The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By definition, these zeta functions are rational functions in one variable, and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non-degenerate with respect to its Newton polyhedron, we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; we give a similar result concerning the global zeta function. Moreover, for any f we show that poles of maximal possible order are always of the form −1/N with N a positive integer.

This paper concerns rigidity of the mapping class groups. It is shown that any homomorphism $\varphi\,{:}\,\mcg_g\,{\to}\,\mcg_h$ between mapping class groups of closed orientable surfaces with distinct genera $g\,{>}\,h$ is trivial if $g\,{\geq}\, 3$, and has finite cyclic image for all $g\,{\geq}\, 1$.

Some implications are drawn for more general homomorphs of these groups.

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.

We characterize the extremal functions for Heisenberg type inequalities of the form

(formula here)

The method also yields an alternative and elementary derivation of the sharp constant in Nash's inequality, originally found by Carlen and Loss.

Necessary and sufficient conditions for Lefschetz fibrations over D2 and S2 to carry spin structures are presented. The conditions are given in terms of vanishing cycles and homology sections.

This paper aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. It emphasizes the relationships between the different descriptions of these spaces. The final two sections of the paper outline results of the author and Kirwan on the cohomology rings of certain of the spaces described earlier (formulas for intersection numbers that were discovered by Witten (Commun. Math. Phys. 141 (1991) 153–209 and J. Geom. Phys. 9 (1992) 303–368) and given a mathematical proof by the author and Kirwan (Ann. of Math. 148 (1998) 109–196)).

We show that the zero curvature limit of the space of hyperbolic monopoles gives the Euclidean monopoles, settling a conjecture of Atiyah. We also study the infinite curvature limit of the space of hyperbolic monopoles, and show that the associated rational maps appear explicitly here.

The famous Burnside–Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latter, Schur rings are introduced over a finite commutative ring, and an analogue of this statement is proved for them. Also, the finite local commutative rings are characterized in permutation group terms.

Throughout this paper, we shall let Σ be a subset of [0, 1] having cardinality N. We shall consider Σ to be a set of slopes, and for any s∈Σ, we shall let es be the unit vector of slope s in ℝ2. Then, following [7], we define the maximal operator on ℝ2 associated with the set Σ by

formula here

The history of the bounds obtained on [Mfr ]0Σ is quite curious. The earliest study of related operators was carried out by Cordoba [2]. He obtained a bound of C√(1+logN) on the L2 operator norm of the Kakeya maximal operator over rectangles of length 1 and eccentricity N. This operator is analogous to [Mfr ]0Σ with

formula here

However, for arbitrary sets Σ, the best known result seems to be C(1+logN). This follows from Lemma 5.1 in [1], but a point of view which produces a proof appears already in [8]. However, in this paper, we prove the following.

Despite a true antipathy to the subject, Hardy contributed deeply to modern probability. His work with Ramanujan begat probabilistic number theory. His work on Tauberian theorems and divergent series has probabilistic proofs and interpretations. Finally, Hardy spaces are a central ingredient in stochastic calculus. This paper reviews his prejudices and accomplishments, through these examples.

Let $ S_n(x)=\sum_{k=1}^{n}({\sin(kx)})/{k} $ be Fejér's sine polynomial. We prove the following statements.

1. The inequality $ (S_n(x+y))^{\alpha} (x+y)^{\beta} \leq (S_n(x))^{\alpha}x^{\beta}+ (S_n(y))^{\alpha}y^{\beta} {(n\in \mathbb{N}; \; \alpha, \beta \in \mathbb{R})}$ holds for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\geq 0$ and $\alpha+\beta \leq 1$.

2. The converse of the above inequality is valid for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\leq 0$ and $\alpha+\beta \geq 1$.

3. For all $n\in\mathbb{N}$ and $x,y \in [0,\pi]$ we have $ 0\leq S_n(x)+S_n(y)- S_n(x+y)\leq \frac{3}{2}\sqrt{3}.$ Both bounds are best possible.

Let [Gfr ] be a finite-dimensional semisimple Lie algebra over the complex numbers. Let A be the finite-dimensional algebra of a (regular or singular) block of the BGG-category [Oscr ] . By results of Soergel, A has a combinatorial description in terms of a subalgebra C0 of the coinvariant algebra C. König and Mazorchuk have constructed an embedding from C0-mod into the category [Fscr ](Δ) of A-modules having a Verma flag. This is the main tool for the classification of [Fscr ] (Δ) into finite, tame and wild representation types presented here. As a consequence a classification of A-mod into finite, tame and wild representation types is obtained, thus re-proving a recent result of Futorny, Nakano and Pollack.

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m<n$, then a homomorphism ${\rm Aut}(F_n)\to {\rm Aut}(F_m)$ can have image of cardinality at most 2. More generally, this is true of homomorphisms from ${\rm Aut}(F_n)$ to any group that does not contain an isomorphic image of the symmetric group $S_{n+1}$. Strong restrictions are also obtained on maps to groups that do not contain a copy of $W_n=({\mathbb Z}/2)^n\rtimes S_{n}$, or of ${\mathbb Z}^{n-1}$. These results place constraints on how ${\rm Aut}(F_n)$ can act. For example, if $n\ge 3$, any action of ${\rm Aut}(F_n)$ on the circle (by homeomorphisms) factors through ${\rm det} : {\rm Aut}(F_n)\to{\mathbb Z}_2$.The research of the first author was supported by an EPSRC Advanced Fellowship. The research of the second author is supported in part by NSF grant DMS-9307313.

The Manin–Mumford conjecture asserts that if K is a field of characteristic zero, C a smooth proper geometrically irreducible curve over K, and J the Jacobian of C, then for any embedding of C in J, the set C(K) ∩ J(K)tors is finite. Although the conjecture was proved by Raynaud in 1983, and several other proofs have appeared since, a number of natural questions remain open, notably concerning bounds on the size of the intersection and the complete determination of C(K) ∩ J(K)tors for special families of curves C. The first half of this survey paper presents the Manin–Mumford conjecture and related general results, while the second describes recent work mostly dealing with the above questions.

We extend a recent result on the existence of wandering domains of polynomial functions defined over the $p$-adic field ${\mathbb{C}_p}$ to any algebraically closed complete non-archimedean field ${\mathbb{C}_K}$ with residue characteristic $p>0$. We also prove that polynomials with wandering domains form a dense subset of a certain one-dimensional family of degree $p+1$ polynomials in ${\mathbb{C}_K}[z]$.

Algebraic higher-rank actions on connected groups are often remarkably rigid in their topological and measurable structure. In contrast to this, the author of this paper constructs uncountably many closed invariant sets and uncountably many invariant measures with positive entropy for irreducible algebraic ${\mathbb{Z}^d}$-actions on zero-dimensional groups.

Three short results are given concerning the elliptic Harnack inequality, in the context of random walks on graphs. The first is that the elliptic Harnack inequality implies polynomial growth of the number of points in balls, and the second that the elliptic Harnack inequality is equivalent to an annulus-type Harnack inequality for Green's functions. The third result uses the lamplighter group to give a counter-example concerning the relation of coupling with the elliptic Harnack inequality

Using techniques from geometric measure theory and descriptive set theory, we prove a general result concerning sets in the plane which meet each straight line in exactly two points. As an application, we show that no such ‘two-point’ set can be expressed as the union of countably many rectifiable sets together with a set with Hausdorff 1-measure zero. Also, as a corollary, we show that no analytic set can be a two-point set provided that every purely unrectifiable set meets some line in at least three points. Some generalizations are given to ‘n-point’ sets and some other geometric constructions.

Given an oriented manifold and its immersion in a euclidean space, we compute the oriented cobordism class of the manifold of [sum ]1r singular points of the projection of the immersion to a hyperplane. For immersions of non-oriented manifolds, we show that the cobordism class of the domain manifold determines those of all [sum ]1r singularity manifolds of the hyperplane projection. Finally, we investigate the possible (algebraic) number of cusps (that is, [sum ]1,1 singular points) of generic maps of oriented 4t-manifolds in R6t−1.

Klass has established decoupling inequalities for discrete time processes with independent increments. We extend his result to continuous time processes with independent increments. This extension enables us to obtain BDG inequalities for exponential functions instead of moderate functions.