Journal of the London Mathematical Society-Second Series最新文献

Generalising the uniform companion for large fields with a single derivation, we construct a theory $��U�C_D�$ of fields of characteristic 0 with free operators—operators determined by a homomorphism from the field to its tensor product with $�D$, a finite-dimensional $�Q$-algebra—which is the model companion of any theory of a field with free operators whose associated difference field is difference large and model complete. Under the assumption that $�D$ is a local ring, we show that simplicity is transferred from the theory of the underlying field to the theory of the field with operators, and we use this to study the model theory of bounded, PAC fields with free operators.

Let $�G$ be a finite group, $�p$ a prime number and $�P$ a Sylow $�p$-subgroup of $�G$. Recently, Malle, Navarro, and Tiep conjectured that the number of $�p$-Brauer characters of $�G$ coincides with that of the normalizer $��N_G��\left(�P�\right)��$ if and only if $�P$ is normal in $�G$. We reduce this conjecture to a question about finite simple groups and prove it for the prime $��p�=�2�$. As a by-product of our work, we prove a reduction theorem for the blockwise version of Alperin's lower bound on $�p$-Brauer characters and prove it for 2-blocks of maximal defect. This improves recent results obtained by Malle, Navarro, and Tiep.

We consider the fluctuations of the number of eigenvalues of $��n�\times �n�$ random normal matrices depending on a potential $�Q$ in a given set $�A$. The eigenvalues of random normal matrices are known to form a determinantal point process, and are known to accumulate on a compact set called the droplet under mild conditions on $�Q$. When $�A$ is a Borel set strictly inside the droplet, we show that the variance of the number of eigenvalues $�N_A^{�\left(�n�\right)�}$ in $�A$ has a limiting behavior given by

We develop novel methods for constructing nearly Hamilton cycles in sublinear expanders with good regularity properties, as well as new techniques for finding such expanders in general graphs. These methods are of independent interest due to their potential for various applications to embedding problems in sparse graphs. In particular, using these tools, we make substantial progress towards a 20-year-old conjecture of Verstraëte, which asserts that for any given graph $�F$, nearly all vertices of every $�d$-regular graph $�G$ can be covered by vertex-disjoint $�F$-subdivisions. This significantly extends previous work on the conjecture by Kelmans, Mubayi and Sudakov, Alon and Kühn and Osthus. Additionally, we present applications of our methods to two other problems.

A toric amplitude is a rational function associated with a simplicial polyhedral fan. The definition is inspired by scattering amplitudes in particle physics. We prove algebraic properties of such amplitudes and study the geometry of their zero loci. These hypersurfaces play the role of Warren's adjoint via a dual volume interpretation. We investigate their Fano schemes and singular loci using the nef cone and toric irrelevant ideal of the fan.

In this paper, we study the Chern–Hamilton energy functional on compact cosymplectic manifolds, fully classifying in dimension 3 those manifolds admitting a critical compatible metric for this functional. This is the case if and only if either the manifold is co-Kähler or if it is a mapping torus of the 2-torus by a hyperbolic toral automorphism and equipped with a suspension cosymplectic structure. Moreover, any critical metric has minimal energy among all compatible metrics. We also exhibit examples of manifolds with first Betti number $��b_1�⩾�2�$ admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.

Twistor spaces are certain compact complex three-folds with an additional real fibre bundle structure. We focus here on twistor spaces over $��P^2�\#�P^2�\#�P^2�$. Such spaces are either small resolutions of double solids or they can be described as modifications of conic bundles. The last type is the more special one: they deform into double solids. We give an explicit description of this deformation, in a more general context.