Stationary Gromov–Witten Invariants of Projective Spaces

We represent stationary descendant Gromov–Witten invariants of projective space, up to explicit combinatorial factors, by polynomials. One application gives the asymptotic behaviour of the large degree behaviour of stationary descendant Gromov–Witten invariants in terms of intersection numbers over the moduli space of curves. We also show that primary Gromov–Witten invariants are"virtual" stationary descendants and hence the string and divisor equations can be understood purely in terms of stationary invariants.     

Orbifold Gromov–Witten Invariants of Weighted Blow-up at Smooth Points

In this paper, one considers the change of orbifold Gromov–Witten invariants under weighted blow-up at smooth points. Some blow-up formula for Gromov–Witten invariants of symplectic orbifolds is proved. These results extend the results of manifolds case to orbifold case.     

Scattering Diagrams,Sheaves, and Curves

We review the recent proof of the N. Takahashi's conjecture on genus 0 Gromov–Witten invariants of(P~2, E), where E is a smooth cubic curve in the complex projective plane P~2. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov–Witten invariants of(P~2, E) and the world of moduli spaces of coherent sheaves on P~2. Using this bridge, the N. Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on P~2. This survey is based on a three hours lecture series given as part of the Beijing–Zurich moduli workshop in Beijing, 9–12 September 2019.     

A Quantum Modification of Relative Chen–Ruan Cohomology

In this paper, by using the de Rham model of Chen–Ruan cohomology, we define the relative Chen–Ruan cohomology ring for a pair of almost complex orbifold(G, H) with H being an almost sub-orbifold of G. Then we use the Gromov–Witten invariants ofG, the blow-up of G along H,to give a quantum modification of the relative Chen–Ruan cohomology ring H*CR(G, H) when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.     

Chamber Structure for Some Equivariant Relative Gromov–Witten Invariants of P~1 in Genus 0

In this paper, we study genus 0 equivariant relative Gromov–Witten invariants of P1 whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighbouring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber structure here shares some similarities to that of double Hurwitz numbers.     

Topological recursion relations from Pixton relations

We propose an algorithm to derive tautological relations from Pixton relations. We carry out this algorithm explicitly to derive some results in genus 0, 1, 2, 3 and analyze the possibility to generalize to higher genera. As an application, some results about reconstruction of Gromov–Witten invariants can be derived.     

Classification of hypersurfaces with parallel Mbius second fundamental form in S~(n+1)

Let M~n(n≥2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S~(n+1). Then M~n is associated with a so-called Mobius metric g,and a Mobius second fundamental form B which are invariants of M~n under the Mobiustransformation group of S~(n+1). In this paper, we classify all umbilic-free hypersurfaces withparallel Mobius second fundamental form.     

The Aubry Set and Mather Set in the Embedded Contact Hamiltonian System

We embed the Aubry set and Mather set in the Tonelli Hamiltonian system to the contact Hamiltonian system. We find the embedded Aubry set is the set of non-wandering points of the contact Hamiltonian system and the Mather set is the support of set of the invariant Borel probability measures for the contact Euler–Lagrange flow. From this viewpoint, we can conclude that Aubry set and Mather set are symplectic invariants.     

The Poincaré Series of Relative Invariants of Finite Pseudo-Reflection Groups

Let F be a field with characteristic 0,V=F~n the n-dimensional vector space over F and let G be a finite pseudo-reflection group which acts on V.Let χ:G→F~* be a 1-dimensional representation of G.In this article we show that X(g)=(detg)~α(0≤α≤r-1),where g∈G and r is the order of g.In addition,we characterize the relation between the relative invariants and the invariants of the group G,and then we use Molien's Theorem of invariants to compute the Poincaré series of relative invariants.     

Every Sub-Riemannian Manifold Is the Gromov–Hausdorff Limit of a Sequence Riemannian Manifolds

In this paper, we will show that every sub-Riemannian manifold is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds.     

CALCULATION OF α_3 COEFFICIENT OF SCALAR FIELD BY POINT SEPARATION METHOD

The α_3 coefficient of scalar field in arbitrary Riemann space-time background is calculated by point separation method in this article. The primary expression of α_3 involves 25 invariants. Using the identities derived in the appendix, we can reduce α_3 from 25 invariants to 17 invariants which agree with Gilkey's formula for α_s coefficient of the scalar field derived by geometric methods and reasons. Recently we derived two important invariant identilies, using these identities we can reduce α_3 from 17 invariants to 15 invariants which would much simplify the calculation of the energymomentum tensor of vacuum polarization.     

ON A COMPLETE SET OF COMBINATORIAL INVARIANTS OF A TREE

The present paper will give out a representation sequence of path lengths anda set of combinatorial invariants of a tree.And it is shown that they are closelyrelated.Then a complete set of combinatorial invariants of a tree is obtained. First,we define the sequence of path lengths of a tree T of order n     

EQUIVARIANT MINIMAL IMMERSIONS FROM S~3 INTO ■

Associated with an immersion φ : S~3→■, we can define a canonical bundle endomorphism F : TS~3→ TS~3 by the pull back of the K?hler form of ■. In this article,related to F we study equivariant minimal immersions from S~3 into ■ under the additional condition(?_XF)X = 0 for all X ∈ ker(F). As main result, we give a complete classification of such kinds of immersions. Moreover, we also construct a typical example of equivariant non-minimal immersion φ : S~3→■ satisfying(?_XF)X = 0 for all X ∈ ker(F), which is neither Lagrangian nor of CR type.     

The Metastable Behavior of the Two Dimensional Ising Model

There are continuous efforts in recent years in order to have a better understan-ding of metastable bekavior of a statistical mechanics system. We consider it in twodimensions, solve problems proposed in [Sch]. By the idea of [FW] we are able toshapen the previous results [NS], and to identify the critical droplet naturally.Let Λ be a 2-dimensional lattice torus, and X = { - 1, + 1}~Λ. For η∈X assignthe Hamiltonian     

Nonexistence of the NNSC-cobordism of Bartnik data

In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(∑_1~(n-1),γ_1,H_1) and(∑_2~(n-1),γ_2,H_2).We prove that given two metrics γ_1 and γ_2 on S~(n-1)(3≤n ≤ 7)with H_1 fixed,then(S~(n-1),γ_1,H_1) and(S~(n-1),γ_2,H_2) admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough(see Theorem 1.3).Moreover,we show that for n=3,a much weaker condition that the total mean curvature ∫_(s~2) H_2 dpγ_2 is large enough rules out NNSC-cobordisms(see Theorem 1.2);if we require the Gaussian curvature of γ_2 to be positive,we get a criterion for nonexistence of the trivial NNSCcobordism by using the Hawking mass and the Brown-York mass(see Theorem 1.1).For the general topology case,we prove that(∑_1~(n-1),γ_1,0) and(∑_2~(n-1),γ_2,H_2) admit no NNSC-cobordism provided the prescribed mean curvature H_2 is large enough(see Theorem 1.5).     

单位球面上具有非负M bius截面曲率的超曲面

Let x:M→S~(n 1)be a hypersurface in the (n 1)-dimensional unit sphere S~(n 1)without umbilic point. The M(?)bius invariants of x under the M(?)bius transformation group of S~(n 1) are M(?)bius metric,M(?)bius form,M(?)bius second fundamental form and Blaschke tensor.In this paper,we prove the following theorem: Let x:M→S~(n 1)(n＞2)be an umbilic free hypersurface in S~(n 1) with nonnegative M(?)bius sectional curvature and with vanishing M(?)bius form.Then x is locally M(?)bius equivalent to one of the following hypersurfaces:(i)the torus S~k(a)×S~(n-k)((1-a~2)~(1/2))with 1≤k≤n-1;(ii)the pre-image of the stereographic projection of the standard cylinder S~k×R~(n-k)(?)R~(n 1) with 1≤k≤n-1;(iii)the pre-image of the stereographic projection of the cone in R~(n 1):(?)(u,v,t)=(tu,tv), where(u,v,t)∈S~k(a)×S~(n-k-1)((1-a~2)~(1/2))×R~ .     

A Bishop–Phelps–Bollobás Theorem for Asplund Operators

By characterizing Asplund operators through Fréchet differentiability property of convex functions, we show the following Bishop–Phelps–Bollobás theorem: Suppose that X is a Banach space,T : X → C(K) is an Asplund operator with ║T║= 1, and that x_0 ∈ S_X, 0 ε satisfy ║T(x_0)║ 1-ε~2/2.Then there exist x_ε∈ S_X and an Asplund operator S : X → C(K) of norm one so that ║S(x_ε)║ = 1, x_0-x_ε ε and ║T-S║ ε.Making use of this theorem, we further show a dual version of Bishop–Phelps–Bollobás property for a strong Radon–Nikodym operator T : ?_1 → Y of norm one: Suppose that y_0~*∈ S_(Y~*), ε≥ 0 satisfy T~*(y_0~*) 1-ε~2/2. Then there exist y_ε~*∈ S_(Y~*), x_ε∈(±e_n), y_ε∈ S_Y, and a strong Radon–Nikodym operator S : ?_1 → Y of norm one so that (ⅰ)║S(x_ε)║= 1;(ⅱ) S(x_ε) = y_ε;(ⅲ)║T-S║ ε;(ⅳ)║S~*(y_ε~*)║=y_ε~*, y_ε= 1;(ⅴ)║y_0~*-y_ε~*║ ε and (ⅵ)║T~*-S~*║ ε,where(e_n) denotes the standard unit vector basis of ?_1.     

有限伪反射群的相对不变式的 Poincar\'e级数

Let F be a field with characteristic 0, V = Fn the n-dimensional vector space over F and let G be a finite pseudo-reflection group which acts on V . Let χ ： G→ F＊ be a 1- dimensional representation of G. In this article we show that χ（g） = （detg）α（0 ≤ α ≤ r - 1）, where g ∈ G and r is the order of g. In addition, we characterize the relation between the relative invariants and the invariants of the group G, and then we use Molien’s Theorem of invariants to compute the Poincar′e series of relative invariants.     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.     

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

We prove the existence of trajectories shadowing chains of heteroclinic orbits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system.The results are quite different for real and complex eigenvalues. General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate ε. If the frozen autonomous system has a hyperbolic equilibrium possessing transverse homoclinic orbits, we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order ε and changing with average rate of orderε| ln ε|. This provides a partial multidimensional extension of the results of A. Neishtadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.