A stability property of symplectic packing

We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H ²(M, Q) there exists a number N ₀ such that for every N≥N ₀, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N ₀. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of Gromov invariants. Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999     

Poisson Traces and D-Modules on Poisson Varieties

To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism ${\phi : X \to Y}To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism f: X ? Y{\phi : X \to Y} and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M_f(X,N){M_\phi(X,N)} on X, and prove that it is holonomic if X has finitely many symplectic leaves, f{\phi} is finite, and N is coherent.     

On the intersection form of surfaces

Given a closed, oriented surface M, the algebraic intersection of closed curves induces a symplectic form Int(., .) on the first homology group of M. If M is equipped with a Riemannian metric g, the first homology group of M inherits a norm, called the stable norm. We study the norm of the bilinear form Int(., .), with respect to the stable norm.     

Counterexamples to the Kneser conjecture in dimension four

We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM ₀#M ₁ unlessM ₀ orM ₁ is homeomorphic toS ⁴. LetN be the nucleus of the minimal elliptic Enrique surfaceV ₁(2, 2) and putM=N∪ _∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S ²×S²) is diffeomorphic toM ₀#M ₁ for non-simply connected closed smooth four-manifoldsM ₀ andM ₁ if and only ifk≥8. On the other hand we show thatM is homeomorphic toM ₀#M ₁ for closed topological four-manifoldsM ₀ andM ₁ withπ ₁(M_i)=ℤ/2.     

Submersions on open symplectic manifolds

Let M be an open manifold with a symplectic form Ω, and N a manifold with dimN<dimM. We prove that submersions with symplectic fibres satisfy the h-principle. Such submersions define Dirac manifold structures on the given manifold. As an application to this result we show that CPn?CPk−1 admits a submersion into R2(2k−n) with symplectic fibres for n/2<k?n.     

Relaxation Methods for Non-Hermitian Linear Systems

For the matrix equation Ax = b, we consider here two splittings A = M₁ ? N₁ = M₂ ? N₂ of the matrix A, where M ₁ ? (A + A*)/2 is the Hermitian part of A, and M ₂ ? I + (A ? A*)/2 is the identity plus the skew-Hermitian part of A. To these two splittings of A, we apply an extrapolation, with extrapolation factor ω, and we find associated regions for ω, in the complex plane, for which these extrapolated splittings yield convergent iterative methods. From this, further applications to semiiterative methods are indicated.     

Spectral flexibility of symplectic manifolds <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">M</Emphasis>

We consider Riemannian metrics compatible with the natural symplectic structure on T ² × M, where T ² is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ₁. We show that λ₁ can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic question.     

Numerical shadows: Measures and densities on the numerical range

For any operator M acting on an N-dimensional Hilbert space H_N we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of M. The shadow of M at point z is defined as the probability that the inner product (Mu, u) is equal to z, where u stands for a random complex vector from H_N, satisfying ||u||=1. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian M its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional B-spline. In the case of a normal M the numerical shadow corresponds to a shadow of a transparent solid simplex in R^N-1 onto the complex plane. Numerical shadow is found explicitly for Jordan matrices J_N, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.     

Hearing the weights of weighted projective planes

Which properties of an orbifold can we “hear,” i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kähler orbifolds: weighted projective planes \(M := {\mathbb{C}}P^2(N_1, N_2, N_3)\) with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on M determine the weights N ₁, N ₂, and N ₃. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of N ₁, N ₂, and N ₃, we can hear whether M is endowed with an extremal Kähler metric.     

An algorithm for associative bilinear forms

We establish the equivalence between the problem of existence of associative bilinear forms and the problem of solvability in commutative power-associative finite-dimensional nil-algebras. We use the tensor product to find sufficient and necessary conditions to assure the existence of associative bilinear forms in an algebra A. The result gives us an algorithm to describe the space of associative bilinear forms for an algebra when its constants of structure γ_i,j,k for i,j,k=1,…,n are known.     

Normal Forms of Symplectic Matrices

Abstract In this paper, we prove that for every symplectic matrix M possessing eigenvalues on the unit circle, there exists a symplectic matrix P such that P ⁻¹ MP is a symplectic matrix of the normal forms defined in this paper. Partially supported by the NSF, MCSEC of China, and the Qiu Shi Sci. Tech. Foundation * Associate Member of the ICTP     

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold(M,ω) into a smooth symplectic manifold . Then we study how the formality and the Lefschetz property of are compared with that of (M,ω). We also study the formality of the symplectic blow-up of (M,ω) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

Δ-Monotone arrangements of real numbers

Given a set of M × N real numbers, can these always be labeled as x_i,j; i = 1,…, M; j = 1,…, N; such that x_i+1,j+1 ? x_i+1,j ? x_i,j+1 + x_ij ≥ 0, for every (i, j) where 1 ≤ i ≤ M ? 1, 1 ≤ j ≤ N ? 1? For M = N = 3, or smaller values of M, N it is shown that there is a “uniform” rule. However, for max(M, N) > 3 and min(M, N) ≥ 3, it is proved that no uniform rule can be given. For M = 3, N = 4 a way of labeling is demonstrated. For general M, N the problem is still open although, for a special case where all the numbers are 0's and 1's, a solution is given.     

Graded Poisson structures on the algebra of differential forms

We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field.     

The extrinsic holonomy Lie algebra of a parallel submanifold

We investigate parallel submanifolds of a Riemannian symmetric space N. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full, complete, parallel submanifolds of N, usually called “1-fullness” of M. Furthermore, for every parallel submanifold \({M\subset N}\) we consider the pullback bundle T N| _M with the linear connection induced by \({\nabla^N}\) . Then there exists a distinguished parallel subbundle \({\mathcal {O}M}\) , usually called the “second osculating bundle” of M. Given a parallel isometric immersion from a symmetric space M into N, we can describe the “extrinsic” holonomy Lie algebra of \({\mathcal {O} M}\) by means of the second fundamental form and the curvature tensor of N at some fixed point. If moreover N is simply connected and M is even a full symmetric submanifold of N, then we will calculate the “extrinsic” holonomy Lie algebra of T N| _M in an explicit form.     

Elementary maps on nest algebras

Let A₁, A₂ be algebras and let M:A₁→A₂, M^∗:A₂→A₁ be maps. An elementary map of A₁×A₂ is an ordered pair (M,M^∗) such that

     

Boolean designs and self-dual matroids

A proper splitting of a rectangular matrix A is one of the form A = M ? N, where A and M have the same range and null spaces. This concept was introduced by R. Plemmons as a means of generalizing to rectangular and singular matrices the concept of a regular splitting of a nonsingular matrix as introduced by R. Varga. In consideration of the linear system Ax=b, A. Berman and R. Plemmons used a proper splitting of A into M ? N and showed that the iteration x⁽ⁱ⁺¹⁾=M⁺Nx⁽ⁱ⁾+M⁺b converges to A⁺b, the best least-squares solution to the system, if and only if the spectral radius of M⁺N is less than one. The purpose of this paper is to further develop the characteristics of proper splittings and to extend these previous results by replacing the Moore-Penrose generalized inverse with a least-squares g-inverse, a minimum-norm g-inverse, or a g-inverse. Also, some criteria are given for comparing convergence rates of M_i^?N_i, where A = M₁?N₁ = M₂?N₂, and a method is developed for constructing proper splittings of special types of matrices.     

Weighted sum of the extensions of the representations of quadratic forms

Let L, N and M be positive definite integral \({\mathbb{Z}}\) -lattices. In this paper, we show some relation between the weighted sum of representations of L and N by gen(M) and the weighted sum of extensions of \(\tilde M_{\tilde \sigma}\) in the gen(M _σ) via N _η when M is even and gcd(dL, dM) =  1. As a consequence of the particular case when M is even unimodular, we recapture the Böcherer formula (13) in (Böcherer, Maths Z 183:21–46, 1983) for the relation of the Fourier coefficients between Eisenstein series and Jacobi–Eisenstein series.     

On newforms for Kohnen plus spaces

In this article, we investigate the plus space of level N, where 4^?1 N is a square-free (not necessarily odd) integer. This is a generalization of Kohnen’s work. We define a Hecke isomorphism ${\wp_k}In this article, we investigate the plus space of level N, where 4⁻¹ N is a square-free (not necessarily odd) integer. This is a generalization of Kohnen’s work. We define a Hecke isomorphism ?_k{\wp_k} of M _k+1/2(4M) onto M_k+1/2⁺(8M){M_{k+1/2}^+(8M)} for any odd positive integer M. The methods of the proof of the newform theory are this isomorphism, Waldspurger’s theorem, and the dimension identity.