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We prove that in the d-dimensional Linial–Meshulam stochastic process the (d - 1)st homology group with integer coefficients vanishes exactly when the final maximal (d - 1)-dimensional face is covered by a top-dimensional face. This generalizes the d = 2 case proved recently by Łuczak and Peled [Discrete Comput. Geom. 59 (2018), pp. 131–142] and establishes that p =d \log n/n is the sharp threshold for homology with integer coefficients to vanish in Y_d(n, p), answering a 2003 question of Linial and Meshulam [Combinatorica 26 (2006), pp. 475–487]. [ABSTRACT FROM AUTHOR]

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology. [ABSTRACT FROM AUTHOR]

We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in R-linear abelian categories of motivic nature, where R is any commutative unitary ring. Universal homology theory on the one point category yields "hieratic" R-modules, i.e. the indization of Freyd's free abelian category on R. Grothendieck ∂ -functors and satellite functors are recovered as certain additive relative homologies on an abelian category for which we also show the existence of universal ones. [ABSTRACT FROM AUTHOR]

We propose that symmetry-protected topological (SPT) phases with crystalline symmetry are formulated by an equivariant generalized homology |$h^G_n(X)$| over a real space manifold X with G a crystalline symmetry group. The Atiyah–Hirzebruch spectral sequence unifies various notions in crystalline SPT phases, such as the layer construction, higher-order SPT phases, and Lieb–Schultz–Mattis-type theorems. This formulation is applicable to not only free fermionic systems but also interacting systems with arbitrary onsite and crystal symmetries. [ABSTRACT FROM AUTHOR]

We compare several notions of equivariant formality for complex K-theory with respect to a compact Lie group action, including surjectivity of the forgetful map and the weak equivariant formality of Harada–Landweber, and find all are equivalent under standard hypotheses. As a consequence, we present an expression for the equivariant K-theory of the isotropy action of |$H$| on a homogeneous space |$G/H$| in all the classical cases. The proofs involve mainly homological algebra and arguments with the Atiyah–Hirzebruch–Leray–Serre spectral sequence, but a more general result depends on a map of spectral sequences from Hodgkin's Künneth spectral sequence in equivariant K-theory to that in Borel cohomology that seems not to have been otherwise defined. The hypotheses for the main structure result are analogous to a previously announced characterization of cohomological equivariant formality, first proved here, expanding on the results of Shiga and Takahashi. [ABSTRACT FROM AUTHOR]

Let en be the connective cover of the Morava E-theory spectrum En of height n. In this paper we compute its homology H*(en; Fp) for any prime p and n 4 up to possible multiplicative extensions. In order to accomplish this we show that the Künneth spectral sequence based on an E3-algebra R is multiplicative when the R-modules in question are commutative Salgebras. We then apply this result by working over BP which is known to be an E4-algebra. [ABSTRACT FROM AUTHOR]

We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the K_1-group of the I-adic completion of the group ring \mathbb {Q}\pi _1\Sigma _{g,1}, and prove that its reduction to \widehat {\mathbb {Q}\pi _1\Sigma _{g,1}}/\hat {I}^{d+1} is a finite-type invariant of degree d. We also show that the 1-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study an extended version of bivariant derived algebraic cobordism in which the cycles carry a vector bundle on the source as additional data. We show that, over a field of characteristic 0, this extends the analogous homological theory of Lee and Pandharipande. We then proceed to study in detail the restricted theory where only rank 1 vector bundles are allowed, and employ the obtained structural results to prove a weak version of the projective bundle formula for bivariant cobordism. Since the proof of this theorem works very generally, we introduce the notion of precobordism theories for quasi-projective derived schemes over an arbitrary Noetherian ring of finite Krull dimension as a reasonable class of theories where the proof can be carried out, and prove some of their basic properties. These results can be considered as the first steps towards a Levine-Morel style algebraic cobordism over a base ring that is not a field of characteristic 0. [ABSTRACT FROM AUTHOR]

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Let en be the connective cover of the Morava E-theory spectrum En of height n. In this paper we compute its homology H*(en; Fp) for any prime p and n 4 up to possible multiplicative extensions. In order to accomplish this we show that the Künneth spectral sequence based on an E3-algebra R is multiplicative when the R-modules in question are commutative Salgebras. We then apply this result by working over BP which is known to be an E4-algebra. [ABSTRACT FROM AUTHOR]

Golchin and Rezaei introduced conditions (PWP) and (PWP)w in (Subpullbacks and atness properties of S-posets). In this paper, we introduce conditions (PWPE) and (PWPE)w as generalizations of these conditions, respectively, and show that the relevant implications are strict. In general, we observe that condition (PWPE)w follows from condition (PWPE), but not conversely. Also, we prove that principal weak po-atness follows from condition (PWPE)w, but not conversely. Then, we obtain some general properties of conditions (PWPE) and (PWPE)w, and find sufficient and necessary conditions for the S-poset A(I) to satisfy these conditions. Finally, we find conditions on a pomonoid S under which a cyclic or Rees factor S-poset satisfies condition (PWPE) or condition (PWPE)w. Thereby, we present some homological classifications of pomonoids over which each of the conditions (PWPE) and (PWPE)w implies a specific property, and vice versa, for Rees factor S-posets. [ABSTRACT FROM AUTHOR]

We provide a sufficient condition for splitness of a link in terms of its reduced sl(N)$\mathfrak {sl}(N)$ link homology with arbitrary field coefficients. The proof of sufficiency uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. If N$N$ is prime and the coefficient field is of characteristic N$N$, then the sufficient condition for splitness is also necessary. When N=2$N = 2$, we recover Lipshitz–Sarkar's split link detection result for Khovanov homology with Z/2$\mathbf {Z}/2$ coefficients. [ABSTRACT FROM AUTHOR]

Building on the algebraic framework developed by Hendricks, Manolescu, and Zemke, we introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. Unlike previous approaches, we do not utilize any closed 4-manifold topology or contact topology. Instead, we adapt the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification ΘιZ of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits a Z1-subgroup of strongly nonextendable corks. The group ΘιZ can also be viewed as a refinement of the bordism group of diffeomorphisms. Using our invariants, we furthermore establish several new families of corks and prove that various known examples are strongly nonextendable. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method of constructing such cobordisms via equivariant surgery. [ABSTRACT FROM AUTHOR]

The third real de Rham cohomology of compact homogeneous spaces is studied. Given M = G / K with G compact semisimple, we first show that each bi-invariant symmetric bilinear form Q on g such that Q | k × k = 0 naturally defines a G-invariant closed 3-form H Q on M, which plays the role of the so called Cartan 3-form Q ([ · , · ] , ·) on the compact Lie group G. Indeed, every class in H 3 (G / K) has a unique representative H Q . Second, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie groups), i.e., b 3 (G / K) = s - 1 if G has s simple factors, we give the conditions to be fulfilled by Q and a given G-invariant metric g in order for H Q to be g-harmonic, in terms of algebraic invariants of G/K. As an application, we obtain that any 3-form H Q is harmonic with respect to the standard metric, although for any other normal metric, there is only one H Q up to scaling which is harmonic. Furthermore, among a suitable (2 s - 1) -parameter family of G-invariant metrics, we prove that the same behavior occurs if k is abelian: either every H Q is g-harmonic (this family of metrics depends on s parameters) or there is a unique g-harmonic 3-form H Q (up to scaling). In the case when k is not abelian, the special metrics for which every H Q is g-harmonic depend on 3 parameters. [ABSTRACT FROM AUTHOR]

In the classical Batalin–Vilkovisky formalism, the BV operator Δ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by Δ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin–Vilkovisky algebras where the operator Δ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever Δ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s. [ABSTRACT FROM AUTHOR]

If a knot K in S³ admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either ±2 or ±1=q for some value of q that is explicitly determined by the knot Floer homology of K. Moreover, in the former case the genus of K must be 2, and in the latter case there is a bound relating q to the genus and the Heegaard Floer thickness of K. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to 2. We also show that the conjecture holds for any knot K for which each prime summand of K has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes ±1 and ±2 on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access. [ABSTRACT FROM AUTHOR]

To an Adams-type homology theory we associate the notion of a synthetic spectrum; this is a product-preserving sheaf on the site of finite spectra with projective E-homology. We show that the ∞ -category of synthetic spectra based on E is in a precise sense a deformation of the ∞ -category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the E ∗ -based Adams spectral sequence. We describe a symmetric monoidal functor from the ∞ -category of cellular motivic spectra over Spec (C) into an even variant of synthetic spectra based on MU and show that it induces an equivalence between the ∞ -categories of p-complete objects for all primes p. In particular, it follows that the p-complete cellular motivic category can be described purely in terms of chromatic homotopy theory. [ABSTRACT FROM AUTHOR]

Corrosion of steel wires in cables is a frequent occurrence and is a major factor affecting the durability and safety of cables. Magnetostrictive guided waves have been employed to detect steel wire corrosion. However, due to the complexity of the geometry of the corrosion area, it is difficult to interpret the magnetostrictive guided wave testing signals. Hence, extracting features from testing signals to characterize the corrosion condition is a great challenge. This study proposes a topological feature for corrosion characterization of steel wire, which is extracted from the two-dimensional point cloud with optimal geometry structure of the testing signal based on persistent homology theory. Corrosion experiments were carried out on 30 galvanized steel wires. The experimental results indicate that the topological feature increases roughly linearly with the corrosion depth. Furthermore, compared to other features obtained from the time domain, frequency domain, and joint time–frequency domain, this topological feature can reflect the progression of corrosion more accurately. [ABSTRACT FROM AUTHOR]

The question of whether an orbit space of a compact linear group is a topological manifold and a homology manifold is studied. The case of a simple three-dimensional group is considered. An upper bound is obtained for the sum of integral parts of the halved dimensions of irreducible components for a representation whose quotient is a homology manifold. This strengthens a similar result obtained previously, which gave such a bound in the case where the quotient of the representation is a smooth manifold. Most representations for which the obtained estimate holds have also been considered previously. The argument uses standard considerations of linear algebra and the theory of Lie groups and algebras and their representations. [ABSTRACT FROM AUTHOR]

In this article, we establish Serre-Hochschild spectral sequences of Hom-Lie algebras. By using these spectral sequences, we describe homology groups of finite dimensional multiplicative Hom-Lie algebras in terms of homology groups of Lie algebras and abelian Hom-Lie algebras. [ABSTRACT FROM AUTHOR]

We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side‐effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K‐theory of the C∗$C^*$‐algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK‐conjecture. We also construct explicit maps from the groupoid homology groups to the K‐theory groups of their C∗$C^*$‐algebras in degrees zero and one, and investigate their properties. [ABSTRACT FROM AUTHOR]

These notes cover the lectures of the first named author at 2021 IHES Summer School on "Enumerative Geometry, Physics and Representation Theory" with additional details and references. They cover the definition of Khovanov‐Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro‐geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane. [ABSTRACT FROM AUTHOR]

We give a new proof of the non-triviality of wheel graph homology classes using higher operations on Lie graph homology and a derived version of Koszul duality for modular operads. [ABSTRACT FROM AUTHOR]

We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript. [ABSTRACT FROM AUTHOR]

We associate with every pure flag simplicial complex |$\Delta $| a standard graded Gorenstein |$\mathbb {F}$| -algebra |$R_{\Delta }$| whose homological features are largely dictated by the combinatorics and topology of |$\Delta $|⁠. As our main result, we prove that the residue field |$\mathbb {F}$| has a |$k$| -step linear |$R_{\Delta }$| -resolution if and only if |$\Delta $| satisfies Serre's condition |$(S_k)$| over |$\mathbb {F}$| and that |$R_{\Delta }$| is Koszul if and only if |$\Delta $| is Cohen–Macaulay over |$\mathbb {F}$|⁠. Moreover, we show that |$R_{\Delta }$| has a quadratic Gröbner basis if and only if |$\Delta $| is shellable. We give two applications: first, we construct quadratic Gorenstein |$\mathbb {F}$| -algebras that are Koszul if and only if the characteristic of |$\mathbb {F}$| is not in any prescribed set of primes. Finally, we prove that whenever |$R_{\Delta }$| is Koszul the coefficients of its |$\gamma $| -vector alternate in sign, settling in the negative an algebraic generalization of a conjecture by Charney and Davis. [ABSTRACT FROM AUTHOR]

We study the homological algebra in the category P p of strict polynomial functors of degree p over a field of positive characteristic p. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras of simple functors and Schur functors. We observe that the category P p has a Kazhdan-Lusztig theory and we show that the DG algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of P p as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of p -weight 1 in P e for e > p. [ABSTRACT FROM AUTHOR]

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite Γ1(p∞)-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of the Shimura datum splits at p. This generalizes and strengthens the vanishing result proved in [A. Caraiani et al., Compos. Math. 156 (2020)]. As an application of this vanishing theorem, we prove a result on the codimensions of ordinary completed homology for the same groups, analogous to conjectures of Calegari-Emerton for completed (Borel-Moore) homology. [ABSTRACT FROM AUTHOR]

External privatisation of public education has emerged in Finland in the admission to higher education. A field analysis of thematic interviews (N = 22) with powerful actors in the private educational market and middle-class young people applying for places at universities in the highly competitive disciplines of medicine and law was conducted. The research task was to examine the discourses that construct the limits of the public and private fields, and the kind of homology that emerges. Four discourses were identified: support, business, social responsibility, and personal responsibility. The private field distinguished itself from the public fields through the discourse of business and attached itself to it through the discourse of support. The dominance of the symbolic power of transformed economic capital in the private field mobilised in the public field was misrecognised during the admission process as 'motivation'. The homology between the two fields was strong yet hidden. [ABSTRACT FROM AUTHOR]

Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asym metry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration. [ABSTRACT FROM AUTHOR]

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is, of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity that is doubly exponential (and this is so for almost all data). [ABSTRACT FROM AUTHOR]

In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials MHkℓ(G)⟶MHk−1ℓ−1(G)$\operatorname{MH}^{\ell }_k(G) \longrightarrow \operatorname{MH}^{\ell -1}_{k-1}(G)$ between magnitude homologies of a digraph G$G$, which make them chain complexes. Then we show that its homology MHkℓ(G)$\mathcal {MH}^{\ell }_k(G)$ is non‐trivial and homotopy invariant in the context of 'homotopy theory of digraphs' developed by Grigor'yan–Muranov–S.‐T. Yau et al. (G‐M‐Ys in the following). It is remarkable that the diagonal part of our homology MHkk(G)$\mathcal {MH}^{k}_k(G)$ is isomorphic to the reduced path homologyH∼k(G)$\tilde{H}_k(G)$ also introduced by G‐M‐Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology MHkℓ(G)$\operatorname{MH}^{\ell }_k(G)$, and the second page is isomorphic to our homology MHkℓ(G)$\mathcal {MH}^{\ell }_k(G)$. As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that H∼k(g)=0$\tilde{H}_k(g) = 0$ for k⩾2$k \geqslant 2$ and H∼1(g)≠0$\tilde{H}_1(g) \ne 0$ if any edges of an undirected graph g$g$ is contained in a cycle of length ⩾5$\geqslant 5$. [ABSTRACT FROM AUTHOR]

The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CW-complexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions of well-known results about topological spaces with combinatorial methods used on finite spaces. In the particular case of h-regular spaces, effective and reasonably efficient methods are implemented and the technique of discrete vector fields is applied in order to improve the previous algorithms. [ABSTRACT FROM AUTHOR]

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper, we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface's skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled "whiskers" that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study (ℤ / n ℤ) -nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu's characterization of checkerboard colorability of virtual links with up to four crossings. We also prove a version of the Kauffman–Murasugi–Thistlethwaite theorem that the breadth of an evaluation of the homological arrow polynomial for an "h-reduced" diagram D is 4 (c (D) − g (D) + 1). [ABSTRACT FROM AUTHOR]

We study invariants associated to Smale spaces obtained from an expanding endomorphism on a (closed connected Riemannian) flat manifold. Specifically, the relevant invariants are the K-theory of the associated C*-algebras and Putnam's homology theory for Smale spaces. The latter is isomorphic to the groupoid homology of the groupoids used to construct the C*-algebras. [ABSTRACT FROM AUTHOR]

We apply persistent homology to figure counting/detecting. This paper includes the theoretical construction, the numerical experiments, and the applications to real images. In the theoretical construction, we define some geometric concepts, formulate the counting problem, and prove the theorem that gives the threshold value for distinguishing noise and figures. We use the Rips and Čech complexes and compare the two complexes. In the numerical experiments, we count several model images. Finally, we apply our method to real images. [ABSTRACT FROM AUTHOR]

Under a generalized subquadratic growth condition, brake orbits are guaranteed via the homological link theorem. Moreover, the minimal period estimate is given by Morse index estimate and L 0 -index estimate. [ABSTRACT FROM AUTHOR]

We introduce a metric homotopy theory, which we call moderately discontinuous homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the moderately discontinuous homology theory recently developed by the authors and E. Sampaio. The |$k$| -th MD homotopy group is a group |$MDH^b_{\bullet }$| for any |$b\in [1,\infty ]$| together with homomorphisms |$MD\pi ^b\to MD\pi ^{b^{\prime}}$| for any |$b\geq b^{\prime}$|⁠. We develop all its basic properties including finite presentation of the groups, long homotopy sequences of pairs, metric homotopy invariance, Seifert van Kampen Theorem, and the Hurewicz Isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For |$b=1$|⁠ , it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for |$b=\infty $|⁠ , the |$MD$| -homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-homology in the same case. [ABSTRACT FROM AUTHOR]

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε -th component with ε the identity of the grade group. For Z -graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras. [ABSTRACT FROM AUTHOR]

Let G be a generalized matrix Banach algebra associated with a Morita context (A , B , M , N , Φ , Ψ). In this paper, we investigate biprojectivity and biflatness of G, under certain conditions. Indeed, we obtain a biprojective (resp. biflat) generalized matrix Banach algebra such that M ≠ 0 and N ≠ 0. Moreover, we give a sufficient condition for biprojectivity and biflatness of G, in the case where A = N = 0. Using this fact, we obtain some results about homological properties of bi-amalgamated Banach algebras. As an application, we answer some recent open questions. [ABSTRACT FROM AUTHOR]

Bivariant theory is a unified framework for cohomology and Borel--Moore homology theories. In this paper, we extract an∞-enhanced bivariant homology theory from Gaitsgory--Rozenblyum's six functor formalism. [ABSTRACT FROM AUTHOR]

We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. We begin by combining geographic and electoral data from a districting plan to produce a persistence diagram. Then, to see beyond a particular plan and understand the possibilities afforded by the choices made in redistricting, we build methods to visualize and analyze large ensembles of alternative plans. Our detailed case studies use zero-dimensional homology (persistent components) of filtered graphs constructed from voting data to analyze redistricting in Pennsylvania and North Carolina. We find that, across large ensembles of partitions, the features cluster in the persistence diagrams in a way that corresponds strongly to geographic location, so that we can construct an average diagram for an ensemble, with each point identified with a geographical region. Using this localization lets us produce zonings of each state at Congressional, state Senate, and state House scales, show the regional non-uniformity of election shifts, and identify attributes of partitions that tend to correspond to partisan advantage. The methods here are set up to be broadly applicable to the use of TDA on large ensembles of data. Many studies will benefit from interpretable summaries of large sets of samples or simulations, and the work here on localization and zoning will readily generalize to other partition problems, which are abundant in scientific applications. For the mathematically and politically rich problem of redistricting in particular, TDA provides a powerful and elegant summarization tool whose findings will be useful for practitioners. [ABSTRACT FROM AUTHOR]

In this paper we propose a computationally efficient multiple hypothesis testing procedure for persistent homology. The computational efficiency of our procedure is based on the observation that one can empirically simulate a null distribution that is universal across many hypothesis testing applications involving persistence homology. Our observation suggests that one can simulate the null distribution efficiently based on a small number of summaries of the collected data and use this null in the same way that p-value tables were used in classical statistics. To illustrate the efficiency and utility of the null distribution we provide procedures for rejecting acyclicity with both control of the Family-Wise Error Rate (FWER) and the False Discovery Rate (FDR). We will argue that the empirical null we propose is very general conditional on a few summaries of the data based on simulations and limit theorems for persistent homology for point processes. [ABSTRACT FROM AUTHOR]

We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the ϵ -surjection property, that were recently introduced to characterize the notion of computable type arising from computability theory. For a class of spaces including the finite simplicial complexes, we develop techniques to prove or disprove these properties using homotopy and homology theories, and give applications of these results. In particular, we answer an open question on the computable type property, showing that it is not preserved by taking products. We also observe that computable type is decidable for finite simplicial complexes. • Computable type of a finite simplicial complex is equivalent to a homotopy property of certain maps to spheres. • Computable type is not preserved under products. • Computable type is decidable for finite simplicial complexes. [ABSTRACT FROM AUTHOR]

With the development of network science and graph theory, brain network research has unique advantages in explaining those mental diseases, the neural mechanism of which is unclear. Additionally, it can provide a new perspective in revealing the pathophysiological mechanism of brain diseases from the system level. The selection of threshold plays an important role in brain networks construction. There are no generally accepted criteria for determining the proper threshold. Therefore, based on the topological data analysis of persistent homology theory, this study developed a multi-scale brain network modeling analysis method, which enables us to quantify various persistent topological features at different scales in a coherent manner. In this method, the Vietoris-Rips filtering algorithm is used to extract dynamic persistent topological features by gradually increasing the threshold in the range of full-scale distances. Subsequently, the persistent topological features are visualized using barcodes and persistence diagrams. Finally, the stability of persistent topological features is analyzed by calculating the Bottleneck distances and Wasserstein distances between the persistence diagrams. Experimental results show that compared with the existing methods, this method can extract the topological features of brain networks more accurately and improves the accuracy of diagnostic and classification. This work not only lays a foundation for exploring the higher-order topology of brain functional networks in schizophrenia patients, but also enhances the modeling ability of complex brain systems to better understand, analyze, and predict their dynamic behaviors. [ABSTRACT FROM AUTHOR]

To solve the autonomous navigation problem in complex environments, an efficient motion planning approach is newly presented in this paper. Considering the challenges from large-scale, partially unknown complex environments, a three-layer motion planning framework is elaborately designed, including global path planning, local path optimization, and time-optimal velocity planning. Compared with existing approaches, the novelty of this work is twofold: 1) a novel heuristic-guided pruning strategy of motion primitives is proposed and fully integrated into the state lattice-based global path planner to further improve the computational efficiency of graph search, and 2) a new soft-constrained local path optimization approach is proposed, wherein the sparse-banded system structure of the underlying optimization problem is fully exploited to efficiently solve the problem. We validate the safety, smoothness, flexibility, and efficiency of our approach in various complex simulation scenarios and challenging real-world tasks. It is shown that the computational efficiency is improved by 66.21% in the global planning stage and the motion efficiency of the robot is improved by 22.87% compared with the recent quintic Bézier curve-based state space sampling approach. We name the proposed motion planning framework E $\mathbf {^{3}} $ MoP, where the number 3 not only means our approach is a three-layer framework but also means the proposed approach is efficient in three stages. Note to Practitioners—This paper is motivated by the challenges of motion planning problems of mobile robots. A three-layer motion planning framework is proposed by combining global path planning, local path optimization, and time-optimal velocity planning. For mobile robot navigation applications in semi-structured environments, optimization-based local planners are recommended. Extensive simulation and experimental results show the effectiveness of the proposed motion planning framework. However, due to the non-convexity of the path optimization formulation, the proposed local planner may get stuck in local optima. In future research, we will concentrate on extending the proposed local path optimization approach with the theory of homology classes to maintain several homotopically distinct local paths and seek global optima. [ABSTRACT FROM AUTHOR]

The tension between early and late H 0 is revised in the context of axion dark matter arising naturally from string theoretical integrations of antisymmetric tensor fields over non-trivial cycles. Certain early universe cycles may appear non-trivial from the perspective of a homology analysis focused on the early universe, while they may become trivial when analysed from the perspective of a homology theory reaching out to lower energies and later times. Such phenomena can introduce variations in the axion potential that would explain the observed H 0 tension. The decay of such pseudo-axions when the pseudo-cycles dissipate trigger axion-two-photon (otherwise having an extremely long lifetime) and axion-gravitational processes mediated by Chern–Simons couplings with observable electromagnetic or gravitational wave signals originating in the early universe. [ABSTRACT FROM AUTHOR]

Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map \sigma on each quandle chain complex that lowers the dimensions by one. By using its pull-back \sigma ^\sharp, each 2-cocycle \phi gives us the 3-cocycle \sigma ^\sharp \phi. For oriented classical links in the 3-space, we explore relation between their quandle 2-cocycle invariants associated with \phi and their shadow 3-cocycle invariants associated with \sigma ^\sharp \phi. For oriented surface links in the 4-space, we explore how powerful their quandle 3-cocycle invariants associated with \sigma ^\sharp \phi are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed. [ABSTRACT FROM AUTHOR]