Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure.

Let be a field and [x, y] the ring of polynomials in two variables over . Let f ∈ [x, y] and consider the residue class ring R := [x, y]/f[x, y]. Our first aim is to study digit representations in R, i.e., we ask for which f each element of R admits a digit representation of the form d0 + d1x + ⋅ ⋅ ⋅ + dℓxℓ with digits di ∈ [y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base” x. In particular, we define and characterize digit representations for the ring S := ((x−1, y−1))/f((x−1, y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers of x in their “x-ary” representation. The translates of the fundamental domain induce a tiling of S. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose =q to be a finite field, these unions become finite.

Given a system of diagonal forms over ℚp, we ask how many variables are required to guarantee that the system has a nontrivial zero. We show that if the prime p satisfies p > (largest degree) − (smallest degree) + 1, then there is a bound on the sufficient number of variables which is a polynomial in the degrees of the forms.

We investigate the zeros of q-Bessel functions of the second and third types as well as those of the associated finite q-Hankel transforms. We derive asymptotic relations of the zeros of the q-Bessel functions by comparison with zeros of the theta function. The asymptotics of q-Bessel functions are also given. Zeros of finite q-Hankel transforms of q-summable functions are shown to be real and simple except for a finite number of possible non real zeros. Sufficient conditions are given to guarantee that all zeros are real. We give some applications concerning zeros of combinations of q-Bessel functions.

The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of , that with respect to the height are “very close” to a given subgroup Γ of finite rank of . Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form.

In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

In this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

In earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.

To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.

Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.

This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

The ‘hit problem’ of F. P. Peterson in algebraic topology asks for a minimal generating set for the polynomial algebra P(n) = 2[x1,. . ., xn] as a module over the Steenrod algebra 2. An equivalent problem is to find an 2-basis for the subring K(n) of elements f in the dual Hopf algebra D(n), a divided power algebra, such that Sqk(f)=0 for all k > 0. The Steenrod kernel K(n) is a 2GL(n,2)-module dual to the quotient Q(n) of P(n) by the hit elements +2P(n). A submodule S(n) of K(n) is obtained as the image of a family of maps from the permutation module Fl(n) of GL(n,2) on complete flags in an n-dimensional vector space V over 2. We use the Schubert cell decomposition of the flags to calculate S(n) in degrees , where λ1 > λ2 > ⋅⋅⋅ > λn ≥ 0. When λn = 0, we define a 2GL(n,2)-module map δ: Qd(n) → Q2d+n−1(n) analogous to the well-known isomorphism Qd(n) → Q2d+n(n) of M. Kameko. When λn−1 ≥ 2, we show that δ is surjective and δ*: S2d+n−1(n)→ Sd(n) is an isomorphism.

Stoimenow and Kidwell asked the following question: let K be a non-trivial knot, and let W(K) be a Whitehead double of K. Let F(a, z) be the Kauffman polynomial and P(v, z) the skein polynomial. Is then always max degzPW(K) − 1 = 2 max degzFK? Here this question is rephrased in more general terms as a conjectured relation between the maximum z-degrees of the Kauffman polynomial of an annular surface A on the one hand, and the Rudolph polynomial on the other hand, the latter being defined as a certain Möbius transform of the skein polynomial of the boundary link ∂ A. That relation is shown to hold for algebraic alternating links, thus simultaneously solving the conjecture by Kidwell and Stoimenow and a related conjecture by Tripp for this class of links. Also, in spite of the heavyweight definition of the Rudolph polynomial {K} of a link K, the remarkably simple formula {◯}{L#M} = {L}{M} for link composition is established. This last result can be used to reduce the conjecture in question to the case of prime links.

In this paper we prove an existence result for local and global isometric immersions of semi-Riemannian surfaces into the three dimensional Heisenberg group endowed with a homogeneous left–invariant Lorentzian metric. As a corollary, we prove a rigidity result for such immersions.

Complex hyperbolic packs are hypersurfaces of complex hyperbolic plane H2ℂ which may be considered as dual to the well known bisectors. In this paper we study the geometric aspects associated to packs.

Let f: M → S1 be a Morse map of a closed manifold M into the circle, where a Morse map is a smooth map with only nondegenerate critical points. In this paper, we classify such maps up to fold cobordism. In the course of the classification, we get several fold cobordism invariants for such Morse maps. We also consider a slightly general situation where the source manifold M has boundary and the map f restricted to the boundary has no critical points. Let g: (Rm, 0) → (R2, 0), m ≥ 2, be a generic smooth map germ, where the target R2 is oriented. Using the above-mentioned fold cobordism invariants, we show that the number of cusps with a prescribed index appearing in a C∞ stable perturbation of g, counted with signs, gives a topological invariant of g.

Professor Vicente Muñoz kindly informed us that there is an inaccuracy in Lemma 3.5 of [1]. The correct statement of Lemma 3.5 is now that the fundamental group π1(X′) of the manifold X′ is Z, since the monodromy coming from φ8 does not imply that g4 = g4−1. Therefore, what we have actually constructed in Section 3 of [1] is a closed non-formal 8-dimensional symplectic manifold with π1 = Z whose triple Massey product is non-zero, so that the simply-connectedness in Theorem 1.1 should be dropped. As far as we know, the existence of a simply connected closed non-formal 8-dimensional symplectic manifold whose triple Massey product is non-zero still remains an open question. All other main results, especially Theorem 1.2 and Corollary 1.3, in [1] are not affected by this mistake. Furthermore, the stability of the non-formality under a finite covering as in Subsection 3.3 holds in general. We want to thank Professor Muñoz for his careful reading.