We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number–theoretic argument.

We study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

We prove Davis and Garsia Inequalities for dyadic perturbations of Hardy martingales and show that those inequalities play a substantial role in the proof of Bourgain's [1] embedding L1 ↪ L1/H10. This paper continues [17] on Davis and Garsia Inequalities (DGI).

Let C be a symmetric convex body of volume 1 in ${\mathbb R}^n$. We provide general estimates for the volume and the radius of C ∩ U(C) where U is a random orthogonal transformation of ${\mathbb R}^n$. In particular, we consider the case where C is in the isotropic position or C is the volume normalized Lq-centroid body Zq(μ) of an isotropic log-concave measure μ on ${\mathbb R}^n$.

For a nonsingular projective threefold of general type X over the field of complex numbers, we show that the fourth pluricanonical map ϕ4 is not birational onto its image if and only if X is birationally fibred by (1,2)-surfaces, provided that vol(X) ≥ 303. We also have similar characterization of birationality of ϕ3.

We study the question of when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.

We establish an asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For ℜ(s) = σ satisfying (log T)−1/3+ε ⩽ (2σ − 1) ⩽ (log log T)−2, we show that the number of zeros of ζ′(s) with imaginary part between zero and T and real part larger than σ is asymptotic to T/(2π(σ−1/2)) as T → ∞. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for σ in this range the zeros of ζ′(s) are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

One way of expressing the self-duality $A\cong {\rm Hom}(A,\mathbb{C})$ of Abelian groups is that their character tables are self-transpose (in a suitable ordering). In this paper we extend the duality to some noncommutative groups considering when the character table of a finite group is close to being the transpose of the character table for some other group. We find that groups dual to each other have dual normal subgroup lattices. We show that our concept of duality cannot work for non-nilpotent groups and we describe p-group examples.

Let F:${\mathbb R}$n→${\mathbb R}$n be a polynomial local diffeomorphism and let SF denote the set of not proper points of F. The Jelonek's real Jacobian Conjecture states that if codim(SF) ≥ 2, then F is bijective. In this work we prove a weak version of such Conjecture, but for more general maps than polynomial, namely: the semi-algebraic maps.

In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu's theorem to imaginary quadratic fields, where we prove the existence of “bubbles” containing arbitrarily many primes which are all, up to units, congruent to a modulo q.

In this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].

However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

We develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms.

As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

This paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form

\begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*}

$\log ((9+\sqrt{105})/2)/\log 16\approx .8166$

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer–Kan equivalence between the simplicial localisations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.

We study surface links whose link groups are free abelian, and construct various stimulating and highly non-trivial examples of such surface links.

Let $\mathbb{K}$ be a number field, Γ a finitely generated subgroup of $\mathbb{K}$*, for instance the unit group of $\mathbb{K}$, and κ>0. For an ideal $\mathfrak{a}$ of $\mathbb{K}$ let indΓ($\mathfrak{a}$]></alt-text></inline-graphic>) denote the multiplicative index of the reduction of Γ in <inline-graphic name="S0305004114000206_inline3"><alt-text><![CDATA[$(\mathcal{O}_\mathbb{K}/\mathfrak{a})$* (whenever it makes sense). For a prime ideal $\mathfrak{p}$ of $\mathbb{K}$ and a positive integer γ let $\mathcal{I}_\gamma^\kappa(\mathfrak{p})$ be the average of ${ind}_{\langle a_1,\dots,a_\gamma\rangle}(\mathfrak{p})^\kappa$ over all tupels $(a_1,\dots,a_\gamma)\in{(\mathcal{O}_\mathbb{K}/\mathfrak{p})^*}^\gamma$. Motivated by a problem of Rohrlich we prove, partly conditionally on fairly standard hypotheses, lower bounds for $\sum_{\mathcal{N}{\mathfrak{a}\leq x}{ind}_{\Gamma}({\mathfrak{a})^\kappa$ and asymptotic formulae for $\sum_{\mathcal{N}\mathfrak{p} \leq x} {\mathcal{I}_{\gamma}^\kappa({\mathfrak{p})$.

We compute the asymptotic behaviour of the number of partitions of large vectors (n1, n2) of ℤ+2 in the critical regime n1 ≍ √n2 and in the subcritical regime n1 = o(√n2). This work completes the results established in the fifties by Auluck, Nanda and Wright.

The theory of twist maps is applied to prove the existence of many harmonic and sub-harmonic solutions for certain Newtonian systems of differential equations. The method of proof leads to very precise information on the oscillatory properties of these solutions.

It follows from the work of Artin and Hooley that, under assumption of the generalised Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is a primitive root modulo q can be written as an infinite product ∏p δp of local factors δp reflecting the degree of the splitting field of Xp - r at the primes p, multiplied by a somewhat complicated factor that corrects for the ‘entanglement’ of these splitting fields.

We show how the correction factors arising in Artin's original primitive root problem and several of its generalisations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors.

The method not only applies in the setting of Galois representations of the multiplicative group underlying Artin's conjecture, but also in the GL2-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.

We establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.