Planar Point Sets With Large Minimum Convex Decompositions

We show the existence of sets with $n$ points ( $n\ge 4$ ) for which every convex decomposition contains more than $\frac{35}{32}n-\frac{3}{2}$ polygons, which refutes the conjecture that for every set of $n$ points there is a convex decomposition with at most $n+C$ polygons. For sets having exactly three extreme points we show that more than $n+\sqrt{2(n-3)}-4$ polygons may be necessary to form a convex decomposition.     

Integer-valued polynomials over matrices and divided differences

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$ , the $k$ th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$ . If in addition $D$ has zero Jacobson radical then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$ , that is, conjugate integral elements of degree $n$ over $D$ .     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

Towards a de Bruijn–Erdős Theorem in the L_1-Metric

A well-known theorem of de Bruijn and Erd?s states that any set of $n$ non-collinear points in the plane determines at least $n$ lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of betweenness. In this paper, we prove that the answer is affirmative for sets of $n$ points in the plane with the $L_1$ metric, provided that no two points share their $x$ - or $y$ -coordinate. In this case, either there is a line that contains all $n$ points, or $X$ induces at least $n$ distinct lines. If points of $X$ are allowed to share their coordinates, then either there is a line that contains all $n$ points, or $X$ induces at least $n/37$ distinct lines.     

Lattice-free sets,multi-branch split disjunctions,and mixed-integer programming

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (Discret Optim 5:724–734, 2008). By analyzing $n$ -dimensional lattice-free sets, we prove that for every integer $n$ there exists a positive integer $t$ such that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with $n$ integer variables is a $t$ -branch split cut. We use this result to give a finite cutting-plane algorithm to solve mixed-integer programs. We also show that the minimum value $t$ , for which all facets of polyhedral mixed-integer sets with $n$ integer variables can be generated as $t$ -branch split cuts, grows exponentially with $n$ . In particular, when $n=3$ , we observe that not all facet-defining inequalities are 6-branch split cuts.     

On Sets Defining Few Ordinary Lines

Let $P$ P be a set of $n$ n points in the plane, not all on a line. We show that if $n$ n is large then there are at least $n/2$ n / 2 ordinary lines, that is to say lines passing through exactly two points of $P$ P . This confirms, for large $n$ n , a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $n-C$ n - C ordinary lines for some absolute constant $C$ C . We also solve, for large $n$ n , the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $P$ P . Underlying these results is a structure theorem which states that if $P$ P has at most $Kn$ K n ordinary lines then all but O(K) points of $P$ P lie on a cubic curve, if $n$ n is sufficiently large depending on $K$ K .     

Acyclic Systems of Permutations and Fine Mixed Subdivisions of Simplices

A fine mixed subdivision of a $(d-1)$ -simplex $T$ of size $n$ gives rise to a system of  ${d \atopwithdelims ()2}$ permutations of $[n]$ on the edges of $T$ , and to a collection of $n$ unit $(d-1)$ -simplices inside $T$ . Which systems of permutations and which collections of simplices arise in this way? The Spread Out Simplices Conjecture of Ardila and Billey proposes an answer to the second question. We propose and give evidence for an answer to the first question, the Acyclic System Conjecture. We prove that the system of permutations of $T$ determines the collection of simplices of $T$ . This establishes the Acyclic System Conjecture as a first step towards proving the Spread Out Simplices Conjecture. We use this approach to prove both conjectures for $n=3$ in arbitrary dimension.     

Strong convergence of a splitting projection method for the sum of maximal monotone operators

In this paper, a projective-splitting method is proposed for finding a zero of the sum of $n$ maximal monotone operators over a real Hilbert space $\mathcal{H }$ . Without the condition that either $\mathcal{H }$ is finite dimensional or the sum of $n$ operators is maximal monotone, we prove that the sequence generated by the proposed method is strongly convergent to an extended solution for the problem, which is closest to the initial point. The main results presented in this paper generalize and improve some recent results in this topic.     

A construction of integer-valued polynomials with prescribed sets of lengths of factorizations

For an arbitrary finite non-empty set $S$ of natural numbers greater $1$ , we construct $f\in \text{ Int }(\mathbb{Z })=\{g\in \mathbb{Q }[x]\mid g(\mathbb{Z })\subseteq \mathbb{Z }\}$ such that $S$ is the set of lengths of $f$ , i.e., the set of all $n$ such that $f$ has a factorization as a product of $n$ irreducibles in $\text{ Int }(\mathbb{Z })$ . More generally, we can realize any finite non-empty multi-set of natural numbers greater 1 as the multi-set of lengths of the essentially different factorizations of $f$ .     

Caffarelli–Kohn–Nirenberg inequality on metric measure spaces with applications

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent $n \ge 3$ , then it has exactly the $n$ -dimensional volume growth. As an application, if an $n$ -dimensional Finsler manifold of non-negative $n$ -Ricci curvature satisfies the Caffarelli–Kohn–Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.     

Functional limit theorems for random regular graphs

Consider $d$ uniformly random permutation matrices on $n$ labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree $2d$ on $n$ vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as $n$ grows to infinity, either when $d$ is kept fixed or grows slowly with $n$ . In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein’s method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn–Szemerédi argument for estimating the second largest eigenvalue for all values of $d$ and $n$ .     

Rigidity of Einstein manifolds with positive scalar curvature

We prove that if $M^n(n\ge 4)$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $R_0>\sigma _{n}K_{\max }$ , where $\sigma _n\in (\frac{1}{4},1)$ is an explicit positive constant depending only on $n$ , then $M$ must be isometric to a spherical space form. Moreover, we prove that if an $n(\ge {\!\!4})$ -dimensional compact Einstein manifold satisfies $K_{\min }\ge \eta _n R_0,$ where $\eta _n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is locally symmetric. It should be emphasized that the pinching constant $\eta _n$ is optimal when $n$ is even. We then obtain some rigidity theorems for Einstein manifolds under $(n-2)$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $M$ is an $n(\ge {\!\!4})$ -dimensional compact Einstein submanifold in the simply connected space form $F^{N}(c)$ with constant curvature $c\ge 0$ , and the normalized scalar curvature $R_0$ of $M$ satisfies $R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$ where $A_n=n^3-5n^2+8n$ , and $H$ is the mean curvature of $M$ , then $M$ is isometric to a standard $n$ -sphere.     

Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

A Gelfand–Tsetlin scheme of depth \(N\) is a triangular array with \(m\) integers at level \(m\) , \(m=1,\ldots ,N\) , subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand–Tsetlin schemes with arbitrary fixed \(N\) th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its \(q\) -deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302, 2007). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).     

Syzygies and tensor product of modules

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$ -modules. Assume $n$ is a nonnegative integer and that the tensor product $M\otimes _{R}N$ is an $(n+c)$ th syzygy of some finitely generated $R$ -module. If ${{\mathrm{Tor}}}^{R}_{>0}(M,N)=0$ , then both $M$ and $N$ are $n$ th syzygies of some finitely generated $R$ -modules.     

Dimension Reduction for Finite Trees in \varvec{\ell _1}

We show that every $n$ -point tree metric admits a $(1+\varepsilon )$ -embedding into $\ell _1^{C(\varepsilon ) \log n}$ , for every $\varepsilon > 0$ , where $C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$ . This matches the natural volume lower bound up to a factor depending only on $\varepsilon $ . Previously, it was unknown whether even complete binary trees on $n$ nodes could be embedded in $\ell _1^{O(\log n)}$ with $O(1)$ distortion. For complete $d$ -ary trees, our construction achieves $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

f-Vectors Implying Vertex Decomposability

We prove that if a pure simplicial complex $\Delta $ of dimension $d$ with $n$ facets has the least possible number of $(d-1)$ -dimensional faces among all complexes with $n$ faces of dimension $d$ , then it is vertex decomposable. This answers a question of J. Herzog and T. Hibi. In fact, we prove a generalization of their theorem using combinatorial methods.     

A Remark on The Geometry of Spaces of Functions with Prime Frequencies

For any positive integer r, denote by \({\mathcal{P}_{r}}\) the set of all integers \({\gamma \in \mathbb{Z}}\) having at most r prime divisors. We show that \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) , the space of all continuous functions on the circle \({\mathbb{T}}\) whose Fourier spectrum lies in \({\mathcal{P}_{r}}\) , contains a complemented copy of \({\ell^{1}}\) . In particular, \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) is not isomorphic to \({C(\mathbb{T})}\) , nor to the disc algebra \({A(\mathbb{D})}\) . A similar result holds in the L ¹ setting.     

Convex equipartitions: the spicy chicken theorem

We show that, for any prime power $n$ and any convex body $K$ (i.e., a compact convex set with interior) in $\mathbb{R }^d$ , there exists a partition of $K$ into $n$ convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov–Borsuk–Ulam theorem for convex sets in the model spaces of constant curvature.     

Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Given any smooth toric surface $S$ , we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov–Witten invariants of the $n$ -fold symmetric product stack $[\mathrm{Sym}^n(S)]$ of $S$ to the 3-point extremal Gromov–Witten invariants of the Hilbert scheme $\mathrm{Hilb}^n(S)$ of $n$ points on $S$ . As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert–Chow morphism $\mathrm{Hilb}^n(S) \rightarrow \mathrm{Sym}^n(S)$ and yields a method of reconstructing the cup product for $\mathrm{Hilb}^n(S)$ from the orbifold invariants of $[\mathrm{Sym}^n(S)]$ .     

Canonical subgroups via Breuil–Kisin modules

Let $p>2$ be a rational prime and $K/ \mathbb Q _p$ be an extension of complete discrete valuation fields. Let $\mathcal G $ be a truncated Barsotti–Tate group of level $n$ , height $h$ and dimension $d$ over $\mathcal{O }_K$ with $0<d<h$ . In this paper, we show that if the Hodge height of $\mathcal G $ is less than $1/(p^{n-2}(p+1))$ , then there exists a finite flat closed subgroup scheme of $\mathcal G $ of order $p^{nd}$ over $\mathcal{O }_K$ with standard properties as the canonical subgroup.