Tao has shown that hard summation (summing Fourier-type series by using terms in order of decreasing size of the Fourier coefficients) works for wavelets, but the present author has shown that it fails for classical Fourier series. This paper, which is intended for a general audience, exhibits the underlying ideas in the context of Haar and Walsh series, where many of the proofs simplify.

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 $j_p^*$.

The statement of the numerical values $\kappa$ and $z_0$ on page 167 of [1, Section 3] contained an error. The values that were actually used were (to nine decimal places): $$\begin{eqnarray*} \kappa &=& 2 \omega_1 -\int_1^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&1.859\,185\,431, \\[6pt] z_0&=&2\omega_3+\int_{-1}^\infty (4t^3-4t+1)^{- {1}/{2}}\,dt \\[4pt] &=&0.204\,680\,500+1.225\,694\,691i, \end{eqnarray*}$$ these being shifted, by the periods $2\omega_1$ and $2\omega_3$ respectively, compared with the values given in [1] (with $\omega_1=1.496\,729\,323$ and $\omega_3=1.225\,694\,691i$). With $\tau_0=\tau_1 =1$ and $\sigma (z)$ denoting the sigma function $\sigma (z;g_2,g_3)$ with invariants $g_2=4$, $g_3=-1$ associated with the elliptic curve given by equation (3.2), these values of $\kappa$ and $z_0$ yield $$\begin{eqnarray*} \sigma (\kappa )&=&1.555\,836\,426, \\[4pt] A=\frac{\tau_0}{\si (z_0 )} &=& 0.112\,724\,016-0.824\,911\,686i, \\[4pt] B=\frac{\si (\ka )\si (z_0 )\,\tau_1} {\si (z_0+\ka )\,\tau_0}&=&0.215\,971\,963+0.616\,028\,193i. \end{eqnarray*}$$ and the latter three values all agree with those stated in the paper (apart from rounding down the last digit in the imaginary part of $A$).

In this paper we compute the subgroup zeta functions of nilpotent groups of the form \[G_n := \langle x_1,\dots,x_{n},y_1,\dots,y_{n-1}|\;[x_i,x_n]=y_i,\;1\leq i \leq n-1\], all other [,] trivial 〉 and deduce local functional equations.

Let $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ be a $\mathbb Z_2$-graded simple Lie algebra. Fix a Borel subalgebra $\mathfrak b_0\subset\mathfrak{g}_0$. Let $\mathfrak a\subset\mathfrak{g}_1$ be a $\mathfrak b_0$-stable subalgebra. Then $\mathfrak a$ is automatically commutative. It is known that if $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety, then $G_0{\cdot}\mathfrak a$ is a spherical $G_0$-variety. We describe all $\mathbb Z_2$-gradings having the property that $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety for any $\mathfrak a$.

The rank of a vector space $\mathcal{A}$ of $n\times n$-matrices is by definition the maximal rank of an element of $\mathcal{A}$. The space $\mathcal{A}$ is called rank-critical if any matrix space that properly contains $\mathcal{A}$ has a strictly higher rank. This paper exhibits a sufficient condition for rank-criticality, which is then used to prove that the images of certain Lie algebra representations are rank-critical. A rather counter-intuitive consequence, and the main novelty in this paper, is that for infinitely many $n$, there exists an eight-dimensional rank-critical space of $n \times n$-matrices having generic rank $n-1$, or, in other words: an eight-dimensional maximal space of non-invertible matrices. This settles the question, posed by Fillmore, Laurie, and Radjavi in 1985, of whether such a maximal space can have dimension smaller than $n$. Another consequence is that the image of the adjoint representation of any semisimple Lie algebra is rank-critical; in both results, the ground field is assumed to have characteristic zero.

In a recent paper, Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general $H$-analogue of the Nakayama conjecture where $H$ is a set of positive integers.

In the introduction to [3] it is mistakenly claimed that Mislin's proof uses Carlsson's proof of Segal's Burnside ring conjecture. In fact, Mislin uses instead the work of Carlsson, Miller and Lannes on Sullivan's fixed-point conjecture, and work of Dwyer and Zabrodski [1]. There is, however, a proof by Snaith [2] that depends on a version of the Segal conjecture.

We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one of P. Brinkmann that one can determine whether two cyclic words in a free group are mapped to each other by some power of a given automorphism. We also solve the power conjugacy problem, and give an algorithm to recognize whether two given elements of a finitely generated free group are twisted conjugated to each other with respect to a given automorphism.

To form Riemann sums for generalized Riemann integrals, the domain of integration must be partitioned in a suitable manner. The existence of the required partitions is usually proved by a simple method of repeated bisection of the domain of integration. However, when the domain is the Cartesian product of infinitely many copies of the set of real numbers, this simple method of proof has frequently failed. A proof which works for infinite-dimensional spaces is provided here.

It is known that quadratic polynomials do not have real rational orbits of period four. By using a two-dimensional model for the quadratic family, this result is generalized for complex rational orbits.

A classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value $\infty$. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.

By considering coverings of surfaces by annuli, we extend previous results concerning the Nielsen kernel of topologically finite Riemann surfaces to arbitrary orbifolds. Specifically, we show that the length of a boundary loop in the Nielsen kernel is strictly greater than twice the length of the corresponding boundary loop of its orbifold, and that the infinite Nielsen kernel has empty interior.

Using reparametrizations of linear flows, we show that there exist area-preserving real analytic maps of the three-dimensional torus that are ‘mixing of all orders’ and do not enjoy the monotone shrinking target property. Prior to that, we give a short proof of a result of Kurzweil from 1955: namely, that a translation $T_\alpha$ of the torus $\mathbb{T}^d$ has the monotone shrinking target property if and only if the vector $\alpha$ is badly approximable (that is, of constant type).

Let $K$ be a convex body with boundary of class ${\mathcal{C}}^{\geq 4}$. We prove that, for a sufficiently small $\delta$, the floating body $K_{\delta}$ is homothetic to $K$ if and only if $K$ is an ellipsoid. The proof relies on properties of the affine curvature flow.

The notions of controlled truncations for operators in the Roe algebras $C^*(X)$ of a coarse space $(X, \cal{E})$ with uniformly locally finite coarse structure, and rank distributions on $(X, \cal{E})$ are introduced. It is shown that the controlled propagation operators in an ideal $I$ of $C^*(X)$ are exactly the controlled truncations of elements in $I$. It follows that the lattice of the ideals of $C^*(X)$ in which controlled propagation operators are dense is isomorphic to the lattice of all rank distributions on $(X, \cal{E})$. If $X$ is a discrete metric space with Yu's property A, then the ideal structure of the Roe algebra $C^*(X)$ is completely determined by the rank distributions on $X$.

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).

Professor Sir Henry Peter Francis Swinnerton-Dyer Bt., KBE, FRS, of the University of Cambridge, is awarded the Pólya Prize.