We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

We denote by $B_Y$ the unit ball of a normed linear space $Y$.

Definition. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P{:}\,Y\,{\to}\,X$ such that $P(B_Y)\,{\subset}\,A$.

Minimal-volume sufficient enlargements are determined for two-dimensional spaces. The main results are:

(i) Each minimal-volume sufficient enlargement for a two-dimensional space is a parallelogram or a hexagon.

(ii) If a two-dimensional normed space $X$ has a minimal-volume sufficient enlargement that is not a parallelogram, then $B_X$ is linearly equivalent to the regular hexagon.

In this paper we describe a group G such that for any simple coefficients A and for any i > 0, Hi(G; A) and Hi(G; A) are zero. Other groups with this property have been found by Baumslag and Gruenberg (1). The group G in this paper has cohomological dimension 2 (that is Hi(G; A) = 0 for any i > 2 and any G-module A). G is the fundamental group of an open aspherical 3-dimensional manifold L, and is not finitely generated. The only non-trivial part of this paper is to prove that the fundamental group of the 3-manifold L, which we shall construct, is not the identity group.

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

In an intersting article, ‘Bezout rings and their subrings’ (2) P. M. Cohn has introduced a construction which may also be applied in a slightly different situation, i.e. the imbedding of a Krull ring in a Dedekind ring.

The hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalise this to the simplicial volume of link complements by analysing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the volume in terms of twists of various lengths for links.

In this paper we continue the study of elliptic curves defined over a quadratic field with good reduction at primes not dividing 2 begun in [9] (referred to as I). We extend the results of I to show that such a curve must have a point of order 2 also defined over when d = −7, −3, −2, −1, 2, 3 or 5 and list all such curves over .

We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n, 1), n > 0. Lehmer's question for Perron polynomials is equivalent to one about generalized growth rates of words under free group endomorphisms.

In this paper, we use the results of (3) to show that a condition which is formally much weaker than that given by Sherman in (11) is necessary and sufficient for the commutativity of a C*-algebra. The basic idea behind the proof is then used again to obtain a characterization in order-theoretic terms of the self-adjoint elements in the centre of a C*-algebra. We shall use the notation and terminology of (3).

In this paper we will study 2-knots in the 4-sphere S4 which are obtained by roll-spinning 1-knots in the 3-sphere S3. The process of spinning was introduced by Artin[2] and generalized to twist-spinning by Fox [7] and Zeeman [20]. In [7] Fox introduced another variation of the spinning process, called roll-spinning. He only showed that the roll-spun figure-eight knot cannot be obtained by twist-spinning the figure-eight knot. Finally Litherland [14] described the general process of deform-spinning, and he gave a precise definition of roll-spinning. We remark that Fox's roll-spinning is not the same as Litherland's: Fox's roll-spun figure-eight knot is the symmetry-spun figure-eight knot in terms of [14].

Assuming CH, the continuum hypothesis, holds we show, by completing an attack first discovered by Roy Davies, that for each α between 0 and 1 there is a subring, in fact a subfield, of R with Hausdorff dimension α.

The aim of the paper is to exhibit the structure of products of flows that are suspensions of diffeomorphisms and show their relationship with suspensions of product diffeomorphisms.

The problem is important for applications of the following type. Suppose we have two dynamical systems each with a stable attractor. If the systems are brought into contact, as, for example, in the way a forced oscillation is imposed upon a natural oscillation in synchronization theory, then this is represented by taking the product flow to find the resulting behaviour of the joint system. However, the product of two stable attractors is no longer stable in general, and therefore it is important to analyse why, in order to understand what new stable attractors can arise from perturbations.

Maximum and minimum principles for certain ordinary differential equations of order 2m are derived in a unified manner from the theory of complementary variational principles for multiple operator equations. The minimum principle is known in the literature, but the maximum principle appears to be new.

Throughout we let ($T,{\frak m},k$) denote a commutative Noetherian local ring with maximal ideal ${\frak m}$ and residue field $k$. We let $I\,{\subseteq}\,T$ be an ideal generated by a regular sequence of length $c$ and set $R\,{\colonequal}\,T/I$. In the important paper [1], Avramov addresses the following question. Given a finitely generated $R$-module $M$, when does $M$ have finite projective dimension over a ring of the form $T/J$, where $J$ is generated by part (or all) of a set of minimal generators for $I$? He gives a fairly complete answer to this question that is expressed in terms of the geometry of varieties in affine space defined by annihilators of certain graded modules derived from resolutions over $R$. In an attempt to understand these ideas more fully, we became interested in the idea that one might answer the question at hand by using data about $M$ (orits syzygies) coming from $T$, in particular, information gleaned from various Fitting ideals defined over $T$. The following theorem from Section 3 is one of our main results. We use $\Fitt_T(M)$ to denote the Fitting ideal of $M$.

Let $G$ be a universal Chevalley group defined over an algebraically closed field $F$ of arbitrary characteristic. In this paper we investigate the restrictions to parabolic subgroups of $G$ of the irreducible $FG$-modules corresponding to minuscule highest weights via a combinatorial scheme of “chamber systems” indexed over certain cosets in the Weyl group.

Let A be a local complete intersection ring. Let M, N be two finitely generated A-modules and I an ideal of A. We prove that

$$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$

$$\begin{linenomath}\begin{subeqnarray*} {\rm Ass}_A\left({\rm Ext}_A^{2i}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \\ {\rm Ass}_A\left({\rm Ext}_A^{2i+1}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \end{subeqnarray*}\end{linenomath}$$

It is shown that the differentiability of a space-time is implied by that of its Riemann tensor, assuming a priori only boundedness of the first derivations of the metric. Consequently all the results on space-time singularities proved in earlier papers by the author hold true in C2− space-times.

Consider the series

and let

The series

is not necessarily convergent. Suppose that its Poisson sum exists. This we denote by

The expression

when existent will be said to be the sum (, p) of the series (1).

Let $G$ be a group, $g$ an element of $G$ and $n$ a natural number, $n\geqslant 2$. In [8] B.H. Neumann considered the equation $r_0(t)\deq t^ng=1$ and proved its solvablility in an overgroup. A natural generalisation of this equation is the equation $r(t)=1$, where $r(t)$ is defined as follows: let $W\in G\ast\langle t\rangle$ be a cyclically reduced word of length at least 2 which is not a proper power in $G\ast\langle t\rangle$. Define ${(\ast)}\hspace{4.8cm}r(t)\deq r_0(W)=W^ng$. In this paper we show that for $n\geqslant 3$ the equation $r(t)=1$ is solvable in an overgroup with some structural properties, under certain conditions on $g$ and $W$. It is conjectured in [3, p. 131] that this equation is solvable over an arbitrary group, without conditions on $W$ or $g$.