We study the blow-up profile of radial solutions of a semilinear heat equation with an exponential source term. Our main aim is to show that solutions which can be continued beyond blow-up possess a nonconstant selfsimilar blow-up profile. For some particular solutions we determine this profile precisely.

We define a new class of manifolds called $n$-Sasakian manifolds that enjoy remarkable geometric properties. We furnish examples of such manifolds and make links to the study of isoparametric hypersurfaces. We demonstrate that these examples carry Einstein metrics.

Let $k$ be a perfect field which admits resolution of singularities in the sense of Friedlander and Voevodsky (for example, $k$ of characteristic $0$). Let $X$ be a smooth proper $k$-variety of pure dimension $n$ and $Y,Z$ two disjoint closed subsets of $X$. We prove an isomorphism \[ M(X-Z,Y)\simeq M(X-Y,Z)^*(n)[2n], \] where $M(X-Z,Y)$ and $M(X-Y,Z)$ are relative Voevodsky motives, defined in his triangulated category $\operatorname{DM}_{\rm gm}(k)$.

We show various examples of torsion-free affine connections which preserve volume elements and have definite Ricci curvature tensors.

Let $\boldsymbol{X}$ be an affine toric variety with big torus $\boldsymbol{T}\subset \boldsymbol{X}$ and let $T\subset\boldsymbol{T}$ be a subtorus. The general $T$-orbit closures in $\boldsymbol{X}$ and their flat limits are parametrized by the main component $H_0$ of the toric Hilbert scheme. Further, the quotient torus $\boldsymbol{T}/T$ acts on $H_0$ with a dense orbit. We describe the fan of this toric variety; this leads us to an integral analogue of the fiber polytope of Billera and Sturmfels. We also describe the relation of $H_0$ to the main component of the inverse limit of GIT quotients of $\boldsymbol{X}$ by $T$.

A few affine invariant structures depending only on the second fundamental form relative to arbitrary transversal bundles on submanifolds of the standard affine spaces are introduced. A notion of “local strong convexity” is proposed for arbitrary codimensional submanifolds. In the case of $n$-dimensional submanifolds of $2n$-dimensional real affine spaces, complex structures on the ambient spaces are used as a tool for studying real affine invariants.

We compute the Euler-Poincaré characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.

In a separable complex Hilbert space endowed with an isometric conjugate-linear involution, we study sequences orthonormal with respect to an associated bilinear form. Properties of such sequences are measured by a positive, possibly unbounded angle operator which is formally orthogonal as a matrix. Although developed in an abstract setting, this framework is relevant to a variety of eigenvector interpolation problems arising in function theory and in the study of differential operators.

We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.