Higher Dimensional Analogue Research Articles

Matrix representation of a cross product and related curl-based differential operators in all space dimensions

Abstract A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of Grassmann’s, Jacobi’s and Room’s identities. Moreover, such a view provides a higher dimensional analogue of the decomposition of the vector Laplacian, which itself gives an explicit index-free Helmholtz decomposition in arbitrary dimensions n ≥ 2 n\ge 2 .

N-Exangulated categories (II): Constructions from n-cluster tilting subcategories

In n-Exangulated Categories (I), we introduced the notion of n-exangulated categories for each positive integer n. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a common generalization of n-exact categories in the sense of Jasso and (n+2)-angulated categories in the sense of Geiss-Keller-Oppermann. In this second article we introduce the notion of n-cluster tilting subcategories for extriangulated categories with enough projectives and injectives and show that under certain conditions such n-cluster tilting subcategories are n-exangulated.

Local-move identities for the ℤ[t,t−1]-Alexander polynomials of 2-links, the alinking number, and high-dimensional analogues

A well-known identity [Formula: see text] holds for three 1-links [Formula: see text], [Formula: see text], and [Formula: see text] which satisfy a famous local-move relation, where [Formula: see text] becomes the Alexander–Conway polynomial of [Formula: see text] if we let [Formula: see text]. We prove a new local-move identity for the [Formula: see text]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander–Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the [Formula: see text]-Alexander polynomials of 2-links and the alinking number of 2-links. We produce high-dimensional analogues of these results. Furthermore, we prove that in the 2-dimensional case we cannot normalize the [Formula: see text]-Alexander polynomials to be compatible with our identity but that in a high-dimensional case we can do that to be compatible with our new identity.

Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds

We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger–Fukaya–Gromov theory of collapse with bounded curvature. The construction starts with a complete noncompact asymptotically conical Calabi–Yau 3-fold B and a circle bundle M→B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gϵ on M that collapses with bounded curvature as ϵ→0 to the original Calabi–Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry. We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact G2-metrics were known.

Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice

We introduce an equation defined on a multi-dimensional lattice, which can be considered as an extension to the coprimeness-preserving discrete KdV like equation in our previous paper. The equation is also interpreted as a higher-dimensional analogue of the Hietarinta-Viallet equation, which is famous for its singularity confining property while having an exponential degree growth. As the main theorem we prove the Laurent and the irreducibility properties of the equation in its "tau-function" form. From the theorem the coprimeness of the equation follows. In Appendix we review the coprimeness-preserving discrete KdV like equation whichis a base equation for our main system and prove the properties such as the coprimeness.

Universal arrays

A word on q symbols is a sequence of letters from a fixed alphabet of size q. For an integer k⩾1, we say that a word w is k-universal if, given an arbitrary word of length k, one can obtain it by removing letters from w. It is easily seen that the minimum length of a k-universal word on q symbols is exactly qk. We prove that almost every word of size (1+o(1))cqk is k-universal with high probability, where cq is an explicit constant whose value is roughly qlog⁡q. Moreover, we show that the k-universality property for uniformly chosen words exhibits a sharp threshold. Finally, by extending techniques of Alon (2017) [1], we give asymptotically tight bounds for every higher dimensional analogue of this problem.

Almost uniform domains and Poincaré inequalities

Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincar\'e inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio and Sarvas. Our condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components. In doing so, our results hold true for metric spaces equipped with doubling measures and Poincar\'e inequalities in general, and for the Heisenberg groups in particular. To our knowledge, these are the first examples of such subsets on any step-2 Carnot group. Such subsets also give, in general, new examples of Sobolev extension domains on doubling metric measure spaces. When specialized to the plane, we give general sufficient conditions for planar subsets, possibly with empty interior, to be Ahlfors 2-regular and to satisfy a (1,2)-Poincar\'e inequality. In the Euclidean case, our construction also covers the non-self-similar Sierpi\'nski carpets of Mackay, Tyson, and Wildrick, as well as higher dimensional analogues not treated in the literature. The analysis of the Poincar\'e inequality with exponent p=1, for these carpets and their higher dimensional analogues, includes a new way of proving an isoperimetric inequality on a space without constructing Semmes families of curves.

The Abrikosov vortex in curved space

We study the self-gravitating Abrikosov vortex in curved space with and with-out a (negative) cosmological constant, considering both singular and non-singular solutions with an eye to hairy black holes. In the asymptotically flat case, we find that non-singular vortices round off the singularity of the point particle’s metric in 3 dimensions, whereas singular solutions consist of vortices holding a conical singularity at their core. There are no black hole vortex solutions. In the asymptotically AdS case, in addition to these solutions there exist singular solutions containing a BTZ black hole, but they are always hairless. So we find that in contrast with 4-dimensional ’t Hooft-Polyakov monopoles, which can be regarded as their higher-dimensional analogues, Abrikosov vortices cannot hold a black hole at their core. We also describe the implications of these results in the context of AdS/CFT and propose an interpretation for their CFT dual along the lines of the holographic superconductor.

Higher dimensional formal orbifolds and orbifold bundles in positive characteristic

Kumar and Parameswaran introduced formal orbifold curves defined over an algebraically closed field of positive characteristics. They studied both étale and Nori fundamental group schemes associated to such objects. Our aim here is to study the higher dimensional analog of these objects and their fundamental groups.

Actions of diagonal endomorphisms on conformally invariant measures on the 2-torus

Let \(\nu \) be a probability measure that is ergodic under the endomorphism \((\times p, \times p)\) of the torus \({\mathbb {T}}^2\), such that \(\dim \pi \mu < \dim \mu \) for some non-principal projection \(\pi \). We show that, if both \(m\ne n\) are independent of p, the \((\times m, \times n)\) orbits of \(\nu \) typical points will equidistribute towards the Lebesgue measure. If \(m>p\) then typically the \((\times m, \times p)\) orbits will equidistribute towards the product of the Lebesgue measure with the marginal of \(\mu \) on the y-axis. We also prove results in the same spirit for certain self similar measures. These are higher dimensional analogues of results due (among others) to Host, Lindenstrauss, and Hochman-Shmerkin.

Microscopic entropy of AdS3 black holes revisited

We revisit the microscopic description of AdS3 black holes in light of recent progress on their higher dimensional analogues. The grand canonical partition function that follows from the AdS3/CFT2 correspondence describes BPS and nearBPS black hole thermodynamics. We formulate an entropy extremization principle that accounts for both the black hole entropy and a constraint on its charges, in close analogy with asymptotically AdS black holes in higher dimensions. We are led to interpret supersymmetric black holes as ensembles of BPS microstates satisfying a charge constraint that is not respected by individual states. This interpretation provides a microscopic understanding of the hitherto mysterious charge constraints satisfied by all BPS black holes in AdS. We also develop thermodynamics and a nAttractor mechanism of AdS3 black holes in the nearBPS regime.

Kirchhoff index of simplicial networks

We introduce a high-dimensional analogue of the Kirchhoff index which is also known as total effective resistance. This analogue, which we call the simplicial Kirchhoff index Kf(X), is defined to be the sum of the simplicial effective resistances of all (d+1)-subsets of the vertex set of a simplicial complex X of dimension d. This definition generalizes the Kirchoff index of a graph G defined as the sum of the effective resistance between all pairs of vertices of G. For a d-dimensional simplicial complex X with n vertices, we give formulas for the simplicial Kirchhoff index in terms of the pseudoinverse of the Laplacian LX in dimension d−1 and its eigenvalues:Kf(X)=n⋅trLX+=n⋅∑λ∈Λ+1λ, where LX+ is the pseudoinverse of LX, and Λ+ is the multi-set of non-zero eigenvalues of LX. Using this formula, we obtain an inequality for a high-dimensional analogue of algebraic connectivity and Kirchhoff index, and propose these quantities as measures of robustness of simplicial complexes. In addition, we derive its integral formula and relate this index to a simplicial dynamical system. We present an open problem for a combinatorial proof of our formula by relating the combinatorial interpretation of Rσ to rooted forests in higher dimensions.

Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes

AbstractThe Linial–Meshulam complex model is a natural higher dimensional analog of the Erdős–Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial–Meshulam complexes with an appropriate scaling of the underlying parameter. The present article aims to extend that result to more general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial–Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their Betti numbers. Moreover, we obtain the convergence of the empirical spectral distributions of their Laplacians. A key element in the argument is the local weak convergence of simplicial complexes. Inspired by the work of Linial and Peled, we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.

Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of Homeo([0,1]) has length 2. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of Homeo([0,1]) have length 2. We answer this question in the affirmative.We call f∈Homeo([0,1]d) singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of f∈Homeo([0,1]) has length 2 iff f is singular iff f is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem.We show that for d≥2 the graph of the generic element of Homeo([0,1]d) has infinite d-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of Homeo([0,1]d) with infinite d-dimensional Hausdorff measure is not Haar null. We show that for d≥2 the generic element of Homeo([0,1]d) is singular but not strongly singular. We also show that for d≥2 almost every element of Homeo([0,1]d) is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null).Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them.

Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence

In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

K3 carpets on minimal rational surfaces and their smoothings

In this paper, we study K3 double structures on minimal rational surfaces [Formula: see text]. The results show there are infinitely many non-split abstract K3 double structures on [Formula: see text] parametrized by [Formula: see text], countably many of which are projective. For [Formula: see text] there exists a unique non-split abstract K3 double structure which is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921, to appear in Eur. J. Math.]). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless [Formula: see text] is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on [Formula: see text]. Moreover, we show any embedded projective K3 carpet on [Formula: see text] with [Formula: see text] arises as a flat limit of embeddings degenerating to 2:1 morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on [Formula: see text], embedded by a complete linear series are smooth points if and only if [Formula: see text]. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on [Formula: see text] and embedded by a complete linear series are always smooth. The results in [P. Bangere, F. J. Gallego and M. González, Deformations of hyperelliptic and generalized hyperelliptic polarized varieties, preprint (2020), arXiv:2005.00342] show that there are no higher dimensional analogues of the results in this paper.

Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

This paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of delta -hyperbolicity.

From n-exangulated categories to n-abelian categories

Herschend-Liu-Nakaoka introduced the notion of n-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of n-exact categories in the sense of Jasso and (n+2)-angulated in the sense of Geiss-Keller-Oppermann. Let C be an n-exangulated category with enough projectives and enough injectives, and X a cluster tilting subcategory of C. In this article, we show that the quotient category C/X is an n-abelian category, and it is equivalent to an n-cluster tilting subcategory of an abelian category with enough projectives. These results generalize the work of Jacobsen-Jørgensen and Zhou-Zhu for (n+2)-angulated categories. Moreover, it highlights new phenomena when it is applied to n-exact categories.

Consensus on simplicial complexes: Results on stability and synchronization.

We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.

The Dual Complex of a Semi-log Canonical Surface

Abstract Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $\Delta $ of a log canonical pair $(X,\Delta )$ and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,\Delta ),$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $\Delta $.