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.

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\backslash G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion that involves root systems.

A subset $A$ of the integers is said to be sum-free if there do not exist elements $x,y,z \,{\in}\, A$ with $x \,{+}\, y \,{=}\, z$. It is shown that the number of sum-free subsets of $\{1,\ldots,N\}$ is $O(2^{N/2})$, confirming a well-known conjecture of Cameron and Erdős.

Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non-scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand–Kirillov dimension 2 over a field $F$ all have transcendence degree 1 over $F$. Finally, centralizers of elements in a finitely graded Goldie domain of Gelfand–Kirillov dimension 2 over an algebraically closed field are considered. In this case, it is shown that the centralizer of a non-scalar element is an affine commutative domain of Gelfand–Kirillov dimension 1.

Let $p$ be a monic complex polynomial of degree $n$, and let $K$ be a measurable subset of the complex plane. Then the area of $p(K)$, counted with multiplicity, is at least $\pi n ( \Area (K)/\pi )^n$, and the area of the pre-image of $K$ under $p$ is at most $\pi^{1-1/n} (\Area (K) )^{1/n}$. Both bounds are sharp. The special case of the pre-image result in which $K$ is a disc is a classical result, due to Pólya. The proof is based on Carleman's classical isoperimetric inequality for plane condensers.

Let $p$ be an $m$-homogeneous polynomial on a complex Banach space, and let $(x_n)_n$ be a bounded sequence such that when evaluated in polynomials of degree less than $m$, it converges to zero, but $p(x_n)=1$. It is proved here that there exists a basic sequence $(y_k)_k$ equivalent to a subsequence $(x_{n_k})_k$, for which $p(\sum_{k=1}^{\infty}a_ky_k)=\sum_{k=1}^{\infty}a_k^m$.

The concept of Morita equivalence is generalized to the context of locally C$^{*}$-algebras. This generalizes a well-known theorem of Brown, Green and Rieffel, Pacific J. Math. 71 (1977) 349–363.

In this paper, perturbations of the left and right essential spectra of $2\times 2$ upper triangular operator matrix $M_C$ are studied, where $M_C= \left(\begin{smallmatrix} A & C 0 & B \end{smallmatrix}\right)$ is an operator acting on the Hilbert space ${\cal H}\oplus{\cal K }$. For given operators $A$ and $B$, the sets $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rle}(M_C)$ and $\bigcap_{C\in B({\cal K},{\cal H})}\sigma_{\rre}(M_C)$ are determined, where $ \sigma_{\rle}(T)$ and $\sigma_{\rre}(T)$ denote, respectively, the left essential spectrum and the right essential spectrum of an operator $T$.

Several inequalities for trace norms of sums of $n$ operators with roots of unity coefficients are proved in this paper. When $n=2$, these reduce to the classical Clarkson inequalities and their non-commutative analogues.

Let $G$ be a finitely generated, discrete group of polynomial growth. This paper provides a proof of the boundedness in $L_p(G)$, $1<p<\infty$, of some higher-order Riesz transform operators associated with a probability density on $G$.

Frank Smithies was born in Edinburgh, on 10 March 1912. His father, also Frank Smithies, and his mother, Mary (née Blakemore) had met through their involvement in the socialist movement. They had two children, Frank and his sister Violet, some four years younger.