Characterizing the round sphere by mean distance

We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main result of this paper is Theorem 4 stating that the round sphere of constant curvature 1 has maximal mean distance among Riemannian n-manifolds with Ricci curvature Ric?n−1, and that such a manifold is diffeomorphic to a sphere if the mean distance is close to .     

On eigenvalue pinching in positive Ricci curvature

We shall show that for manifolds with Ric≥n−1 the radius is close to π iff the (n+1)st eigenvalue is close to n. This extends results of Cheng and Croke which show that the diameter is close to π iff the first eigenvalue is close to n. We shall also give a new proof of an important theorem of Colding to the effect that if the radius is close to π, then the volume is close to that of the sphere and the manifold is Gromov-Hausdorff close to the sphere. From work of Cheeger and Colding these conditions imply that the manifold is diffeomorphic to a sphere. Oblatum 29-V-1998 & 4-II-1999 / Published online: 21 May 1999     

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature

Let L=Δ−∇φ⋅∇ be a symmetric diffusion operator with an invariant measure on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry-Émery Ricci curvature satisfying Ricm,n(L)?−(n−1), and therefore generalize a Cheng's result on the Laplacian (S.-Y. Cheng (1975) [8]) to the case of the diffusion operator.     

Three-dimensional Lie group actions on compact (4n+3)-dimensional geometric manifolds

The (4n+3)-dimensional sphere S⁴ⁿ⁺³ can be viewed as the boundary of the quaternionic hyperbolic space and the group PSp(n+1,1) of quaternionic hyperbolic isometries extends to a real analytic transitive action on S⁴ⁿ⁺³. We call the pair (PSp(n+1,1),S⁴ⁿ⁺³) a spherical Q C-C geometry. A manifold M locally modelled on this geometry is said to be a spherical Q C-C manifold. We shall classify all pairs (G,M) where G is a three-dimensional connected Lie group which acts smoothly and almost freely on a compact spherical Q C-C manifold M, preserving the geometric structure. As an application, we shall determine all compact 3-pseudo-Sasakian manifolds admitting spherical Q C-C structures.     

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to minus n+2 hyperplanes.Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accommodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h^n,o(V) spheres Sⁿ.     

Approximation of quantum tori by finite quantum tori for the quantum Gromov-Hausdorff distance

We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (H_n)_n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, then is the quantum Gromov-Hausdorff limit of any sequence for the natural quantum metric structures and when the lifts of σ_n to converge pointwise to σ. This allows us in particular to approximate the quantum tori by finite-dimensional C^*-algebras for the quantum Gromov-Hausdorff distance. Moreover, we also establish that if the length function l is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.     

Disjoint hamiltonian cycles in bipartite graphs

Let G=(X,Y) be a bipartite graph and define . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163-165. MR 28 # 4540] showed that if G is a bipartite graph on 2n vertices such that , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231-249].     

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.     

Harmonic cohomology of symplectic manifolds

Mathieu (Math. Helv. 70 (1995) 1) introduced a canonic filtration in the de Rham cohomology of a symplectic manifold and proved, that the middle filtration space is the space of harmonic cohomology classes. We give an interpretation of the other filtration spaces, prove a Künneth theorem for harmonic cohomology, prove Poincaré duality for harmonic cohomology and show how surjectivity of certain Lefschetz type mappings is related to properties of the filtration. For a closed symplectic manifold M we also introduce symplectic invariants , and show if M is of dimension 2n with n even.     

Operator identities involving the bivariate Rogers-Szegö polynomials and their applications to the multiple q-series identities

In this paper, we first give several operator identities involving the bivariate Rogers-Szegö polynomials. By applying the technique of parameter augmentation to the multiple q-binomial theorems given by Milne [S.C. Milne, Balanced summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997) 93-187], we obtain several new multiple q-series identities involving the bivariate Rogers-Szegö polynomials. These include multiple extensions of Mehler's formula and Rogers's formula. Our U(n+1) generalizations are quite natural as they are also a direct and immediate consequence of their (often classical) known one-variable cases and Milne's fundamental theorem for An or U(n+1) basic hypergeometric series in Theorem 1.49 of [S.C. Milne, An elementary proof of the Macdonald identities for , Adv. Math. 57 (1985) 34-70], as rewritten in Lemma 7.3 on p. 163 of [S.C. Milne, Balanced summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997) 93-187] or Corollary 4.4 on pp. 768-769 of [S.C. Milne, M. Schlosser, A new An extension of Ramanujan's summation with applications to multilateral An series, Rocky Mountain J. Math. 32 (2002) 759-792].     

The canonical Kaehler potential on the parameter space of the versal family of CR structures

Let (M,⁰T″) be a compact strongly pseudo convex CR structure with dim_RM=2n−1. Then, in [Michigan Math. J. 50 (2002) 517-549], the construction of the versal family of CR structures is settled. The purpose of this paper is to introduce a canonical Kaehler metric for the parameter space of this versal family if the CR structure admits a normal vector field and a non-vanishing CR n-form with the condition and

     

On the singularity probability of discrete random matrices

Let n be a large integer and Mn be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that Mn is singular. For a constant 0<p<1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of Mn satisfy this property, then the probability that Mn is singular is at most (p1/r+on(1)). All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of Mn are “fair coin flips” (taking the values +1,−1 each with probability 1/2), our general bound implies that the probability that Mn is singular is at most , improving on the previous best upper bound of , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In the special case where the entries of Mn are “lazy coin flips” (taking values +1,−1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that Mn is singular is at most , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223-240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603-628], which was obtained using tools from additive combinatorics.     

Polygon dissections and some generalizations of cluster complexes

Let W be a Weyl group corresponding to the root system An−1 or Bn. We define a simplicial complex in terms of polygon dissections for such a group and any positive integer m. For m=1, is isomorphic to the cluster complex corresponding to W, defined in [S. Fomin, A.V. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (2003) 977-1018]. We enumerate the faces of and show that the entries of its h-vector are given by the generalized Narayana numbers , defined in [C.A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005) 179-196]. We also prove that for any m?1 the complex is shellable and hence Cohen-Macaulay.     

Hyperkähler structures and infinite-dimensional Grassmannians

In this paper, we describe an example of a hyperkähler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach space), the quotient space obtained is a Hilbert manifold, which can be furthermore identified either with the cotangent space of a connected component (j∈Z), of the restricted Grassmannian or with a natural complexification of this connected component, thus proving that these two manifolds are isomorphic hyperkähler manifolds. Moreover, Kähler potentials associated with the natural complex structure of the cotangent space of and with the natural complex structure of the complexification of are computed using Kostant-Souriau's theory of prequantization.     

Filtrations and local syzygies of multiplier ideals II

Let a be a non-zero ideal sheaf on a smooth affine variety X of dimension d and let c be a positive rational number. Let x be a closed point of X and let m_x be the maximal ideal sheaf at x. In [Robert Lazarsfeld, Kyungyong Lee, Local syzygies of multiplier ideals, Invent. Math. 167 (2007) 409-418] the authors studied the local syzygies of the multiplier ideal J(a^c). Motivated by their result, the asymptotic behavior of the local syzygies of the multiplier ideal at x for k≥d−2 was studied in [Seunghun Lee, Filtrations and local syzygies of multiplier ideals, J. Algebra (2007) 629-639]. In this note, we study the local syzygies of at x for 1≤k≤d−3. As a by-product we give a different proof of the main theorem in the former reference cited above.     

A matroid invariant via the K-theory of the Grassmannian

Let G(d,n) denote the Grassmannian of d-planes in Cn and let T be the torus n(C^∗)/diag(C^∗) which acts on G(d,n). Let x be a point of G(d,n) and let be the closure of the T-orbit through x. Then the class of the structure sheaf of in the K-theory of G(d,n) depends only on which Plücker coordinates of x are nonzero - combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from K^○(G(d,n)) to Z[t]. Letting gx(t) denote the image of , gx behaves nicely under the standard constructions of matroid theory, such as direct sum, two-sum, duality and series and parallel extensions. We use this invariant to prove bounds on the complexity of Kapranov's Lie complexes [M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (2) (1993) 29-110], Hacking, Keel and Tevelev's very stable pairs [P. Hacking, S. Keel, E. Tevelev, Compactification of the moduli space of hyperplane arrangements, J. Algebraic Geom. 15 (2006) 657-680] and the author's tropical linear spaces when they are realizable in characteristic zero [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527-1558]. Namely, in characteristic zero, a Lie complex or the underlying (d−1)-dimensional scheme of a very stable pair can have at most strata of dimensions n−i and d−i, respectively. This prove the author's f-vector conjecture, from [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527-1558], in the case of a tropical linear space realizable in characteristic 0.     

A description of continued fraction expansions of quadratic surds represented by polynomials

The principal thrust of this investigation is to provide families of quadratic polynomials , where e_k²−f_k²C=n (for any given nonzero integer n) satisfying the property that for any , the period length of the simple continued fraction expansion of is constant for fixed k and lim_k→∞?_k=∞. This generalizes, and completes, numerous results in the literature, where the primary focus was upon |n|=1, including the work of this author, and coauthors, in Mollin (Far East J. Math. Sci. Special Vol. 1998, Part III, 257-293; Serdica Math. J. 27 (2001) 317) Mollin and Cheng (Math. Rep. Acad. Sci. Canada 24 (2002) 102; Internat Math J 2 (2002) 951) and Mollin et al. (JP J. Algebra Number Theory Appl. 2 (2002) 47).