M^2\cdot f(\tau (y)) \end{equation*} for all f\in G and all y\in \partial H. The constant M^2 in the inequality above is optimal. A similar argument yields a generalization of the classical Banach-Stone Theorem to isometries defined from an additive subgroup of C_0(X) into C_0(Y) for locally compact Hausdorff spaces X and Y. [ABSTRACT FROM AUTHOR]

A set E in a Banach space X is compactivorous if for every compact set K in X there is a nonempty, (relatively) open subset of K which can be translated into E. In a separable Banach space, this is a sufficient condition which guarantees the Haar nonnegligibility of Borel subsets. We give some characterisations of this property in both separable and nonseparable Banach spaces and prove an extension of the main theorem to countable products of locally compact Polish groups. [ABSTRACT FROM AUTHOR]

Given a Banach space we consider the \sigma-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC (i.e. Zermelo-Fraenkel set theory with the axiom of choice) for every nonseparable Banach space X: (1) X can be covered by \omega _1-many of its hyperplanes; (2) All subsets of X of cardinalities less than \operatorname {cf}([\operatorname {dens}(X)]^\omega) can be covered by countably many hyperplanes. We prove (1) and (2) for all Banach spaces in many well-investigated classes and that they are consistent with any possible size of the continuum. (1) is related to the problem whether every compact Hausdorff space which has small diagonal is metrizable and (2) to large cardinals. [ABSTRACT FROM AUTHOR]

Given a space X we study the topology of the space of embeddings of X into Rd through the combinatorics of triangulations of X. We give a simple combinatorial formula for upper bounds for the largest dimension of a sphere that antipodally maps into the space of embeddings. This result summarizes and extends results about the nonembeddability of complexes into Rd, the nonexistence of nonsingular bilinear maps, and the study of embeddings into Rd up to isotopy, such as the chirality of spatial graphs. [ABSTRACT FROM AUTHOR]