Many Empty Triangles have a Common Edge

Given a finite point set $X$ X in the plane, the degree of a pair $\{x,y\} \subset X$ { x , y } ? X is the number of empty triangles $t=\mathrm {conv} \{x,y,z\},$ t = conv { x , y , z } , where empty means $t\cap X=\{x,y,z\}.$ t ∩ X = { x , y , z } . Define $\deg X$ deg X as the maximal degree of a pair in $X.$ X . Our main result is that if $X$ X is a random sample of $n$ n independent and uniform points from a fixed convex body, then $\deg X \ge cn/\ln n$ deg X ≥ cn / ln n in expectation.     

2-Nilpotent real section conjecture

We show a $2$ -nilpotent section conjecture over $\mathbb{R }$ : for a geometrically connected curve $X$ over $\mathbb{R }$ such that each irreducible component of its normalization has $\mathbb{R }$ -points, $\pi _0(X(\mathbb{R }))$ is determined by the maximal $2$ -nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for $X$ smooth and proper, $X(\mathbb{R })^{\pm }$ is determined by the maximal $2$ -nilpotent quotient of $\mathrm{Gal }(\mathbb{C }(X))$ with its $\mathrm{Gal }(\mathbb{R })$ action, where $X(\mathbb{R })^{\pm }$ denotes the set of real points equipped with a real tangent direction, showing a $2$ -nilpotent birational real section conjecture.     

On the bicanonical map of primitive varieties with q(X) = \mathrm{dim }X: the degree and the Euler number

Let $X$ be a variety of maximal Albanese dimension and of general type. Assume that $q(X) = \mathrm{dim }X$ , the Albanese variety $\mathrm {Alb} (X)$ is a simple abelian variety, and the bicanonical map is not birational. We prove that the Euler number $\chi (X, \omega _X)$ is equal to 1, and $|2K_X|$ separates two distinct points over the same general point on $\mathrm {Alb} (X)$ via $\mathrm {alb}_X$ (Theorem 1.1).     

Étale cohomology of a DM curve-stack with coefficients in \mathbb G _m

We compute étale cohomology groups $H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb G _m)$ in several cases, where $X$ is a connected smooth tame Deligne–Mumford stack of dimension $1$ over an algebraically closed field. We have complete results for orbicurves (and, more generally, for twisted nodal curves) and in the case all stabilizers are cyclic; we give partial results and examples in the general case. In particular, we show that if the stabilizers are abelian then $H_{\acute{\mathrm{e}}\mathrm{t}}^2(X, \mathbb{G }_m)$ does not depend on $X$ but only on the underlying orbicurve $Y$ and on the generic stabilizer. We show with two examples that, in general, the higher cohomology groups $H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb{G }_m)$ cannot be computed knowing only the base of the gerbe $X \rightarrow Y$ and the banding group.     

Constructible \nabla -modules on curves

Let $\mathcal{V }$ be a complete discrete valuation ring of mixed characteristic with perfect residue field. Let $X$ be a geometrically connected smooth proper curve over $\mathcal{V }$ . We introduce the notion of constructible convergent $\nabla $ -module on the analytification $X_{K}^{\mathrm{an}}$ of the generic fiber of $X$ . A constructible module is an $\mathcal{O }_{X_{K}^{\mathrm{an}}}$ -module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $X_{k}$ of $X$ . The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $\nabla $ -modules to the category of $\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$ -modules. We show that specialization induces an equivalence between constructible $F$ - $\nabla $ -modules and perverse holonomic $F$ - $\mathcal{D }^\dagger _{\hat{X} \mathbf{Q }}$ -modules.     

Maximal Equilateral Sets

A subset of a normed space $X$ X is called equilateral if the distance between any two points is the same. Let $m(X)$ m ( X ) be the smallest possible size of an equilateral subset of $X$ X maximal with respect to inclusion. We first observe that Petty’s construction of a $d$ d - $X$ X of any finite dimension $d\ge 4$ d ≥ 4 with $m(X)=4$ m ( X ) = 4 can be generalised to give $m(X\oplus _1\mathbb R )=4$ m ( X ⊕ 1 R ) = 4 for any $X$ X of dimension at least $2$ 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set $\Gamma $ Γ , $m(\ell _p(\Gamma ))$ m ( ? p ( Γ ) ) is finite and bounded above by a function of $p$ p , for all $1\le p<2$ 1 ≤ p < 2 . Also, for all $p\in [1,\infty )$ p ∈ [ 1 , ∞ ) and $d\in \mathbb N $ d ∈ N there exists $c=c(p,d)>1$ c = c ( p , d ) > 1 such that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $c$ c from $\ell _p^d$ ? p d . Using Brouwer’s fixed-point theorem we show that $m(X)\le d+1$ m ( X ) ≤ d + 1 for all $d$ d - $X$ X with Banach–Mazur distance less than $3/2$ 3 / 2 from $\ell _\infty ^d$ ? ∞ d . A graph-theoretical argument furthermore shows that $m(\ell _\infty ^d)=d+1$ m ( ? ∞ d ) = d + 1 . The above results lead us to conjecture that $m(X)\le 1+\dim X$ m ( X ) ≤ 1 + dim X for all finite-normed spaces $X$ X .     

A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem $$\begin{aligned} \begin{array}{ll} \min \limits _{X\in \mathbb{R }^{m\times n}}&\mu _1\Vert \sigma (\mathcal{F }(X)-G)\Vert _\alpha +\mu _2\Vert \mathcal{C }(X)-d\Vert _\beta ,\\ \text{ subject} \text{ to}&\mathcal{A }(X)-b\in \mathcal{Q }, \end{array} \end{aligned}$$ where $\sigma (X)$ denotes the vector of singular values of $X \in \mathbb{R }^{m\times n}$ , the matrix norm $\Vert \sigma (X)\Vert _{\alpha }$ denotes either the Frobenius, the nuclear, or the $\ell _2$ -operator norm of $X$ , the vector norm $\Vert .\Vert _{\beta }$ denotes either the $\ell _1$ -norm, $\ell _2$ -norm or the $\ell _{\infty }$ -norm; $\mathcal{Q }$ is a closed convex set and $\mathcal{A }(.)$ , $\mathcal{C }(.)$ , $\mathcal{F }(.)$ are linear operators from $\mathbb{R }^{m\times n}$ to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all $\epsilon >0$ , the FALC iterates are $\epsilon $ -feasible and $\epsilon $ -optimal after $\mathcal{O }(\log (\epsilon ^{-1}))$ iterations, which require $\mathcal{O }(\epsilon ^{-1})$ constrained shrinkage operations and Euclidean projection onto the set $\mathcal{Q }$ . Surprisingly, on the problem sets we tested, FALC required only $\mathcal{O }(\log (\epsilon ^{-1}))$ constrained shrinkage, instead of the $\mathcal{O }(\epsilon ^{-1})$ worst case bound, to compute an $\epsilon $ -feasible and $\epsilon $ -optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.     

Local linear estimation for stochastic processes driven by \alpha -stable L\acute{\mathbf{e}}vy motion

The $\alpha $ α -stable L $\acute{\mathrm{e}}$ e ´ vy motion together with the Poisson process and Brownian motion are the most important examples of L $\acute{\mathrm{e}}$ e ´ vy processes, which form the first class of stochastic processes being studied in the modern spirit. In this paper, the stochastic processes driven by $\alpha $ α -stable L $\acute{\mathrm{e}}$ e ´ vy motion are considered, local linear estimator of the drift function for these processes is discussed. Under mild conditions, we derive consistency of the local linear estimator of the drift function. The performance of the proposed estimator is assessed by simulation study.     

The 2-Page Crossing Number of K_{n}

Around 1958, Hill described how to draw the complete graph $K_n$ K n with $$\begin{aligned} Z(n) :=\frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \end{aligned}$$ Z ( n ) : = 1 4 ? n 2 ? ? n ? 1 2 ? ? n ? 2 2 ? ? n ? 3 2 ? crossings, and conjectured that the crossing number ${{\mathrm{cr}}}(K_{n})$ cr ( K n ) of $K_n$ K n is exactly $Z(n)$ Z ( n ) . This is also known as Guy’s conjecture as he later popularized it. Towards the end of the century, substantially different drawings of $K_{n}$ K n with $Z(n)$ Z ( n ) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line $\ell $ ? (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by  $\ell $ ? . The 2-page crossing number of $K_{n} $ K n , denoted by $\nu _{2}(K_{n})$ ν 2 ( K n ) , is the minimum number of crossings determined by a 2-page book drawing of $K_{n}$ K n . Since ${{\mathrm{cr}}}(K_{n}) \le \nu _{2}(K_{n})$ cr ( K n ) ≤ ν 2 ( K n ) and $\nu _{2}(K_{n}) \le Z(n)$ ν 2 ( K n ) ≤ Z ( n ) , a natural step towards Hill’s Conjecture is the weaker conjecture $\nu _{2}(K_{n}) = Z(n)$ ν 2 ( K n ) = Z ( n ) , popularized by Vrt’o. In this paper we develop a new technique to investigate crossings in drawings of $K_{n}$ K n , and use it to prove that $\nu _{2}(K_{n}) = Z(n) $ ν 2 ( K n ) = Z ( n ) . To this end, we extend the inherent geometric definition of $k$ k -edges for finite sets of points in the plane to topological drawings of $K_{n}$ K n . We also introduce the concept of ${\le }{\le }k$ ≤ ≤ k -edges as a useful generalization of ${\le }k$ ≤ k -edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of $K_{n}$ K n in terms of its number of ${\le }k$ ≤ k -edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of $K_{n}$ K n and show that, up to equivalence, they are unique for $n$ n even, but that there exist an exponential number of non homeomorphic such drawings for $n$ n odd.     

Minimal generating and normally generating sets for the braid and mapping class groups of \mathbb{D }^{2}, \mathbb S ^{2} and \mathbb{R P^2}

We consider the (pure) braid groups $B_{n}(M)$ and $P_{n}(M)$ , where $M$ is the $2$ -sphere $\mathbb S ^{2}$ or the real projective plane $\mathbb R P^2$ . We determine the minimal cardinality of (normal) generating sets $X$ of these groups, first when there is no restriction on $X$ , and secondly when $X$ consists of elements of finite order. This improves on results of Berrick and Matthey in the case of $\mathbb S ^{2}$ , and extends them in the case of $\mathbb R P^2$ . We begin by recalling the situation for the Artin braid groups ( $M=\mathbb{D }^{2}$ ). As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show that for $M=\mathbb S ^{2}$ or $\mathbb R P^2$ , the induced action of $B_n(M)$ on $H_3(\widetilde{F_n(M)};\mathbb{Z })$ is trivial, $F_{n}(M)$ being the $n^\mathrm{th}$ configuration space of $M$ .     

Half-flat structures on S^3\times S^3

We describe left-invariant half-flat $ \mathrm{SU }(3) $ -structures on $ S^3\times S^3$ using the representation theory of $ \mathrm SO (4) $ and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy $ \mathrm G _2$ obtained on $ 7 $ -manifolds with equidistant $ S^3\times S^3$ hypersurfaces. The generic case is analysed numerically.     

On the largest conjugacy class length of a finite group

Let $G$ be a finite group and $\mathrm{bcl}(G)$ the largest conjugacy class length of $G$ . In this note we slightly improve He and Shi’s lower bound for $\mathrm{bcl}(G)$ , showing that $|\mathrm{bcl}(G)|\ge p^{\frac{1}{p}}(|G:O_{p}(G)|_{p})^{\frac{p-1}{p}}$ .     

Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary $p$ -groups in the groups concerned. In some cases, including ${\mathrm{Sp}}(2n,\mathbb Z )$ , the bounds we obtain are sharp: if $X$ is a generalized $\mathbb Z /3$ -homology sphere of dimension less than $2n-1$ or a $\mathbb Z /3$ -acyclic $\mathbb Z /3$ -homology manifold of dimension less than $2n$ , and if $n\ge 3$ , then any action of ${\mathrm{Sp}}(2n,\mathbb Z )$ by homeomorphisms on $X$ is trivial; if $n=2$ , then every action of ${\mathrm{Sp}}(2n,\mathbb Z )$ on $X$ factors through the abelianization of ${\mathrm{Sp}}(4,\mathbb Z )$ , which is $\mathbb Z /2$ .     

Characterization of projective linear groups by the order of the normalizer of a Sylow subgroup

Let $r$ be a prime and $G$ be a finite group, and let $R, \,S$ be Sylow $r$ -subgroups of $G$ and $\text{ PGL }(2, r)$ respectively. We prove the following results: (1) If $|G|=|\text{ PGL }(2, r)|$ and $|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)} (S)|$ and $r$ is not a Mersenne prime, then $G$ is isomorphic to $\text{ PSL } (2, r) \times C_{2}, \,\text{ SL }(2, r)$ or $\text{ PGL }(2, r)$ . (2) If $|G|=|\text{ PGL }(2, r)|, \,|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)}(S)|$ where $r>3$ is a Mersenne prime and $r$ is an isolated vertex of the prime graph of $G$ , then $G\cong \text{ PGL }(2, r)$ .     

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ X in $\mathbb{R }^2$ R 2 that is not an axis-aligned rectangle and for any positive integer $k$ k produces a family $\mathcal{F }$ F of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$ X , such that no three sets in $\mathcal{F }$ F pairwise intersect and $\chi (\mathcal{F })>k$ χ ( F ) > k . This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

On the existence of Evans potentials

It is shown that, for every noncompact parabolic Riemannian manifold $X$ and every nonpolar compact $K$ in  $X$ , there exists a positive harmonic function on $X\setminus K$ which tends to $\infty $ at infinity. (This is trivial for $\mathbb{R }$ , easy for  $\mathbb{R }^2$ , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space  $X$ , where constants are the only positive superharmonic functions and, for every nonpolar compact set  $K$ , there is a symmetric (positive) Green function for $X\setminus K$ . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines $\left[0,\infty \right)\times \{0\}, \left[0,\infty \right)\times \{1\}$ , and the line segments $\{n\}\times [0,1], n=0,1,2,\dots $ ).     

Nef line bundles on flag bundles on a curve over {\overline{{\mathbb F}}_p}

Let E be a vector bundle of rank r over an irreducible smooth projective curve X defined over the field ${\overline{{\mathbb F}}_p}$ F ¯ p . For fixed integers ${r_1\, , \ldots\, , r_\nu}$ r 1 , ... , r ν with ${1\, \leq\, r_1\, <\, \cdots\, <\, r_\nu\, <\, r}$ 1 ≤ r 1 < ? < r ν < r , let ${\text{Fl}(E)}$ Fl ( E ) be the corresponding flag bundle over X associated to E. Let ${\xi\, \longrightarrow \, {\rm Fl}(E)}$ ξ ? Fl ( E ) be a line bundle such that for every pair of the form ${(C\, ,\phi)}$ ( C , ? ) , where C is an irreducible smooth projective curve defined over ${\overline{\mathbb F}_p}$ F ¯ p and ${\phi\, :\, C\, \longrightarrow\, {\rm Fl}(E)}$ ? : C ? Fl ( E ) is a nonconstant morphism, the inequality ${{\rm degree}(\phi^* \xi)\, > \, 0}$ degree ( ? ? ξ ) > 0 holds. We prove that the line bundle ${\xi}$ ξ is ample.     

Sharpening the Norm Bound in the Subspace Perturbation Theory

Let $A$ be a (possibly unbounded) self-adjoint operator on a separable Hilbert space $\mathfrak H .$ Assume that $\sigma $ is an isolated component of the spectrum of $A$ , that is, $\mathrm{dist}(\sigma ,\Sigma )=d>0$ where $\Sigma =\mathrm spec (A)\setminus \sigma .$ Suppose that $V$ is a bounded self-adjoint operator on $\mathfrak H $ such that $\Vert V\Vert <d/2$ and let $L=A+V$ , $\mathrm{Dom }(L)=\mathrm{Dom }(A).$ Denote by $P$ the spectral projection of $A$ associated with the spectral set $\sigma $ and let $Q$ be the spectral projection of $L$ corresponding to the closed $\Vert V\Vert $ -neighborhood of $\sigma .$ Introducing the sequence $$\begin{aligned} \varkappa _n=\frac{1}{2}\left(1-\frac{(\pi ^2-4)^n}{(\pi ^2+4)^n}\right), \quad n\in \{0\}\cup {\mathbb N }, \end{aligned}$$ we prove that the following bound holds: $$\begin{aligned} \arcsin (\Vert P-Q\Vert )\le M_\star \left(\frac{\Vert V\Vert }{d}\right), \end{aligned}$$ where the estimating function $M_\star (x)$ , $x\in \bigl [0,\frac{1}{2}\bigr )$ , is given by $$\begin{aligned} M_\star (x)=\frac{1}{2}\,\,n_{_\#}(x)\,\arcsin \left(\frac{4\pi }{\pi ^2+4}\right) +\frac{1}{2}\,\arcsin \left(\frac{\pi ( x-\varkappa _{n_{_\#}(x)})}{1-2\varkappa _{n_{_\#}(x)})}\right), \end{aligned}$$ with $n_{_\#}(x)=\max \left\{ n\,\bigr |\,\,n\in \{0\}\cup {\mathbb N }\,, \varkappa _n\le x\right\} $ . The bound obtained is essentially stronger than the previously known estimates for $\Vert P-Q\Vert .$ Furthermore, this bound ensures that $\Vert P-Q\Vert <1$ and, thus, that the spectral subspaces $\mathrm{Ran }(P)$ and $\mathrm{Ran }(Q)$ are in the acute-angle case whenever $\Vert V\Vert <c_\star \,d$ , where $$\begin{aligned} c_\star =16\,\,\frac{\pi ^6-2\pi ^4+32\pi ^2-32}{(\pi ^2+4)^4}=0.454169\ldots . \end{aligned}$$ Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic $\sin 2\theta $ estimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.