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

This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point $�O$ inside an elliptic billiard table, one considers the family of rays emanating from $�O$ and the caustic $�{\Gamma }_n$ of the reflected family after $�n$ reflections off the ellipse, for each positive integer $�n$. It is known that $�{\Gamma }_n$ has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of $�{\Gamma }_n$, though we do not prove that these are the only cusps.

We study graphs with nonnegative Bakry–Émery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov–Hausdorff convergence, we prove that the space of bounded harmonic functions is finite dimensional and, as a corollary, the number of nonparabolic ends is finite.

We study obstructions to rationality on a nodal Fano threefold $�M$ that is a double cover of a smooth quadric threefold ramified over an intersection with a quartic threefold in $�P^4$. We prove that if $�M$ admits an Artin–Mumford obstruction to rationality then it lies in one of three explicitly described families. Conversely, a general element of any of these families admits an Artin–Mumford obstruction to rationality. Only one of these three families was known before; other two families of nodal Fano threefolds with obstructions to rationality are new.

We fix two mistakes in the paper “The average analytic rank of elliptic curves with prescribed torsion” and remove the moment conditions of the main theorem therein.

Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to $�k$ times the canonical divisor for $��k�\in �Z_{�⩾�1�}�$. In this article, we give a purely combinatorial criterion to determine if such a pair arises as the tropicalization of a pair consisting of a smooth algebraic curve over a non-Archimedean field with algebraically closed residue field of characteristic 0 together with an effective pluri-canonical divisor. To do so, we introduce tropical normalized covers as special instances of cyclic tropical Hurwitz covers and reduce the realizability problem for pluri-canonical divisors to the realizability problem for normalized covers. Our main result generalizes the work of Möller–Ulirsch–Werner on realizability of tropical canonical divisors and incorporates the recent progress on compactifications of strata of $�k$-differentials in the work of Bainbridge–Chen–Gendron–Grushevsky–Möller.

There are a number of well-known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3-colouring such that for every vertex $�v$, at most half of the out-neighbours of $�v$ have the same colour as $�v$. As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every $�d$, all but finitely many $�d$-regular graphs have a partition into two non-empty parts such that for every vertex $�v$, at least half of the neighbours of $�v$ lie in the same part as $�v$. We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for Erdős–Rényi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of ‘exceptional vertices’. Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a personality-changing scheme: we ‘forget’ certain information based on the state of a Markov chain, giving us more independence to work with.

We extend the recent paradigm “Integrability via Geometry” from dimensions 3 and 4 to higher dimensions, relating dispersionless integrability of partial differential equations to curvature constraints of the background geometry. We observe that in higher dimensions on any solution manifold, the symbol defines a vector distribution equipped with a subconformal structure, and the integrability imposes a certain compatibility between them. In dimension 5, we express dispersionless integrability via the vanishing of a certain curvature of this subconformal structure. We also obtain a “master equation” governing all second-order dispersionless integrable equations in 5D, and count their functional dimension. It turns out that the obtained background geometry is parabolic of the type $��(�A_3�,�P_{13}�)�$. We provide its Cartan-theoretic description and compute the harmonic curvature components via the Kostant theorem. Then, we relate it to 3D projective and 4D conformal geometries via twistor theory, discuss symmetry reductions and nested Lax sequences, as well as give another interpretation of dispersionless integrability in 5D through Levi-degenerate CR structures in 7D.