We show that any embedding $\mathbb {R}^d \to \mathbb {R}^{2d+2^{\gamma (d)}-1}$ inscribes a trapezoid or maps three points to a line, where $2^{\gamma (d)}$ is the smallest power of $2$ satisfying $2^{\gamma (d)} \geq \rho (d)$, and $\rho (d)$ denotes the Hurwitz–Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $3$-regular maps, for infinitely many dimensions $d$, without resorting to sophisticated algebraic techniques.

We investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn–Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to f divergences of the cone measures of the convex body and its polar, namely the Kullback–Leibler divergence and the Rényi divergence.

We prove an improvement on Schmidt’s upper bound on the number of number fields of degree n and absolute discriminant less than X for $6\leq n\leq 94$ . We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].

We give a short proof of the Torelli theorem for $ALH^*$ gravitational instantons using the authors’ previous construction of mirror special Lagrangian fibrations in del Pezzo surfaces and rational elliptic surfaces together with recent work of Sun-Zhang. In particular, this includes an identification of 10 diffeomorphism types of $ALH^*_b$ gravitational instantons.

We present a representation formula for translating soliton surfaces to the mean curvature flow in Euclidean space ${\mathbb {R}}^{4}$ and give examples of conformal parameterisations for translating soliton surfaces.

The purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. Let X be a smooth proper curve over a finite field $\mathbb {F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $\rho :\pi _1^{et}(V) \to \mathbb {C}^{\times }$ . In this article, we prove a lower bound on the Newton polygon of the L-function $L(\rho ,s)$ . The estimate depends on monodromy invariants of $\rho $ : the Swan conductor and the local exponents. Under certain nondegeneracy assumptions, this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne’s prediction that the irregular Hodge filtration would force p-adic bounds on L-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).

We study orthogonal projections of generic embedded hypersurfaces in ℝ4 with boundary to 2-spaces. Therefore, we classify simple map germs from ℝ3 to the plane of codimension less than or equal to 4 with the source containing a distinguished plane which is preserved by coordinate changes. We also go into some detail on their geometrical properties in order to recognize the cases of codimension less than or equal to 1.

In this paper, we prove some Bernstein type results for n-dimensional minimal Lagrangian graphs in quaternion Euclidean space Hn ≅ R4n. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in Cn.

A skeletal structure (M, U) in ${\mathbb R}^{n+1}$ is a special type of n-dimensional Whitney stratified set M on which is defined a multivalued ‘radial vector field’ U. This is an extension of the notion of the Blum medial axis of a region in ${\mathbb R}^{n+1}$ with generic smooth boundary. For such a skeletal structure an ‘associated boundary’ $\mathcal{B}$ is defined. In part I of this paper, we introduced radial and edge shape operators, which are geometric invariants of the radial vector field U on M, and a ‘radial flow’ from M to $\mathcal{B}$. In this paper, in the partial Blum case we derive formulas for the differential geometric shape operator of the boundary (and hence all curvature invariants) in terms of the shape operators on the medial axis. We further derive the effects of a diffeomorphism of the skeletal structure on the radial and edge shape operators using a distortion operator which is computed from the second derivative of the diffeomorphism evaluated on the unit radial vector field. This allows one to compute the geometry of the boundary associated to a deformed skeletal structure purely in terms of operators defined on the original skeletal set.

It is proved that, given a convex polytope $P$ in $\mathbb{R}^n$, together with a collection of compact convex subsets in the interior of each facet of $P$, there exists a smooth convex body arbitrarily close to $P$ that coincides with each facet precisely along the prescribed sets, and has positive curvature elsewhere.

We study two kinds of generalized umbilics on smoothly embedded n-manifolds in ℝn+1. A sectional umbilic occurs where two of the principal curvatures are equal, and a split sectional umbilic is a more general notion.