This paper provides a survey of both old and new results about minimal surfaces and submanifolds.

We determine the intersection theory on the Igusa compactification and the second Voronoi compactification of ${\cal A}_4$.

For any $n \geq 3$, let $F \in \mathbb{Z}[X_0,\ldots,X_n]$ be a form of degree $d\geq 5$ that defines a non-singular hypersurface $X \subset \mathbb{P}^{n}$. The main result in this paper is a proof of the fact that the number $N(F;B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $N(F;B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}),$ for any $\varepsilon >0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$. New estimates are also obtained for the number of representations of a positive integer as the sum of three $d$th powers, and for the paucity of integer solutions to equal sums of like polynomials.

A description is given of serial rings whose maximal quotient rings are quasi-Frobenius (QF). Every serial ring is a factor of a serial ring whose maximal quotient ring is a QF-ring. This result is used to give a new, conceptual proof for the selfduality of serial rings, emphasising the importance of weakly symmetric rings.

A complete structural characterization of submonoids $S$ of a polycyclic-by-finite group such that the semigroup algebra $K[S]$ over a field $K$ is right noetherian is obtained. It follows that such algebras are also left noetherian.

Let $S={\mathop{\rm Sym}(\Omega)$ be the group of all permutations of an infinite set $\Omega$. Extending an argument of Macpherson and Neumann, it is shown that if $U$ is a generating set for $S$ as a group, then there exists a positive integer $n$ such that every element of $S$ may be written as a group word of length at most $n$ in the elements of $U$. Likewise, if $U$ is a generating set for $S$ as a monoid, then there exists a positive integer $n$ such that every element of $S$ may be written as a monoid word of length at most $n$ in the elements of $U$. Some related questions and recent results are noted, and a brief proof is given of a result of Ore's on commutators, which is used in the proof of the above result.

Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property $O$. It is deduced both that these groups are exact and satisfy Yu's Property $A$.

A unipotent linear group over a skew field of characteristic 2 is nilpotent.

A finitary orbit equivalence mapping between the binary and ternary odometers is constructed.

The main result in this paper is a representation theorem for homogeneous orthogonally additive polynomials on Banach lattices. The representation theorem is used to study the linear span of the set of zeros of homogeneous real-valued orthogonally additive polynomials. It is shown that in certain lattices every element can be represented as the sum of two or three zeros or, at least, can be approximated by such sums. It is also indicated how these results can be used to study weak topologies induced by orthogonally additive polynomials on Banach lattices.

We give a short proof of the fundamental result that the critical probability for bond percolation in the planar square lattice ${\mathbb Z}^2$ is equal to $1/2$. The lower bound was proved by Harris, who showed in 1960 that percolation does not occur at $p=1/2$. The other, more difficult, bound was proved by Kesten, who showed in 1980 that percolation does occur for any $p>1/2$.

In this paper, it is proved that for $n\geq 2$, any horizontally homothetic submersion $\varphi :\mathbb{R}^{n+1}\longrightarrow (N^{n}, h)$ is a Riemannian submersion up to a homothety. It is also shown that if $\varphi :\mathbb{S}^{n+1}\longrightarrow (N^{n}, h)$ is a horizontally homothetic submersion, then $n=2m$, $(N^{n}, h)$ is isometric to $\mathbb{C}P^{m}$ and, up to a homothety, $\varphi$ is a standard Hopf fibration $\mathbb{S}^{2m+1} \longrightarrow \mathbb{C}P^{m}$.

In this paper we point out how some recent developments in the theory of constant scalar curvature Kähler metrics can be used to clarify the existence issue for such metrics in the special case of (geometrically) ruled complex surfaces.

We construct an infinite family of simply connected, pairwise nondiffeomorphic 4-manifolds, all homeomorphic to $3{\mathbb {CP}}^2 \# 9 {\overline {{\mathbb C}{\mathbb P}^2}}$. Similar ideas provide examples of 4-manifolds with $b_2^+=3$, $b_2^-=8$, vanishing first homology and nontrivial Seiberg–Witten invariants.

Shiing-Shen Chern was born on October 26th, 1911 in Jia Xin, Chekiang Province, in China. His father practised law and worked for the government. At Fu Luen Middle School in Tsientsin he first showed his mathematical ability by doing all the exercises in classical English textbooks on algebra and trigonometry, and then went at the age of fifteen to Nankai University — a one-man Department run by Li-Fu Chiang, a student of Coolidge. As a result he studied a great deal of geometry, reading Coolidge, Salmon, Castelnuovo and Staude. He then became a postgraduate in 1930 at Tsinghua University in Beijing (or Peiping as it was then called) and came under the influence of Dan Sun, one of the few mathematicians in China writing research papers. During this period he became seriously interested in Sun's subject, projective differential geometry, and studied in detail the works of Wilhelm Blaschke. It was also at Tsinghua that he met his wife Shih-Ning, the daughter of a professor. After Blaschke paid a visit to Tsinghua in 1932 and lectured on differential-geometric invariants, Chern won a fellowship to study with him in Hamburg for two years. In 1936 he received his DSc there for work on the theory of webs. While in Hamburg, he also attended the lectures of Kähler on what is now called Cartan–Kähler theory, and the following year he spent in Paris studying with Cartan himself.