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.

In this paper we use the Hausdorff metric to prove that two compact metric spaces are homeomorphic if and only if their canonical complements are uniformly homeomorphic. We take one of the two steps needed to prove that the difference between the homotopical and topological classifications of compact connected ANRs depends only on the difference between continuity and uniform continuity of homeomorphisms in their canonical complements, which are totally bounded metric spaces. The more important step was provided by the Chapman complement and the Curtis–Schori–West theorems. We also improve the multivalued description of shape theory given by J. M. R. Sanjurjo but only in the class of locally connected compacta.

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).

The Fredholm properties of Toeplitz operators $T_a$ on Hardy spaces $H^p$ ($1<p<\infty$) with continuous symbols $a$ are well understood. We consider $T_a$ acting on $H^1$, where the operator is bounded provided that $a$ belongs to the class of symbols given by Janson and Stegenga's result on the pointwise multipliers on $H^1$. A necessary and sufficient condition for $T_a$ to be a Fredholm operator is given when $a$ is continuous and satisfies a mild additional condition (much weaker than Hölder continuity). A formula for the index of $T_a$ is also derived. In addition, we study the case of matrix-valued symbols and Toeplitz operators on $\rm{BMO}_A$.

Using the twistorial approach and some previous results, we prove the conjecture that the dimension of the moduli space of harmonic maps of area $4 \pi d$ from the 2-sphere to the $2n$-sphere is $2 d+n^2$ for the particular case $n=3$.

We provide a criterion for the energy measures of regular Dirichlet forms on post-critically finite self-similar sets to be singular with respect to every self-similar measure. A probabilistic interpretation of the singularity is also discussed.

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.

Real flag manifolds are the isotropy orbits of noncompact symmetric spaces $G/K$. Any such manifold $M$ is acted on transitively by the (noncompact) Lie group $G$, and it is embedded in euclidean space as a taut submanifold. The aim of this paper is to show that the gradient flow of any height function is a one-parameter subgroup of $G$, where the gradient is defined with respect to a suitable homogeneous metric $s$ on $M$; this generalizes the Kähler metric on adjoint orbits (the so-called complex flag manifolds).

A Radon measure $\mu$ on ${mathbb R}^n$ is said to be $k$-monotone if $r\mapsto{\mu(B(x,r))}/{r^k}$ is a non-decreasing function on $(0,\infty)$ for every $x\in {\mathbb R}^n$. (If $\mu$ is the $k$-dimensional Hausdorff measure restricted to a $k$-dimensional minimal surface then this important property is expressed by the monotonicity formula.) We give an example of a 1 -monotone measure $\mu$ in ${\mathbb R}^2$ with non-unique and non-conical tangent measures at a point. Furthermore, we show that $\mu$ can be the one-dimensional Hausdorff measure restricted to a closed set $A\subset {\mathbb R}^2$.

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.

We consider the uniqueness of the positive solution to a semilinear elliptic equation with Dirichlet boundary condition and the nonlinearity satisfying $f(0)<0$ and having asymptotic sublinear growth rate. A similar idea is also applied to the nonexistence of a positive solution to a superlinear problem.

We give a general method for expressing various characteristic numbers of the multiple-point manifolds of an immersion.

Ali Fröhlich was one of the truly great mathematicians of his generation. In addition to making fundamental contributions, he is one of the very few of whom it can be said, “he founded a new subject”: namely that area now known as Galois module theory. In addition to establishing a theoretical basis for the subject, he also formulated some very deep and wonderful conjectures that would thrust him onto the world's stage. He also built up a school of researchers that played a central part in establishing British algebraic number theory as a world force. The outstanding nature of his contributions to mathematics was recognised in 1992 when the London Mathematical Society awarded him the De Morgan Medal: their highest and most distinguished prize, which is awarded triennially.

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 $X$.

Regular homotopy classes of immersions of homotopy $n$-spheres into $(n+q)$-space form an Abelian group under connected summation. The subgroup of immersions of those homotopy spheres which bound parallelizable manifolds is computed for $n=4k-1$ and arbitrary codimension $q$. A consequence of this computation is that in codimensions $q<2k+1$, the group of immersed homotopy $(4k-1)$-spheres is not the direct sum of the group of immersed standard spheres and the group of homotopy spheres. Also, our results shed light on Brieskorn's famous equations: for any exotic sphere (bounding a parallelizable manifold), a countable number of distinct equations give rise to codimension-two embeddings of this exotic sphere. It follows from our results that these embeddings lie in pairwise distinct regular homotopy classes, and that any regular homotopy class containing embeddings is representable by a map arising from such an equation.

Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space $W_{1, p}(D; \rho)$, where the weight function $\rho = \rho(z; \partial D)$ is the distance of the variable point $z$ from the boundary $\partial D$ raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved.

We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng–Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville theorem for the heat equation does not hold even in ${\mathbb R}^n$ without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds.