Bulletin of the London Mathematical Society最新文献

We show that every graph $�G$ on $�n$ vertices with $��\delta �\left(�G�\right)�⩾�(�1�/�2�+�\epsilon �)�n�$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $��\Omega �\left(�\log �n�\right)�$. The proof is based on a recently introduced approach for finding vertex-spanning substructures via blow-up covers.

What is the probability that a random walk in the free group ends in a proper power? Or in a primitive element? We present a formula that computes the exponential decay rate of the probability that a random walk on a regular tree ends in a given subset, in terms of the exponential decay rate of the analogous probability of the non-backtracking random walk. This generalizes the well-known cogrowth formula of Grigorchuk, Cohen and Northshield. We also extend the formula to arbitrary subsets of the biregular tree.

We formulate and answer Gorenstein projective, flat, and injective analogs of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in 2007, was for integral group rings, in this paper, we deal with more general commutative base rings. We make use of the vast developments that have happened in the field of Gorenstein homological algebra over group rings in recent years, and we also improve and generalize several existing results from this area along the way.

Positive geometries are semialgebraic sets equipped with a canonical differential form whose residues mirror the boundary structure of the geometry. Every full-dimensional projective polytope is a positive geometry. Motivated by the canonical forms of polytopes, we construct a canonical form for any tope of an oriented matroid inside the Orlik–Solomon algebra of the underlying matroid. Using these canonical forms, we construct bases for the Orlik–Solomon algebra of a matroid and for the Aomoto cohomology. These bases of canonical forms are a foundational input in the theory of matroid amplitudes introduced by the second author.

In 1972, Cornalba and Shiffman showed that the number of zeros of an order zero holomorphic function in two or more variables can grow arbitrarily fast. We generalize this finding to the setting of complex dynamics, establishing that the number of isolated primitive periodic points of an order zero holomorphic function in two or more variables can grow arbitrarily fast as well. This answers a recent question posed by Buhovsky et al.

We prove that Hinich's construction of the Day convolution operad of two $�O$-monoidal $�\infty$-categories is an exponential in the $�\infty$-category of $�\infty$-operads over $�O$, and use this to give an explicit description of the formation of algebras in the Day convolution operad as a bivariant functor.

We study a variation of Kac's question, “Can one hear the shape of a drum?” if we allow ourselves access to some additional information. In particular, we allow ourselves to “hear” the local Weyl counting function at each point on the manifold and ask if this is enough to uniquely recover the Riemannian metric. This is physically equivalent to asking whether one can determine the shape of a drum if one is allowed to knock at any place on the drum. We show that the answer to this question is “yes” provided the Laplace–Beltrami spectrum of the drum is simple. We also provide a counterexample illustrating why this hypothesis is necessary. As a corollary, we give a short proof that manifolds with simple Laplace–Beltrami spectra have finite isometry groups.

We study stable commutator length on free $�Q$-groups. We prove that every non-identity element has positive stable commutator length, and that the corresponding free group embeds isometrically. We deduce that a non-abelian free $�Q$-group has an infinite-dimensional space of homogeneous quasimorphisms modulo homomorphisms, answering a question of Casals–Ruiz, Garreta and de la Nuez González. We conjecture that stable commutator length is rational on free $�Q$-groups. This is connected to the long-standing problem of rationality on surface groups: indeed, we show that free $�Q$-groups contain isometrically embedded copies of non-orientable surface groups.

In this paper, we construct a model structure for $��(�\infty �,�1�)�$-categories on the category of simplicial spaces, whose fibrant objects are the Segal spaces. In particular, we show that it is Quillen equivalent to the models of $��(�\infty �,�1�)�$-categories given by complete Segal spaces and Segal categories. We furthermore prove that this model structure has desirable properties: it is cartesian closed and left proper. As applications, we get a simple description of the inclusion of categories into $��(�\infty �,�1�)�$-categories and of homotopy limits of $��(�\infty �,�1�)�$-categories.