A notion of Property (T) is defined for an arbitrary unital C*-algebra $A$ admitting a tracial state. This is extended to a notion of Property (T) for a pair $(A,B),$ where $B$ is a C*-subalgebra of $A.$ Let $\Gamma$ be a discrete group and ${C}^*_{\rm r}(\Gamma)$ its reduced algebra. We show that $C^*_{\rm r}(\Gamma)$ has Property (T) if and only if the group $\Gamma$ has Property (T). More generally, given a subgroup $\Lambda$ of $\Gamma$, the pair $(C^*_{\rm r}(\Gamma),C^*_{\rm r}(\Lambda)) $ has Property (T) if and only if the pair of groups $(\Gamma, \Lambda)$ has Property (T).

Every complex manifold carries a canonical orientation, and it is natural to wonder when the underlying topological or smooth manifold carries a complex structure compatible with the other orientation. In [2], Beauville raised this question for compact complex surfaces. He noted that there are a lot of examples, like products of curves, or Hopf surfaces, where the underlying manifold admits an orientation-reversing selfdiffeomorphism. This implies that the signature is zero. Beauville asked if there are any examples of non-zero signature.

We show that every μ-constant family of isolated hypersurface singularities of type F(x, t) =f(x)+tg(x), where t is a parameter, is topologically trivial. The proof uses only the curve selection lemma, and hence, for an appropriately translated statement, also works over the reals and for some families of non-isolated singularities. Some applications to study the singularities at infinity of complex polynomials are given.

Hodge theory for a smooth algebraic curve includes both the Hodge structure (period matrix) on cohomology and the use of that Hodge structure to study the geometry of the curve, via the Jacobian variety. Hodge extended the theory of the period matrix to smooth algebraic varieties of any dimension, defining in general a Hodge structure on the cohomology of the variety. He gave a few applications to the geometry of the variety, but these did not attain the richness of the Jacobian variety. In recent years, Hodge theory has been successfully extended to arbitrary varieties, and to families of varieties. In this expository paper, some of these developments are reviewed, with special emphasis on instances where these extensions can be used to study the geometry – especially the algebraic cycles – on the variety.

First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,jR2ij [les ] r2/(n−1), where [sum ]i,jR2ij is the squared norm of the Ricci curvature tensor.

The existence of Salem numbers of every trace is proved; the nontrivial part of this result concerns Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction, using pairs of polynomials whose zeros interlace on the unit circle, of polynomials of specified negative trace having one factor a Salem polynomial, with any other factors being cyclotomic. The second is an upper bound for the exponent of a maximal torsion coset of an algebraic torus in a variety defined over the rationals. This second result, which may be of independent interest, has enabled the construction to be refined so as to avoid cyclotomic factors, giving a Salem polynomial of any specified trace, with a trace-dependent bound for its degree. It is also shown how this new interlacing construction can be easily adapted to produce Pisot polynomials, giving a simpler, and more explicit, construction for Pisot numbers of arbitrary trace than was previously known.

Let Ω be a bounded connected open set in ℝN, N[ges ]2, and let −ΔΩ[ges ]0 be the Dirichlet Laplacian defined in L2(Ω). Let λΩ>0 be the smallest eigenvalue of −ΔΩ, and let ϕΩ>0 be its corresponding eigenfunction, normalized by ∥ϕΩ∥2=1. For sufficiently small ε>0 we let R(ε) be a connected open subset of Ω satisfying

formula here

Let −Δε[ges ]0 be the Dirichlet Laplacian on R(ε), and let λε>0 and ϕε>0 be its ground state eigenvalue and ground state eigenfunction, respectively, normalized by ∥ϕε∥2=1. For functions f defined on Ω, we let Sεf denote the restriction of f to R(ε). For functions g defined on R(ε), we let Tεg be the extension of g to Ω satisfying

formula here

It is shown that if (vj)n1 is a sequence of norm 1 vectors in a complex Hilbert space and (tj)n1 is a sequence of non-negative numbers satisfying

[formula here]

then there is a unit vector z for which

[formula here]

for every j. The result is a strong, complex analogue of the author's real plank theorem.

Let (Pn) be a sequence of polynomials which converges with a geometric rate on some arc in the complex plane to an analytic function. It is shown that if the sequence has restricted growth on a closed plane set E which is non-thin at ∞, then the limit function has a maximal domain of existence, and (Pn) converges with a locally geometric rate on this domain. If (snk) is a sequence of partial sums of a power series, a similar growth restriction on E forces the power series to have Ostrowski gaps. Moreover, the requirement of non-thinness of E at ∞ is necessary for these conclusions.

The well-known inequality of Hardy for Fourier coefficients of functions f(t)∼[sum ]∞n=−∞bneint in the real Hardy space is [sum ]∞n= −∞[mid ]bn[mid ]/([mid ]n[mid ]+1)<∞. We shall establish analogues of this inequality for the Hermite function expansions and also for the Laguerre function expansions.

Given a set $\pi$ of odd primes and a $\pi$-solvable finite group $G$, one can choose three (not necessarily distinct) Hall $\pi$-subgroups of $G$ whose intersection is $O_{\pi}(G)$. This statement relies on a result concerning long orbits in coprime actions of groups of odd order.

We derive an identity connecting a theta function and a sum of Lambert series. As a consequence of this identity, we deduce a number of results of Jacobi, Dirichlet, Lorenz, Ramanujan and Rademacher.

Suppose that m is a positive integer, τ=(τ1, …, τm) ∈ℝm+ is a vector of strictly positive numbers, and Q is an infinite set of positive integers. Let WQ(m; τ) be the set

formula here

In this paper we obtain the Hausdorff dimension of this set. We also consider a generalization of the set WQ(m; τ), where the error terms q−τi in the inequalities are replaced by ψi(q), for general functions ψi satisfying a certain condition at infinity.

A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

Let B be a complete ccc Boolean algebra and let $\tau_s$ be the topology on B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonempty open set in $(B,\tau_s)$ is topologically dense.

2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.

3. The topological space $(B,\tau_s)$ is sequentially compact if and only if the generic extension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but not independent reals.

Semigroups (with zero) and inverse semigroups (with zero) are constructed from subshifts. The semigroups and inverse semigroups that are obtained by this construction are characterized and classified.

Under very general conditions, a proof is given, via Morse theory, of the existence of solutions for asymptotically ‘linear’ $p$-Laplacian equations, where the asymptotic limit may be greater than the second eigenvalue. The existence of nonzero solutions is also considered.

It is shown that every ascending HNN extension of a finitely generated free group is Hopfian. An important ingredient in the proof is that under certain hypotheses on the group H, if G is an ascending HNN extension of H, then cd(G) = cd(H) + 1.

Let $p$ be a prime for which the congruence group $\Gamma_0(p)^*$ is of genus zero, and let $j_p^*$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on $\Gamma_0(4p)$ which satisfies some congruence condition on its Fourier coefficients. We interpret $f$ as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to $j_p^*$ and relate them to Zagier's trace formula for the singular values of $j_p^*$.

For arbitrary infinite directed graphs $E$, the characterisation of the (not necessarily simple) Cuntz–Krieger algebras $C^*(E)$ which are purely infinite in the sense of Kirchberg–Rørdam is given. It is also shown that $C^*(E)$ has real rank zero if and only if the graph $E$ satisfies Condition (K).