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

We provide a new description of the hom functor on weak $�\omega$-categories, and show that it admits a left adjoint that we call the suspension functor. We then show that the hom functor preserves the property of being free on a computad, in contrast to the hom functor for strict $�\omega$-categories. Using the same technique, we define the opposite of an $�\omega$-category with respect to a set of dimensions, and show that this construction also preserves the property of being free on a computad. Finally, we show that the constructions of opposites and homs commute.

To begin the paper, we revisit a cohomological localization result of Jones–Rawnsley which was subsequently improved by Farber, further generalizing the result. We then proceed to improve a previous result of the author on complete intersections of dimension $��8�k�$ with a Hamiltonian $�S^1$-action in two directions. First, in dimension 8 we remove the assumption on the fixed-point set. Second, in any dimension we prove the result under an analogous assumption on the fixed-point set. We also give some applications toward the unimodality of Betti numbers of symplectic manifolds having a Hamiltonian $�S^1$-action, and discuss the relation to symplectic rationality problems.

We study the geometry and arithmetic of the curves $��C�:�y^3�=�x^4�+�a�x^2�+�b�$ and their associated Prym abelian surfaces $�P$. We prove a Torelli-type theorem in this context and give a geometric proof of the fact that $�P$ has quaternionic multiplication by the quaternion order of discriminant 6. This allows us to describe the Galois action on the geometric endomorphism algebra of $�P$. As an application, we classify the torsion subgroups of the Mordell–Weil groups $��P�\left(�Q�\right)�$, as both abelian groups and $��\mathrm{End}�\left(�P�\right)�$-modules.

Let $�p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as $�p^{�1�/�4�+�\kappa �}$ for any $��\kappa �>�0�$. Our methods capitalize on the relationship between characters mod $�p$ and characters over finite field extensions as well as bounds on the multiplicative energy of sets in products of finite fields.

We investigate Riemannian manifolds $��(�M^n�,�g�)�$ whose curvature operator of the second kind $�\mathring{R}$ satisfies the condition

The Carrollian fluid equations arise from the equations for relativistic fluids in the limit as the speed of light vanishes, and have recently experienced a surge of interest in the theoretical physics community in the context of asymptotic symmetries and flat-space holography. In this paper, we initiate the rigorous systematic analysis of these equations by studying them in one space dimension in the $�C^1$ setting. We begin by proposing a notion of isentropic Carrollian equations, and use this to reduce the Carrollian equations to a $��2�\times �2�$ system of conservation laws. Using the scheme of Lax, we then classify when $�C^1$ solutions to the isentropic Carrollian equations exist globally, or blow up in finite time. Our analysis assumes a Carrollian analogue of a constitutive relation for the Carrollian energy density, with exponent in the range $��\gamma �\in �(�1�,�3�]�$.

In the setting of length PI spaces satisfying a suitable deformation property, it is known that each isoperimetric set has an open representative. In this paper, we construct an example of a length PI space (without the deformation property) where an isoperimetric set does not have any representative whose topological interior is nonempty. Moreover, we provide a sufficient condition for the validity of the deformation property, consisting in an upper Laplacian bound for the squared distance functions from a point. Our result applies to essentially nonbranching $��\mathrm{MCP}�(�K�,�N�)�$ spaces, thus in particular to essentially nonbranching $��\mathrm{CD}�(�K�,�N�)�$ spaces and to many Carnot groups and sub-Riemannian manifolds. As a consequence, every isoperimetric set in an essentially nonbranching $��\mathrm{MCP}�(�K�,�N�)�$ space has an open representative, which is also bounded whenever a uniform lower bound on the volumes of unit balls is assumed.