Bulletin of the London Mathematical Society最新文献

We show that there is a constant $�c$ such that any 3-uniform hypergraph $�H$ with $�n$ vertices and at least $��c�n^{�5�/�2�}�$ edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of Kupavskii, Polyanskii, Tomon and Zakharov. Furthermore, our work, combined with prior results, asymptotically determines the Turán number of all surfaces.

We prove a version of Shelah's categoricity conjecture for arbitrary deconstructible classes of modules. Moreover, we show that if $�A$ is a deconstructible class of modules that fits in an abstract elementary class $��(�A�,�⪯�)�$ such that (1) $�A$ is closed under direct summands and (2) $�⪯$ refines direct summands, then $�A$ is closed under arbitrary direct limits. In the Appendix, we prove that the assumption (2) is not needed in some models of ZFC.

We strengthen a result of Bagaria and Magidor (Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857–1877)  about the relationship between large cardinals and torsion classes of abelian groups, and prove that

Herglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well-known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi-connected domain $�\Omega$, a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator-valued functions on a multi-connected domain.

We show that under mild assumptions the Segre product of two graded cluster algebras has a natural cluster algebra structure.

We study finitely generated pairs of groups $��H�⩽�G�$ such that the Schreier graph of $�H$ has at least two ends and is narrow. Examples of narrow Schreier graphs include those that are quasi-isometric to finitely ended trees or have linear growth. Under this hypothesis, we show that $�H$ is a virtual fiber subgroup if and only if $�G$ contains infinitely many double cosets of $�H$. Along the way, we prove that if a group acts essentially on a finite-dimensional CAT(0) cube complex with no facing triples, then it virtually surjects onto the integers with kernel commensurable to a hyperplane stabiliser.

We generalise two facts about finite-dimensional algebras to finite-dimensional differential graded algebras. The first is the Nakayama lemma and the second is that the simples can detect finite projective dimension. We prove two dual versions which relate to Gorenstein differential graded algebras and Koszul duality, respectively. As an application, we prove a corepresentability result for finite-dimensional differential graded algebras.

Suppose that a Hilbert scheme of points on a K3 surface $�S$ of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville–Mukai system on a Fourier–Mukai partner of $�S$. We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.