Bulletin of the London Mathematical Society最新文献

In the standard category of directed graphs, graph morphisms map edges to edges. By allowing graph morphisms to map edges to finite paths (path homomorphisms of graphs), we obtain an ambient category in which we determine subcategories enjoying covariant functors to categories of algebras given by constructions of path algebras, Cohn path algebras, and Leavitt path algebras, respectively. Thus, we obtain new tools to unravel homomorphisms between Leavitt path algebras and between graph C*-algebras. In particular, a graph-algebraic presentation of the inclusion of the C*-algebra of a quantum real projective plane into the Toeplitz algebra allows us to determine a quantum CW-complex structure of the former. It comes as a mixed-pullback theorem where two $�\ast$-homomorphisms are covariantly induced from path homomorphisms of graphs and the remaining two are contravariantly induced by admissible inclusions of graphs. As a main result and an application of new covariant-induction tools, we prove such a mixed-pullback theorem for arbitrary graphs whose all vertex-simple loops have exits, which substantially enlarges the scope of examples coming from noncommutative topology.

In this article, we study a certain Galois property of subextensions of $��k�\left(�A_{\mathrm{tors}}�\right)�$, the minimal field of definition of all torsion points of an abelian variety $�A$ defined over a number field $�k$. Concretely, we show that each subfield of $��k�\left(�A_{\mathrm{tors}}�\right)�$ that is Galois over $�k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of $�k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, that is, does not contain any infinite set of algebraic numbers of bounded height.

The cross-ratio degree problem counts configurations of $�n$ points on $�P^1$ with $��n�-�3�$ prescribed cross-ratios. Cross-ratio degrees arise in many corners of combinatorics and geometry, but their structure is not well-understood in general. Interestingly, examining various special cases of the problem can yield combinatorial structures that are both diverse and rich. In this paper, we prove a simple closed formula for a class of cross-ratio degrees indexed by triangulations of an $�n$-gon; these degrees are connected to the geometry of the real locus of $�M_{�0�,�n�}$, and to positive geometry.

We consider rotated $�n$-Laplace systems on the unit ball $��B_1�\subset �R^n�$ of the form

In this note, the partial differential equation (PDE) approach of Guo–Phong–Tong and Guo–Phong–Tong–Wang adapted to prove an $�L^{\infty }$ estimate for transverse complex Monge–Ampère equations on homologically orientable transverse Kähler manifolds. As an application, a purely PDE-based proof of the regularity of Calabi–Yau cone metrics on $�Q$-Gorenstein $�T$-varieties is obtained.

We first prove a Selberg-type identity for the Shimura lift of the Rankin–Cohen bracket of a normalized Hecke eigenform and the theta function. We then discuss its relationship with the nonvanishing of central values of $�L$-functions associated to Hecke eigenforms.

Let $��(�X�,�B�)�$ be a pair of a normal surface over a perfect field of characteristic $��p�>�0�$ and an effective $�Q$-divisor $�B$ on $�X$. We prove that Steenbrink-type vanishing holds for $��(�X�,�B�)�$ if it is log canonical and $��p�>�5�$, or it is $�F$-pure. We also show that rational surface singularities satisfying the vanishing are $�F$-injective.

Stanley asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach Stanley's problem by studying the relationship between the chromatic symmetric function and other invariants. First, we prove Crew's conjecture that the chromatic symmetric function of a tree determines its generalized degree sequence, which enumerates vertex subsets by cardinality and the numbers of internal and external edges. Second, we prove that the restriction of the generalized degree sequence to subtrees contains exactly the same information as the subtree polynomial, which enumerates subtrees by cardinality and number of leaves. Third, we construct arbitrarily large families of trees sharing the same subtree polynomial, proving and generalizing a conjecture of Eisenstat and Gordon.

The aim of this paper is to apply the framework developed by Sam and Snowden to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton, and we prove that it is a finitely generated functor (on graphs of bounded genus). More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs.