Volume 21, Issue 1

Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T * Q is an h -cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

Improving a result of Hajela, we show for every function f with lim n →∞ f ( n ) = ∞ and f ( n ) = o ( n ) that there exists n 0 = n 0 ( f ) such that for every n ⩾ n 0 and any S ⊆ {–1, 1} n with cardinality | S | ⩽ 2 n / f ( n ) one can find orthonormal vectors x 1 , …, x n ∈ ℝ n satisfying ∥ ε 1x1+⋯ +ε nxn∥ ∞ ⩾ clog⁡ f(n)$\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all ( ε 1 , …, ε n ) ∈ S . We obtain analogous results in the case where x 1 , …, x n are independent random points uniformly distributed in the Euclidean unit ball B2n$\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the ℓ ∞ n$\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$-norm is replaced by an arbitrary norm on ℝ n .

An ordinary hypersphere of a set of points in real d -space, where no d + 1 points lie on a ( d - 2)-sphere or a ( d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a ( d + 2)- point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d -dimensional spherical analogue of the Dirac–Motzkin conjecture for d ⩾ 3. We also find the maximum number of ( d + 2)-point hyperspheres in even dimensions, solving the d -dimensional spherical analogue of the orchard problem for even d ⩾ 4.

We study birational projective models of 𝓜 2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2 (𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2 (𝓩).

We show that the Grothendieck group associated to integral polytopes in ℝ n is free-abelian, by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of ℝ n .

We focus on the description of the automorphism group Γ ∥ of a Clifford-like parallelism ∥ on a 3-dimensional projective double space (ℙ( H F ), ∥ ℓ , ∥ r ) over a quaternion skew field H (of any characteristic). We compare Γ ∥ with the automorphism group Γ ℓ of the left parallelism ∥ ℓ , which is strictly related to Aut( H ). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose ∥ in such a way that Γ ∥ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ∥ ≠ ∥ ℓ .

We prove a generalization of Pál's conjecture from 1921: if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q . We also prove a lower bound of Ω ( m n 2 ) on the number of combinatorially distinct maximal placements of a convex m -gon P in a convex n -gon Q . This matches the upper bound proven by Agarwal et al.

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.

Let S be a surface of genus g at least 2. A representation ρ:π1S→PSL2R$ \rho:\pi_1 S\to{\mathrm{PSL}_2\mathbb{R}} $ is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S . In this work we characterise them completely, giving necessary and sufficient conditions.

For a surface immersed in a three-dimensional space endowed with a smooth norm instead of an inner product, one can define analogous concepts of curvature and metric. With such concepts in mind, various questions immediately appear. The aim of this paper is to propose and answer some of those questions. In this framework we prove several characterizations of minimal surfaces in normed spaces, and respective analogues of some global theorems (e.g., Hadamard-type theorems) are also derived. A result on the curvature of surfaces having constant Minkowskian width is given, and finally we study the ambient metric induced on the surface, proving an extension of the classical Bonnet theorem.

It is well-known that every polyhedral cone is finitely generated (i.e. polytopal), and vice versa. Surprisingly, the two notions differ almost always for non-commutative versions of such cones, as was recently proved by different authors. In this note we give a direct and constructive proof of the statement. Our proof yields a new and surprising quantitative result: the difference of the two notions can always be seen at the first level of non-commutativity, i.e. for matrices of size 2, independent of dimension and complexity of the initial convex cone.

In this note we prove a ‘maximum principle’ for circle-valued solutions of the (time-dependent) heat equation on closed Riemannian manifolds.

For a smooth manifold X equipped with a volume form, let 𝓛 0 ( X ) be the Lie algebra of volume preserving smooth vector fields on X . Lichnerowicz proved that the abelianization of 𝓛 0 ( X ) is a finite-dimensional vector space, and that its dimension depends only on the topology of X . In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties with trivial canonical bundle, which include certain algebraic surfaces and linear algebraic groups. The proofs are based on a remarkable result of Grothendieck on the cohomology of affine varieties, and some techniques that were introduced with the purpose of extending the Andersén–Lempert theory, which was originally developed for the complex spaces ℂ n , to the larger class of Stein manifolds that satisfy the density property.

In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.