Recent zbMATH articles in MSC 55https://zbmath.org/atom/cc/552023-09-22T14:21:46.120933ZWerkzeugDominance complexes and vertex cover numbers of graphshttps://zbmath.org/1517.051332023-09-22T14:21:46.120933Z"Matsushita, Takahiro"https://zbmath.org/authors/?q=ai:matsushita.takahiroSummary: The dominance complex \(D(G)\) of a simple graph \(G = (V,E)\) is the simplicial complex consisting of the subsets of \(V\) whose complements are dominating. We show that the connectivity of \(D(G)\) plus 2 is a lower bound for the vertex cover number \(\tau (G)\) of \(G\).Perfect matching complexes of honeycomb graphshttps://zbmath.org/1517.051382023-09-22T14:21:46.120933Z"Bayer, Margaret"https://zbmath.org/authors/?q=ai:bayer.margaret-m"Jelić Milutinović, Marija"https://zbmath.org/authors/?q=ai:milutinovic.marija-jelic"Vega, Julianne"https://zbmath.org/authors/?q=ai:vega.julianneSummary: The perfect matching complex of a graph is the simplicial complex on the edge set of the graph with facets corresponding to perfect matchings of the graph. This paper studies the perfect matching complexes, \(\mathcal{M}_p(H_{k \times m\times n})\), of honeycomb graphs. For \(k = 1\), \(\mathcal{M}_p(H_{1\times m\times n})\) is contractible unless \(n\geqslant m=2\), in which case it is homotopy equivalent to the \((n-1)\)-sphere. Also, \(\mathcal{M}_p(H_{2\times 2\times 2})\) is homotopy equivalent to the wedge of two 3-spheres. The proofs use discrete Morse theory.On the number of generators of an algebra over a commutative ringhttps://zbmath.org/1517.160142023-09-22T14:21:46.120933Z"First, Uriya A."https://zbmath.org/authors/?q=ai:first.uriya-a"Reichstein, Zinovy"https://zbmath.org/authors/?q=ai:reichstein.zinovy-b"Williams, Ben"https://zbmath.org/authors/?q=ai:williams.benLet \(k\) be an infinite field, and \(R\) a \(k\)-ring (meaning commutative associative unital \(k\)-algebra. \textit{O. Forster} [Math. Z. 84, 80--87 (1964; Zbl 0126.27303)] proved the following result: if \(R\) is noetherian with Krull dimension \(k\), then any projective \(R\)-module of rank \(n\) can be generated by \(k+n\) elements. The authors create the following setting for Forster type Theorems. An \(R\)-algebra is an \(R\)-module \(B\) with a \(B\)-bilinear map \(B^2\to B\). An \(R\)-form of a \(k\)-algebra \(A\) is an \(R\)-algebra \(B\) such that there exists a faithfully flat \(R\)-ring \(S\) such that \(A\otimes_k S\) and \(B\otimes_RS\) are isomorphic as \(S\)-algebras. In this way projective \(R\)-modules of rank \(n\) are forms of \(k^n\) with zero multiplication. Other examples of forms include finite étale \(R\)-algebras, Azumaya algebras, octonion algebras and Albert algebras. An even more general setting arises when one considers so-called multialgebras. An \(R\)-multialgebra is an \(R\)-module \(A\) together with a family of \(R\)-linear maps \(m_i:\ A^{n_i}\to A\). In fact an algebra with unit, or an algebra with an involution can be considered as a multialgebra.
For an \(R\)-(multi)algebra \(B\), let \(\mathrm{gen}_R(B)\) be the smallest cardinality of a set of generators of \(B\) as an \(R\)-algebra. The aim of the present paper is to find upper and lower bounds of \(\mathrm{gen}_R(B)\) in the situation where \(B\) is an \(R\)-form of a given \(R\)-algebra \(A\). There are three main result, Theorems 1.3, 1.4 and 1.5. Theorem 1.3 brings an upper bound in the case where \(A\) is an \(n\)-dimensional \(k\)-algebra, and \(R\) is a finite type \(k\)-ring. Theorem 1.4 gives a lower bound for certain rings \(R\) and certain forms of \(A\), under the assumption that \(\mathrm{Aut}_k(A)\) is not unipotent. Theorems 1.3 and 1.4 also hold for multialgebras. In Theorem 1.5, both the upper and lower bound can be sharpened in the situation where \(A\) is a matrix algebra (and \(B\) is an \(R\)-Azumaya algebra).
Reviewer: Stefaan Caenepeel (Brussels)Categorical models for path spaceshttps://zbmath.org/1517.180082023-09-22T14:21:46.120933Z"Minichiello, Emilio"https://zbmath.org/authors/?q=ai:minichiello.emilio"Rivera, Manuel"https://zbmath.org/authors/?q=ai:rivera.manuel"Zeinalian, Mahmoud"https://zbmath.org/authors/?q=ai:zeinalian.mahmoudThis paper introduces a natural transformation \(Sz\colon \mathfrak{C}\Rightarrow G,\) given in full detail, and as the nerve of a map of poset-enriched categories. Here, \(\mathfrak{C}\colon \mathrm{sSet}\to\mathrm{sCat}\) is the rigidification functor from simplicial sets to simplicial categories that underlies the simplicial nerve construction, and \(G\colon \mathrm{sSet}\to\mathrm{sCat}\) is the left-adjoint to the Dwyer-Kan classifying space functor \(\overline{W}\). In particular, after localising categories to groupoids, \(\mathcal{L}\colon \mathrm{Cat}\to\mathrm{Gpd}\), one has the Kan loop groupoid functor \(G^{\mathrm{Kan}}\cong\mathcal{L}G\colon \mathrm{sSet}\to\mathrm{sGpd}\) to simplicial groupoids.
A technically useful and clarifying result of the paper is that the map \(Sz_{\Delta^n}\colon\mathfrak{C}(\Delta^n)\to G(\Delta^n)\) of simplicial categories coincides, up to an \(\mathrm{op}\)-taking, with an earlier construction due to Hinich concerned with simplicial constructions in his deformation theory. The \(\mathrm{op}\)-difference is because the two nerves, simplicial versus homotopy coherent, are related by taking opposites on the posets of paths whose nerves constitute the resulting mapping spaces.
The Dugger-Spivak description of the mapping spaces under the rigidification functor via `necklaces' features crucially in one of the main technical results in the first part of the paper -- that \(Sz\) induces a weak equivalence of simplicial sets given by simplicial groupoids along Dwyer-Kan's \(\overline{W}\) on one side and along the homotopy coherent nerve \(\mathfrak{N}\) on the other. More precisely, the necklace description is shown to imply that \(Sz_{S^n}\colon \mathfrak{C}(S^n)\xrightarrow{\sim} G(S^n)\), where \(S^n=\Delta^n/\partial\Delta^n\), is given essentially by a weak equivalence induced by the obvious collapsing map \((S^1)^{\wedge (n-1)}\to S^{n-1}\).
The authors relate these two constructions to the locally Kan path category \(\mathbb{P}(X)\) of a simplicial set \(X\), obtained by applying the singular chains functor to the topological path category of the geometric realisation of \(X\). Realisation makes morphisms invertible, and indeed \(\mathcal{L}G\) and \(\mathcal{L}\mathfrak{C}\), forgotten to functors into \(\mathrm{sCat}\), are shown to be weakly equivalent to \(\mathbb{P}\).
The second part of the paper, concerned with algebraic models, makes a number of worthwhile contributions. Inspired by work of Larazev-Holstein on categorical Koszul duality, the authors introduce `categorical coalgebras' (over some ambient commutative ring) which form the domain of a many-object cobar construction \(\mathbf{\Omega}\) with target connective dg categories. This \(\mathbf{\Omega}\) is shown, more particularly, to send a certain natural coalgebraic structure, a `\(B_{\infty}\)-coalgebra structure' on the categorical coalgebra \(\widetilde{C}^{\Delta}_*(X)\) of normalised chains (up to a modification) of a simplicial set \(X\) into a category structure enriched over dg coalgebras. The overarching result is that there is a natural quasi-equivalence \(\widehat{\mathbf{\Omega}}(\widetilde{\mathbf{C}}_*(X))\simeq {C}^{\Delta}_*(\mathbb{P}(X))\) of categories enriched over dg coalgebras, where \(\widetilde{\mathbf{C}}_*\) lifts \(\widetilde{C}^{\Delta}_*(X)\) to produce \(B_{\infty}\)-coalgebras, and \(\widehat{\mathbf{\Omega}}\) formally inverts set-like elements in mapping dg coalgebras after applying \(\mathbf{\Omega}\). On the right-hand side, \(C^{\Delta}_*\) takes ordinary normalised chains. This results combines with the results from the first part mentioned above in the obvious way.
One may single out the constructions \(Sz\), which gives a comparison between the rigidification model and the Kan loop groupoid model based on simplicial operators introduced in 1961 by Szczarba, and the many-object cobar construction \(\mathbf{\Omega}\) and its `geometric version', \(\widehat{\mathbf{\Omega}}\) or \(\widehat{\mathbf{\Omega}}\widetilde{\mathbf{C}}\), as particularly useful due to their explicit presentation. Further work remains desirable to find further uses of enhancing the algebraic model of the path category from a dg category to a category enriched over dg coalgebras.
Reviewer: Ödül Tetik (Zürich)Cubical models of higher categories without connectionshttps://zbmath.org/1517.180092023-09-22T14:21:46.120933Z"Doherty, Brandon"https://zbmath.org/authors/?q=ai:doherty.brandonThe author proves that each of the model structures for (\(n\)-trivial, saturated) comical sets on the category of marked cubical sets having only faces and degeneracies (without connections) is Quillen equivalent to the corresponding model structure for (\(n\)-trivial, saturated) complicial sets on the category of marked simplicial sets, as well as to the corresponding comical model structures on cubical sets with connections. As a consequence, it is proved that the cubical Joyal model structure on cubical sets without connections is equivalent to its analogues on cubical sets with connections and to the Joyal model structure on simplicial sets. It is also proved that any comical set without connections may be equipped with connections via lifting, and that this can be done compatibly on the domain and codomain of any fibration or cofibration of comical sets.
Reviewer: Philippe Gaucher (Paris)Bounded cohomology of classifying spaces for families of subgroupshttps://zbmath.org/1517.200792023-09-22T14:21:46.120933Z"Li, Kevin"https://zbmath.org/authors/?q=ai:li.kevin-x|li.kevin-wThe author introduces a generalization of bounded cohomology as bounded version of Bredon cohomology for groups relative to a family of subgroups, i.e. a collection of subgroups of a group G such that it is closed under conjugation and under taking subgroups. This generalization of bounded cohomology, like the Bredon cohomology, is well-behaved with respect to normal subgroups and admits a topological interpretation in terms of classifying spaces for families.
Analogous to \textit{B. E. Johnson}'s characterization of amenability [Cohomology in Banach algebras. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0256.18014)], the author proves a characterization of relatively amenable groups in terms of bounded Bredon cohomology. He also provides a characterization of relative amenability in terms of relatively injective modules. Analogous to \textit{I. Mineyev}'s characterization of hyperbolicity [Q. J. Math. 53, No. 1, 59--73 (2002; Zbl 1013.20048)], the relative hyperbolicity is characterized in terms of this generalized bounded cohomology.
Reviewer: Mădălina Buneci (Targu-Jiu)Morse-Novikov cohomology for blow-ups of complex manifoldshttps://zbmath.org/1517.320772023-09-22T14:21:46.120933Z"Meng, Lingxu"https://zbmath.org/authors/?q=ai:meng.lingxuThis paper is dedicated to the study of Morse-Novikov cohomology for blow-ups of complex manifolds through the weight \(\theta\)-sheaf \(\underline{\mathbb{R}}_{X,\theta}\). The author gives elementary properties of the weight \(\theta\)-sheaf \(\underline{\mathbb{R}}_{X,\theta}\) and also establishes several theorems of Künneth and Leray-Hirsch types. Then the computation of Morse-Novikov cohomologies of projective bundles as well as the independence between the \(\theta\)-Lefschetz number and \(\theta\) are given. Finally blow-up formulae on complex manifolds are investigated.
Reviewer: Guokuan Shao (Zhuhai)Odd-dimensional GKM-manifolds of non-negative curvaturehttps://zbmath.org/1517.530432023-09-22T14:21:46.120933Z"Escher, Christine"https://zbmath.org/authors/?q=ai:escher.christine-m"Goertsches, Oliver"https://zbmath.org/authors/?q=ai:goertsches.oliver"Searle, Catherine"https://zbmath.org/authors/?q=ai:searle.catherineThe study of Riemannian manifolds with positive or non-negative curvature has been a central subject in Riemannian geometry for a long time. Since this is a rather difficult problem, most of the research is focused on spaces with large isometry groups.
The paper is a continuation of the previous papers [\textit{O. Goertsches} and \textit{M. Wiemeler}, Int. Math. Res. Not. 2015, No. 22, 12015--12041 (2015; Zbl 1339.57042); Math. Z. 300, No. 2, 2007--2036 (2022; Zbl 1493.57021)]. It studies the GKM and GKM\(_k\) torus actions on odd-dimensional manifolds with non-negative curvature, while the aforementioned papers are concerned with the more studied even-dimensional case.
A torus action is equivariantly formal if its equivariant cohomology is a free module over the symmetric algebra of the abelian Lie algebra of the torus. In even-dimension, such an action is called GKM (or GKM\(_k\)) if in addition every pair of weights of the action (resp., every \(k\) weights) are linearly independent and the fixed points are isolated and finitely many. In the odd-dimensional case one replaces the condition on the fixed point set, that is now a finite union of circles. The main property of such an action is that the cohomology, both equivariant and rational, can be described in terms of a graph attached to the skeleton of fixed subspaces of dimension up to 2. It is called GKM graph and comes with additional structures -- weight function and connection, which allow a definition of graph cohomology.
The main results of the paper concern the rational cohomology type of odd-dimensional GKM\(_k\) manifolds of positive or non-negative curvature. The first one states that for \(k=3\) this cohomology is a tensor product of the graph cohomology and the cohomology of an odd-dimensional sphere. Then, under some restrictions on the graph, the graph cohomology factor could be identified with the cohomology of a compact rank one Riemannian symmetric space. Similar statements are formulated in the case \(k=4\). As a corollary, the authors show that the rational cohomology of a GKM\(_3\) manifold with positive curvature is the same as the one of an odd-dimensional sphere.
Reviewer: Gueo Grantcharov (Miami)Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomologyhttps://zbmath.org/1517.530772023-09-22T14:21:46.120933Z"Wilkins, Nicholas"https://zbmath.org/authors/?q=ai:wilkins.nicholasThe author discusses the relationship between the construction of the quantum Steenrod square in [\textit{N. Wilkins}, Geom. Topol. 24, No. 2, 885--970 (2020; Zbl 1467.53095)] and the equivariant pair-of-pants product due to \textit{P. Seidel} in [Geom. Funct. Anal. 25, No. 3, 942--1007 (2015; Zbl 1331.53119)], which are both generalizations of the Steenrod square on a topological space \(M\).
From the abstract: ``We relate the quantum Steenrod square to Seidel's equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We define continuation maps between different Hamiltonians in Z/2-equivariant Floer cohomology, and prove expected properties of them. We prove a symplectic Cartan relation for the equivariant pair-of-pants product, pointing out the difficulties in stating it. We give a nonvanishing result for the equivariant pair-of-pants product for some elements of \(SH^*(T^*\mathbb{S}^n)\). We finish by calculating the symplectic square for the negative line bundles \(M = \mathrm{Tot}(\mathcal{O}(-1) \to \mathbb{C}P^m\)), proving an equivariant version of a result due to Ritter.''
Reviewer: Mikhail Malakhal'tsev (Bogotá)Parametrized topological complexity of sphere bundleshttps://zbmath.org/1517.550012023-09-22T14:21:46.120933Z"Farber, Michael"https://zbmath.org/authors/?q=ai:farber.michael-s"Weinberger, Shmuel"https://zbmath.org/authors/?q=ai:weinberger.shmuelIn the present paper the \textit{sectional category} of a fibration \(p:E\to B\), denoted by \(\mathrm{secat}(p)\), is defined to be the smallest \(k\) such that \(B\) admits a cover by open sets \(U_0,\ldots,U_k\), each of which admits a local continuous section. Note that the authors use the reduced version, i.e., \(\mathrm{secat}(p)=0\) if and only if \(p\) admits a global continuous section.
The \textit{height} of a cohomology class \(\alpha\neq 0\), denoted by \(\mathrm{h}(\alpha)\), is the largest integer \(k\) such that its \(k\)-th power is nonzero, \(\alpha^k\neq 0\).
Given a fibration \(p:E\to B\), consider the fibration \(\pi^p:E^{I}_B\to E\times_B E, \gamma\mapsto (\gamma(0),\gamma(1))\), where \(E\times_B E=\{(e_1,e_2)\in E\times E:~p(e_1)=p(e_2)\}\) and \(E^{I}_B\) is the space of continuous paths in \(E\) lying in a single fiber of \(p\). The \textit{parametrized topological complexity} of \(p\) is given by
\[
\mathrm{TC}[p:E\to B]=\mathrm{secat}(\pi^p).
\]
This numerical invariant was introduced in [\textit{D. C. Cohen} et al., SIAM J. Appl. Algebra Geom. 5, No. 2, 229--249 (2021; Zbl 1473.55010)]
Let \(\xi:E\to B\) be an oriented rank \(q\geq 2\) vector bundle equipped with fibrewise scalar product. Let \(S(\xi):S(E)\to B\) denote the unit sphere bundle and \(\xi':E'\to S(E), (x,y)\to x\), where \(E'\subset S(E)\times_B S(E)\) is the space of pairs of mutually orthogonal unit vectors \((x,y)\in S(E)\times_B S(E)\), \(x\perp y\). Note that \(\xi'\) is an oriented locally trivial fibration with fibre sphere of dimension \(q-2\). Consider its Euler class \(e(\xi')\in H^{q-1}(S(E))\).
The authors estimate the parametrized topological complexity \(\mathrm{TC}[S(\xi):S(E)\to B]\). In particular, they show the following statement.
\textbf{Theorem 5.2} One has the estimates
\[
\mathrm{h}(e(\xi'))+1\leq\mathrm{TC}[S(\xi):S(E)\to B]\leq\mathrm{secat}(\xi')+1.
\]
Moreover, if \(B\) is a CW-complex satisfying \(\dim B\leq (q-1)\cdot\mathrm{h}(e(\xi'))\) then
\[
\mathrm{TC}[S(\xi):S(E)\to B]=\mathrm{h}(e(\xi'))+1=\mathrm{secat}(\xi')+1.
\]
Reviewer: Cesar A. Ipanaque Zapata (São Carlos)Stable homotopy groupshttps://zbmath.org/1517.550022023-09-22T14:21:46.120933Z"Wang, Guozhen"https://zbmath.org/authors/?q=ai:wang.guozhen"Xu, Zhouli"https://zbmath.org/authors/?q=ai:xu.zhouliFrom the introduction: In this article, we give a survey of the stable part of the homotopy groups of spheres. We will first recall the notion of stable homotopy, and then discuss an interpretation in terms of the framed cobordism and an application to the classification of exotic spheres. In the last part we discuss some methods for computing these stable homotopy groups.On the homotopy and strong homotopy type of complexes of discrete Morse functionshttps://zbmath.org/1517.570172023-09-22T14:21:46.120933Z"Donovan, Connor"https://zbmath.org/authors/?q=ai:donovan.connor"Lin, Maxwell"https://zbmath.org/authors/?q=ai:lin.maxwell"Scoville, Nicholas A."https://zbmath.org/authors/?q=ai:scoville.nicholas-aThe Morse complex of \(K\), denoted \(\mathcal{M}(K)\), was introduced by \textit{M. K. Chari} and \textit{M. Joswig} [Discrete Math. 302, No. 1--3, 39--51 (2005; Zbl 1091.57025)] as the simplicial complex of all gradient vector fields on \(K\), and its existence is ensured by Forman's discrete Morse theory that yields a method by which one can construct a gradient vector field on \(K\). While, as \textit{N. A. Capitelli} and \textit{E. G. Minian} [Discrete Comput. Geom. 58, No. 1, 144--157 (2017; Zbl 1371.05326)] have proven, the Morse complex of \(K\) is rich enough to reconstruct the isomorphism type of \(K\), at the same time the simple homotopy type of the Morse complex does not determine the simple homotopy type of \(K\). In fact, an explicit computation of the homotopy type of the Morse complex is known for only a very small number of complexes.
In the reviewed paper the authors determine the homotopy types of the Morse complexes of certain collections of simplicial complexes by studying dominating vertices or strong collapses. They also show that if \(K\) contains two leaves that share a common vertex, then its Morse complex is strongly collapsible, thence it has the homotopy type of a point. Furthermore, they prove that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of \textit{R. Ayala} et al. [Topology Appl. 155, No. 17--18, 2084--2089 (2008; Zbl 1185.57002)]. Moreover, they demonstrate that the Morse complex of a disjoint union \(K\sqcup L\) is the Morse complex of the join \(K \ast L\). This last result is used to compute the homotopy type of the Morse complex of certain families of graphs, including so called caterpillar graphs, as well as the automorphism group of a disjoint union for a large collection of disjoint complexes.
Reviewer: Emil Saucan (Karmiel)Fold maps on small dimensional manifolds with prescribed singular sethttps://zbmath.org/1517.570182023-09-22T14:21:46.120933Z"Kalmár, Boldizsár"https://zbmath.org/authors/?q=ai:kalmar.boldizsarFold maps are a generalization of Morse functions. A fold map \(f\) is a map from a closed \((n+q)\) dimensional manifold \(M\) to \(\mathbb R^n\) such that the only singularities it can have are fold singularities (i.e. \(f\) can be expressed in local coordinates around the singular point as the identity in the first \(n-1\) variables and components and a sum of quadratic terms in the last \(q+1\) variables in the last component). The singular set of such a map is an \((n-1)\)-dimensional submanifold of \(M\) and \(f\) restricted to it is an immersion.
The existence of fold maps has been studied by many authors and in the process many relations between the topology of \(M\) and the topology of the singular set have been obtained.
In this paper the author looks for sufficient and necessary conditions for the existence of fold maps with a prescribed singular set on a closed manifold \(M\). In particular the paper deals with the cases when \(4\leq n\leq 7\) and \(q=1\). Amongst many nice results, the author proves that if \(M\) is a closed orientable 7-manifold with second Stiefel-Whitney and first Pontryagin classes equal to 0, such that \(H^4(M)\) has no 2-torsion, then there exist fold maps to \(\mathbb R^6\) with orientable or non-orientable singular sets. Similar results are obtained for \(n=4,5\) which involve other characteristic classes.
In the case of \(n=7\) the author uses symplectic classes and derives conditions about the cobordism class of orientable 8-manifolds with fold maps to \(\mathbb R^7\). In this case, besides a certain obstruction class and the second symplectic class to be 0, the Euler characteristic of \(M\) has to be equal to a signed sum of Euler characteristics of the components of the singular set in order to have a fold map from an 8-manifold to \(\mathbb R^7\).
Reviewer: Raúl Oset Sinha (València)A reduced basis method for Darcy flow systems that ensures local mass conservation by using exact discrete complexeshttps://zbmath.org/1517.651022023-09-22T14:21:46.120933Z"Boon, Wietse M."https://zbmath.org/authors/?q=ai:boon.wietse-m"Fumagalli, Alessio"https://zbmath.org/authors/?q=ai:fumagalli.alessioIn this paper, a three-step solution procedure is proposed for Darcy flow systems based on the exact de Rham complex. The mass conservation equation is first solved and the flux field is subsequently corrected by adding a solenoidal vector field. The computational cost in constructing the correction is reduced by applying reduced basis methods based on proper orthogonal decomposition. In the third step, the pressure field is constructed with discretization methods capable of conserving mass locally. The procedure was extended to the setting of Darcy flow in fractured porous media by employing mixed-dimensional differential operators.
Reviewer: Bülent Karasözen (Ankara)Synthetic topology in homotopy type theory for probabilistic programminghttps://zbmath.org/1517.680722023-09-22T14:21:46.120933Z"Bidlingmaier, Martin E."https://zbmath.org/authors/?q=ai:bidlingmaier.martin-e"Faissole, Florian"https://zbmath.org/authors/?q=ai:faissole.florian"Spitters, Bas"https://zbmath.org/authors/?q=ai:spitters.basSummary: The ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions. However, continuous distributions have to be discretized because the corresponding measures cannot be defined on all subsets of their carriers. This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory. We study the initial \(\sigma\)-frame and the corresponding induced topology on arbitrary sets. Based on these intrinsic topologies, we define valuations and lower integrals on sets and prove versions of the Riesz and Fubini theorems. We then show how the Lebesgue valuation, and hence continuous distributions, can be constructed.Covariant holographic reflected entropy in \(AdS_3/CFT_2\)https://zbmath.org/1517.810652023-09-22T14:21:46.120933Z"Afrasiar, Mir"https://zbmath.org/authors/?q=ai:afrasiar.mir"Chourasiya, Himanshu"https://zbmath.org/authors/?q=ai:chourasiya.himanshu"Raj, Vinayak"https://zbmath.org/authors/?q=ai:raj.vinayak"Sengupta, Gautam"https://zbmath.org/authors/?q=ai:sengupta.gautamSummary: We substantiate a covariant proposal for the holographic reflected entropy in \(CFT\)s dual to non-static \(AdS\) geometries from the bulk extremal entanglement wedge cross section in the literature with explicit computations in the \(AdS_3/CFT_2\) scenario. In this context we obtain the reflected entropy for zero and finite temperature time dependent bipartite mixed states in \(CFT_{1 + 1}\)s with a conserved charge dual to bulk rotating extremal and non-extremal BTZ black holes through a replica technique. Our results match exactly with the corresponding extremal entanglement wedge cross section for these bulk geometries in the literature. This constitutes a significant consistency check for the proposal and its possible extension to the corresponding higher dimensional \(AdS/CFT\) scenario.Twisted cohomotopy implies M5-brane anomaly cancellationhttps://zbmath.org/1517.810842023-09-22T14:21:46.120933Z"Sati, Hisham"https://zbmath.org/authors/?q=ai:sati.hisham"Schreiber, Urs"https://zbmath.org/authors/?q=ai:schreiber.ursSummary: We highlight what seems to be a remaining subtlety in the argument for the cancellation of the total anomaly associated with the M5-brane in M-theory. Then, we prove that this subtlety is resolved under the hypothesis that the C-field flux is charge-quantized in the generalized cohomology theory called J-twisted cohomotopy.Scalar correlators and normal modes in holographic neutron starshttps://zbmath.org/1517.850032023-09-22T14:21:46.120933Z"Canavesi, Tobías"https://zbmath.org/authors/?q=ai:canavesi.tobias"Fierro, Octavio"https://zbmath.org/authors/?q=ai:fierro.octavio"Grandi, Nicolás"https://zbmath.org/authors/?q=ai:grandi.nicolas-esteban"Pisani, Pablo"https://zbmath.org/authors/?q=ai:pisani.pablo-a-gSummary: The holographic neutron star provides a strong coupling description for a highly degenerate metallic state on a sphere. Its phase space can be split into two different sectors, with an unstable region at intermediate degeneracies. We investigate the critical nature of such region, by analyzing the asymptotic behavior of a scalar probe to compute the two-point correlator of the boundary theory. We show that in the stable region the correlator is dominated by the normal modes, whereas it displays a critical power-law behavior as we move into the unstable region.