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 $[\mathrm{Sym}^n(S)]$ .     

Minimal simplices inscribed in a convex body

Measuring how far a convex body $\mathcal{K }$ (of dimension $n$ ) with a base point ${O}\in \,\text{ int }\,\mathcal{K }$ is from an inscribed simplex $\Delta \ni {O}$ in “minimal” position, the interior point ${O}$ can display regular or singular behavior. If ${O}$ is a regular point then the $n+1$ chords emanating from the vertices of $\Delta $ and meeting at ${O}$ are affine diameters, chords ending in pairs of parallel hyperplanes supporting $\mathcal{K }$ . At a singular point ${O}$ the minimal simplex $\Delta $ degenerates. In general, singular points tend to cluster near the boundary of $\mathcal{K }$ . As connection to a number of difficult and unsolved problems about affine diameters shows, regular points are elusive, often non-existent. The first result of this paper uses Klee’s fundamental inequality for the critical ratio and the dimension of the critical set to obtain a general existence for regular points in a convex body with large distortion (Theorem A). This, in various specific settings, gives information about the structure of the set of regular and singular points (Theorem B). At the other extreme when regular points are in abundance, a detailed study of examples leads to the conjecture that the simplices are the only convex bodies with no singular points. The second and main result of this paper is to prove this conjecture in two different settings, when (1) $\mathcal{K }$ has a flat point on its boundary, or (2) $\mathcal{K }$ has $n$ isolated extremal points (Theorem C).     

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.     

Existence of positive periodic solutions for second-order functional differential equations

In this paper, we show the existence of positive $T$ -periodic solutions of second-order functional differential equations $u^{\prime \prime }(t)-\rho ^2u(t)+\lambda g(t)f(u(t-\tau (t)))=0,\ \ t\in \mathbb R , $ where $\rho >0$ is a constant, $g\in C(\mathbb R ,[0,\infty ))$ , $\tau \in C(\mathbb R ,\mathbb R )$ are $T$ -periodic functions, $f\in C([0,\infty ),[0,\infty ))$ and $\lambda $ is a positive parameter. Our approach based on global bifurcation theorem.     

Quasirandom permutations are characterized by 4-point densities

For permutations ${\pi}$ and ${\tau}$ of lengths ${|\pi|\le|\tau|}$ , let ${t(\pi,\tau)}$ be the probability that the restriction of ${\tau}$ to a random ${|\pi|}$ -point set is (order) isomorphic to ${\pi}$ . We show that every sequence ${\{\tau_j\}}$ of permutations such that ${|\tau_j|\to\infty}$ and ${t(\pi,\tau_j)\to 1/4!}$ for every 4-point permutation ${\pi}$ is quasirandom (that is, ${t(\pi,\tau_j)\to 1/|\pi|!}$ for every ${\pi}$ ). This answers a question posed by Graham.     

Subgroup properties of pro-p extensions of centralizers

We prove that a finitely generated pro- $p$ group acting on a pro- $p$ tree $T$ with procyclic edge stabilizers is the fundamental pro- $p$ group of a finite graph of pro- $p$ groups with vertex groups being stabilizers of certain vertices of $T$ and edge groups (when non-trivial) being stabilizers of certain edges of $T$ , in the following two situations: (1) the action is $n$ -acylindrical, i.e., any non-identity element fixes not more than $n$ edges; (2) the group $G$ is generated by its vertex stabilizers. This theorem is applied to obtain several results about pro- $p$ groups from the class $\mathcal L $ defined and studied in Kochloukova and Zalesskii (Math Z 267:109–128, 2011) as pro- $p$ analogues of limit groups. We prove that every pro- $p$ group $G$ from the class $\mathcal L $ is the fundamental pro- $p$ group of a finite graph of pro- $p$ groups with infinite procyclic or trivial edge groups and finitely generated vertex groups; moreover, all non-abelian vertex groups are from the class $\mathcal L $ of lower level than $G$ with respect to the natural hierarchy. This allows us to give an affirmative answer to questions 9.1 and 9.3 in Kochloukova and Zalesskii (Math Z 267:109–128, 2011). Namely, we prove that a group $G$ from the class $\mathcal L $ has Euler–Poincaré characteristic zero if and only if it is abelian, and if every abelian pro- $p$ subgroup of $G$ is procyclic and $G$ itself is not procyclic, then $\mathrm{def}(G)\ge 2$ . Moreover, we prove that $G$ satisfies the Greenberg–Stallings property and any finitely generated non-abelian subgroup of $G$ has finite index in its commensurator.     

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 $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

Euclidean Steiner trees optimal with respect to swapping 4-point subtrees

The Steiner tree problem in Euclidean space $E^3$ asks for a minimum length network $T$ , called a Euclidean Steiner Minimum Tree (ESMT), spanning a given set of points. This problem is NP-hard and the hardness is inherently due to the number of feasible topologies (underlying graph structure of $T$ ) which increases exponentially as the number of given points increases. Planarity is a very strong condition that gives a big difference between the ESMT problem in the Euclidean plane $E^2$ and in Euclidean $d$ -space $E^d (d\ge 3)$ : the ESMT problem in the plane is practically solvable whereas the ESMT problem in $d$ -space is really intractable. The simplest tree rearrangement technique is to repeatedly replace a subtree spanning four nodes in $T$ with another subtree spanning the same four nodes. This technique is referred to as the Swapping 4-point Topology/ Tree technique in this paper. An indicator (or quasi-indicator) of $T$ plays a similar role in the optimization of the length $L(T)$ of $T$ in the discrete topology space (the underlying graph structure of $T$ ) to the derivative of a differentiable function which indicates a fastest direction of descent. $T$ will be called S4pT-optimal if it is optimal with respect to swapping 4-point subtrees. In this paper we first make a complete analysis of 4-point trees in Euclidean space exploring all possible types of 4-point trees and their properties. We review some known indicators of 4-point ESMTs in $E^2$ , and give some simple geometric proofs of these indicators. Then, we translate these indicators to $E^3$ , producing eight quasi-indicators in $E^3$ using computational experiments, the best quasi-indicator $\rho _\mathrm{osr}$ is sifted with an effectiveness for randomly generated 4-point sets as high as 98.62 %. Finally we show how $\rho _\mathrm{osr}$ is used to find an S4pT-optimal ESMT on 14 probability vectors in $4d$ -space with a detailed example.     

Ricci flow of homogeneous manifolds

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset $\mathcal H _{q,n}$ of the variety of $(q+n)$ -dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension $n$ with a $q$ -dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a $2$ -parameter subspace of $\mathcal H _{1,3}$ reaching most of $3$ -dimensional geometries, and a $2$ -parameter family in $\mathcal H _{0,n}$ of left-invariant metrics on $n$ -dimensional compact and non-compact semisimple Lie groups.     

Coset intersection of irreducible plane cubics

In a projective plane $PG(2,\mathbb K )$ over an algebraically closed field $\mathbb K $ of characteristic $p\ge 0$ , let $\Omega $ be a pointset of size $n$ with $5\le n \le 9$ . The coset intersection problem relative to $\Omega $ is to determine the family $\mathbf F$ of irreducible cubics in $PG(2,\mathbb K )$ for which $\Omega $ is a common coset of a subgroup of the additive group $(\mathcal F ,+)$ for every $\mathcal F \in \mathbf F$ . In this paper, a complete solution of this problem is given.     

\alpha -Admissibility for Ritt Operators

Let $T:X\rightarrow X$ be a power bounded operator on Banach space. An operator $C:X\rightarrow Y$ is called admissible for $T$ if it satisfies an estimate $\sum _k\Vert CT^k(x)\Vert ^2\,\le M^2\Vert x\Vert ^2$ . Following Harper and Wynn, we study the validity of a certain Weiss conjecture in this discrete setting. We show that when $X$ is reflexive and $T$ is a Ritt operator satisfying a appropriate square function estimate, $C$ is admissible for $T$ if and only if it satisfies a uniform estimate $(1-\vert \omega \vert ^2)^{\frac{1}{2}}\Vert C(I-\omega T)^{-1}\Vert \,\le K\,$ for $\omega \in \mathbb{C }$ , $\vert \omega \vert <1$ . We extend this result to the more general setting of $\alpha $ -admissibility. Then we investigate a natural variant of admissibility involving $R$ -boundedness and provide examples to which our general results apply.     

Stratonovich’s signatures of Brownian motion determine Brownian sample paths

The signature of Brownian motion in $\mathbb R ^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\ge 2$ , almost all Brownian motion sample paths (running up to time $T$ ) are determined by their signature over $[0,T]$ .     

Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in {\varvec{R}}^{2n}

In this paper, let $n$ be a positive integer and $P=diag(-I_{n-\kappa },I_\kappa ,-I_{n-\kappa },I_\kappa )$ for some integer $\kappa \in [0, n]$ , we prove that for any compact convex hypersurface $\Sigma $ in $\mathbf{R}^{2n}$ with $n\ge 2$ there exist at least two geometrically distinct P-invariant closed characteristics on $\Sigma $ , provided that $\Sigma $ is P-symmetric, i.e., $x\in \Sigma $ implies $Px\in \Sigma $ . This work is shown to extend and unify several earlier works on this subject.     

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.     

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.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Homology of Littlewood complexes

Let $V$ be a symplectic vector space of dimension $2n$ . Given a partition $\lambda $ with at most $n$ parts, there is an associated irreducible representation $\mathbf{{S}}_{[\lambda ]}(V)$ of $\mathbf{{Sp}}(V)$ . This representation admits a resolution by a natural complex $L^{\lambda }_{\bullet }$ , which we call the Littlewood complex, whose terms are restrictions of representations of $\mathbf{{GL}}(V)$ . When $\lambda $ has more than $n$ parts, the representation $\mathbf{{S}}_{[\lambda ]}(V)$ is not defined, but the Littlewood complex $L^{\lambda }_{\bullet }$ still makes sense. The purpose of this paper is to compute its homology. We find that either $L^{\lambda }_{\bullet }$ is acyclic or it has a unique nonzero homology group, which forms an irreducible representation of $\mathbf{{Sp}}(V)$ . The nonzero homology group, if it exists, can be computed by a rule reminiscent of that occurring in the Borel–Weil–Bott theorem. This result can be interpreted as the computation of the “derived specialization” of irreducible representations of $\mathbf{{Sp}}(\infty )$ and as such categorifies earlier results of Koike–Terada on universal character rings. We prove analogous results for orthogonal and general linear groups. Along the way, we will see two topics from commutative algebra: the minimal free resolutions of determinantal ideals and Koszul homology.     

Privileged Regions in Critical Strips of Non-lattice Dirichlet Polynomials

This paper shows, by means of Kronecker’s theorem, the existence of infinitely many privileged regions called $r$ -rectangles (rectangles with two semicircles of small radius $r$ ) in the critical strip of each function $L_{n}(z)\!:=\!$ $1-\sum _{k=2}^{n}k^{z}$ , $n\!\ge \!2$ , containing exactly $\left[ \dfrac{T\log n}{2\pi }\right] +1$ zeros of $L_{n}(z)$ , where $T$ is the height of the $r$ -rectangle and $\left[\cdot \right]$ represents the integer part.     

Explicit Constructions of Centrally Symmetric k-Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric $2$ -neighborly $d$ -dimensional polytopes with about $3^{d/2}\approx (1.73)^d$ vertices and of centrally symmetric $k$ -neighborly $d$ -polytopes with about $2^{{3d}/{20k^2 2^k}}$ vertices. Using this result, we construct for a fixed $k\ge 2$ and arbitrarily large $d$ and $N$ , a centrally symmetric $d$ -polytope with $N$ vertices that has at least $\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$ faces of dimension $k-1$ , where $\gamma _2=1/\sqrt{3}\approx 0.58$ and $\gamma _k = 2^{-3/{20k^2 2^k}}$ for $k\ge 3$ . Another application is a construction of a set of $3^{\lfloor d/2 -1\rfloor }-1$ points in $\mathbb R ^d$ every two of which are strictly antipodal as well as a construction of an $n$ -point set (for an arbitrarily large $n$ ) in $\mathbb R ^d$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.     

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.