It is well known in Kähler geometry that the infinite-dimensional symmetric space of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds of Bergman metrics of height k. Then it is natural to ask whether geodesics in can be approximated by Bergman geodesics in . For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.

Given k ≥ 2, we construct a (2k − 2)-parameter family of properly embedded minimal surfaces in ℍ2 × ℝ invariant by a vertical translation T, called saddle towers, which have total intrinsic curvature 4π(1 − k), genus zero and 2k vertical Scherk-type ends in the quotient by T. Each of those examples is obtained from the conjugate graph of a Jenkins–Serrin graph over a convex polygonal domain with 2k edges of the same (finite) length. As limits of saddle towers, we obtain properly embedded minimal surfaces, called minimal k-noids, which are symmetric with respect to a horizontal slice (in fact they are vertical bi-graphs) and have total intrinsic curvature 4π(1 − k), genus zero and k vertical planar ends.

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

Schneider posed the problem of determining the maximal value of the affine invariant ∣ΠK∣/∣K∣d−1, where ΠK is the projection body of the d-dimensional convex body K. Some three-dimensional conjectures of Brannen, related to Schneider’s problem, are confirmed. Namely, we determine the maximal value of ∣ΠK∣/∣K∣2 in the class of three-dimensional zonoids, cones and double cones. Equality cases are, also, investigated. Moreover, results related to a conjecture of Petty, concerning the minimal value of the above quantity, are obtained. In particular, we provide a negative answer to a question of Martini and Mustafaev.

The tools of zero biasing are adapted to yield a general result suitable for analysing the behaviour of certain growth processes. The main theorem is applied to prove a central limit theorem, with explicit error terms in the L1 metric, for a natural statistic of the Jack measure on partitions.

The level l Fock space admits canonical bases and . They correspond to and -module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in ℕ[v]. Restriction to the highest-weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki–Koike algebras.

The set of non-constant holomorphic mappings between two given compact Riemann surfaces of genus greater than 1 is always finite. This classical statement was made by de Franchis. Furthermore, bounds on the cardinality of the set depending only on the genera of the surfaces have been obtained by a number of mathematicians. The analysis is carried over in this paper to the case of Riemann surfaces of finite analytic type (i.e. compact Riemann surfaces minus a finite set of points) so that the finiteness result, together with a crude but explicit bound depending only on the topological data, may be extended for the number of holomorphic mappings between such surfaces.

An old result by Shearer relates the Lovász local lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard-core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lovász local lemma. As an application we obtain tighter bounds on conditions for the existence of Latin transversal matrices.

We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is, the Coxeter groups of types An, Dn and En, and show that these are naturally arrived at purely through consideration of certain natural actions of symmetric groups. We go on to use these techniques to provide explicit representations of these groups.

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.