Let $X$ be an infinite-dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exist a constant $\lambda \gt 0$ and an infinite sequence $\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$ such that $\Vert {x}_{i} - {x}_{j} \Vert = \lambda $ for all $i\not = j$.

We solve a randomized version of the following open question: is there a strictly convex, bounded curve $\gamma \subset { \mathbb{R} }^{2} $ such that the number of rational points on $\gamma $, with denominator $n$, approaches infinity with $n$? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson process that simulates the refined rational lattice $(1/ d){ \mathbb{Z} }^{2} $, which we call ${M}_{d} $, for each natural number $d$. The main result here is that with probability $1$ there exists a strictly convex, bounded curve $\gamma $ such that $\vert \gamma \cap {M}_{d} \vert \rightarrow + \infty , $ as $d$ tends to infinity. The methods include the notion of a generalized affine length of a convex curve as defined by F. V. Petrov [Estimates for the number of rational points on convex curves and surfaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 344 (2007), 174–189; Engl. transl. J. Math. Sci. 147(6) (2007), 7218–7226].

A singularity is said to be weakly exceptional if it has a unique purely log terminal blow-up. This is a natural generalization of the surface singularities of types Dn, E6, E7 and E8. Since this idea was introduced, quotient singularities of this type have been classified in dimensions up to at most 4. This note extends that classification to dimension 5.

This paper suggests a new approach to questions of rationality of 3-folds based on category theory. Following work by Ballard et al., we enhance constructions of Kuznetsov by introducing Noether–Lefschetz spectra: an interplay between Orlov spectra and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where the above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H3(X, ℤ). This is a novelty: after the classical example of Artin and Mumford, this is the second example of a Fano 3-fold with a torsion in the third integer homology group. In particular, X is non-rational. We consider other examples as well: V10 with 10 singular points, and the double covering of a quadric ramified in an octic with 20 nodal singular points. After analysing the geometry of their Landau–Ginzburg models, we suggest a general non-rationality picture based on homological mirror symmetry and category theory.

We study a notion of ‘b-stability’, introduced previously by the author in connection with the existence of constant scalar curvature Kähler, and Kähler-Einstein, metrics. The main result is Theorem 1.2, which makes progress towards a statement that the existence of such metrics implies b-stability. The proof is a modification of an argument of Stoppa, taking account of the birational transformations involved in the definition of b-stability.

We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor can be split by a grading fpqc-locally on the base scheme.

This paper consists mainly of a review and applications of our old results relating to the title. We discuss how many elliptic fibrations and elliptic fibrations with infinite automorphism groups (or Mordell–Weil groups) an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with a finite number of non-singular rational curves, with a finite number of Enriques involutions, and with naturally arithmetic automorphism groups.

Let ${\mathrm{OT} }_{d} (n)$ be the smallest integer $N$ such that every $N$-element point sequence in ${ \mathbb{R} }^{d} $ in general position contains an order-type homogeneous subset of size $n$, where a set is order-type homogeneous if all $(d+ 1)$-tuples from this set have the same orientation. It is known that a point sequence in ${ \mathbb{R} }^{d} $ that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in ${ \mathbb{R} }^{d} $. Two famous theorems of Erdős and Szekeres from 1935 imply that ${\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )$ and ${\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)} $. For $d\geq 3$, we give new bounds for ${\mathrm{OT} }_{d} (n)$. In particular, we show that ${\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} } $, answering a question of Eliáš and Matoušek, and, for $d\geq 4$, we show that ${\mathrm{OT} }_{d} (n)$ is bounded above by an exponential tower of height $d$ with $O(n)$ in the topmost exponent.

We examine the minimal magnitude of perturbations necessary to change the number $N$ of static equilibrium points of a convex solid $K$. We call the normalized volume of the minimally necessary truncation robustness and we seek shapes with maximal robustness for fixed values of $N$. While the upward robustness (referring to the increase of $N$) of smooth, homogeneous convex solids is known to be zero, little is known about their downward robustness. The difficulty of the latter problem is related to the coupling (via integrals) between the geometry of the hull $\mathrm{bd} \hspace{0.167em} K$ and the location of the center of gravity $G$. Here we first investigate two simpler, decoupled problems by examining truncations of $\mathrm{bd} \hspace{0.167em} K$ with $G$ fixed, and displacements of $G$ with $\mathrm{bd} \hspace{0.167em} K$ fixed, leading to the concept of external and internal robustness, respectively. In dimension 2, we find that for any fixed number $N= 2S$, the convex solids with both maximal external and maximal internal robustness are regular $S$-gons. Based on this result we conjecture that regular polygons have maximal downward robustness also in the original, coupled problem. We also show that in the decoupled problems, three-dimensional regular polyhedra have maximal internal robustness, however, only under additional constraints. Finally, we prove results for the full problem in the case of three-dimensional solids. These results appear to explain why monostatic pebbles (with either one stable or one unstable point of equilibrium) are found so rarely in nature.

We explicitly calculate some Gromov–Witten correspondences determined by maps of labelled curves of genus 0 to the moduli spaces of labelled curves of genus 0. We consider these calculations as the first step towards studying the self-referential nature of motivic quantum cohomology.

Generalized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1, 3). Our main result is that F is a Cayley form if and only if Z = G(1, 3) ∩ {F = 0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quadratic equations, since there is a unique representative F0 + QF1 of F, with F0, F1 harmonic, such that the harmonic projection of the Cayley equation is identically 0. We also give new equations for honest Cayley forms, but show, with some calculations, that the variety of honest Cayley forms does not seem to be defined by quadratic and cubic equations.

We prove a result on the inversion of adjunction for log canonical pairs that generalizes Kawakita's result to log canonical centres of arbitrary codimension.

Using orbifold metrics of the appropriately signed Ricci curvature on orbifolds with a negative or numerically trivial canonical bundle and the two-dimensional log minimal model programme, we prove that the fundamental group of special compact Kähler 3-folds is almost abelian. This property was conjectured in all dimensions by Campana in 2004, and also for orbifolds in 2007, where the notion of specialness was introduced. We briefly recall the definition, basic properties and the role of special manifolds in classification theory.

Let L be a nef line bundle on a projective scheme X in positive characteristic. We prove that the augmented base locus of L is equal to the union of the irreducible closed subsets V of X such that L∣V is not big. For a smooth variety in characteristic 0, this was proved by Nakamaye using vanishing theorems.

We show that some of the properties of log canonical centres of a log canonical pair also hold for certain subvarieties that are close to being a log canonical centre. As a consequence, we obtain that, in working with deformations of pairs where all the coefficients of the boundary divisor are bigger than ½, embedded points never appear on the boundary divisor.

Let $P$ and $Q$ be non-zero integers. The generalized Fibonacci sequence $\{ {U}_{n} \} $ and Lucas sequence $\{ {V}_{n} \} $ are defined by ${U}_{0} = 0$, ${U}_{1} = 1$ and ${U}_{n+ 1} = P{U}_{n} + Q{U}_{n- 1} $ for $n\geq 1$ and ${V}_{0} = 2, {V}_{1} = P$ and ${V}_{n+ 1} = P{V}_{n} + Q{V}_{n- 1} $ for $n\geq 1$, respectively. In this paper, we assume that $Q= 1$. Firstly, we determine indices $n$ such that ${V}_{n} = k{x}^{2} $ when $k\vert P$ and $P$ is odd. Then, when $P$ is odd, we show that there are no solutions of the equation ${V}_{n} = 3\square $ for $n\gt 2$. Moreover, we show that the equation ${V}_{n} = 6\square $ has no solution when $P$ is odd. Lastly, we consider the equations ${V}_{n} = 3{V}_{m} \square $ and ${V}_{n} = 6{V}_{m} \square $. It has been shown that the equation ${V}_{n} = 3{V}_{m} \square $ has a solution when $n= 3, m= 1$, and $P$ is odd. It has also been shown that the equation ${V}_{n} = 6{V}_{m} \square $ has a solution only when $n= 6$. We also solve the equations ${V}_{n} = 3\square $ and ${V}_{n} = 3{V}_{m} \square $ under some assumptions when $P$ is even.

We provide an alternative constructive proof of the classical Brauer theorem for finite groups based on the well-known description of the complex irreducible representations of the symmetric groups Sn. The theorem is first proved for Sn and then for general G by embedding in Sn and applying the Mackey subgroup theorem.