Showing papers in "Mathematical Research Letters in 2004"

Fuglede's conjecture is false in 5 and higher dimensions

Abstract: We give an example of a set Ω ⊂ R 5 which is a finite union of unit cubes, such that L 2 (Ω) admits an orthonormal basis of exponentials { 1 |Ω|1/2 e 2πiξj ·x :

A power structure over the Grothendieck ring of varieties

Abstract: Let R be either the Grothendieck semiring (semigroup with multiplication) of complex quasi-projective varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class \L of the complex affine line. We define a power structure over these (semi)rings. This means that, for a power series A(t)=1+∑i=1∞[Ai]ti with the coefficients [Ai] from R and for [M]∈R, there is defined a series (A(t))[M], also with coefficients from R, so that all the usual properties of the exponential function hold. In the particular case A(t)=(1−t)−1, the series (A(t))[M] is the motivic zeta function introduced by M. Kapranov. As an application we express the generating function of the Hilbert scheme of points, 0-dimensional subschemes, on a surface as an exponential of the surface.

An example of neckpinching for Ricci flow on $S^{n+1}$

Abstract: We give an example of a class of metrics on Sn+1 that evolve under the Ricci Flow into a “neckpinch.” We show that the solution has a Type I singularity, and that the length of the neck, i.e. the region where |Rm| ∼ (T−t)−1, is bounded from below by c √ (T − t)| log(T − t)| for some c > 0.

Connections with torsion, parallel spinors and geometry of Spin(7) manifolds

Abstract: We show that on every Spin(7)-manifoldthere always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor andthe Spin(7) structure. We express its torsion andthe Riemannian scalar cur- vature in terms of the fundamental 4-form. We present an explicit formula for the Riemannian covariant derivative of the fundamental 4-form in terms of its exterior differential. We show the vanishing of the ˆ A-genus andobtain a linear relation between Betti numbers of a compact Spin(7) manifoldwhich is locally but not globally conformally equivalent to a space with closedfund amental 4-form. A general solution to the Killing spinor equations is presented.

Contact Structures with Distinct Heegaard Floer Invariants

Abstract: We prove that the contact structures on Y= dX induced by non-homotopic Stein structures on the 4-manifold X have distinct Heegaard Floer invariants.

Dehn surgery, the fundamental group and SU(2)

Abstract: Let K be a non-trivial knot in the 3-sphere and let Y(r) be the 3-manifold obtained by surgery on K with surgery-coefficient a rational number r. We show that there is a homomorphism from the fundamental group of Y(r) to SU(2) with non-cyclic image if r is less than or equal to 2.

Almost orthogonality and a class of bounded bilinear pseudodifferential operators

Abstract: Several results and techniques that generate bilinear alternatives of a celebrated theorem of Calderón and Vaillancourt about the L2 continuity of linear pseudodifferential operators with symbols with bounded derivatives are presented. The classes of bilinear pseudodifferential symbols considered are shown to produce continuous operators from L2 × L2 into L1.

Finite dimensional quasi-Hopf algebras with radical of codimension 2

Abstract: It is shown in math.QA/0301027 that a finite dimensional quasi-Hopf algebra with radical of codimension 1 is semisimple and 1-dimensional. On the other hand, there exist quasi-Hopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known that these are exactly the Nichols Hopf algebras H_{2^n} of dimension 2^n, n\ge 1 (one for each value of n). The main result of this paper is that if H is a finite dimensional quasi-Hopf algebra over C with radical of codimension 2, then H is twist equivalent to a Nichols Hopf algebra H_{2^n}, n\ge 1, or to a lifting of one of the four special quasi-Hopf algebras H(2), H_+(8), H_-(8), H(32) of dimensions 2, 8, 8, and 32, defined in Section 3. As a corollary we obtain that any finite tensor category which has two invertible objects and no other simple object is equivalent to \Rep(H_{2^n}) for a unique n\ge 1, or to a deformation of the representation category of H(2), H_+(8), H_-(8), or H(32). As another corollary we prove that any nonsemisimple quasi-Hopf algebra of dimension 4 is twist equivalent to H_4.

On the Geometry of Weil-Petersson Completion of Teichmüller Spaces

Abstract: Given a surface of higher genus, we will look at the Weil-Petersson completion of the Teichmüller space of the surface, and will study the geometry induced by the Weil-Petersson distance functional. Although the completion is no longer a Riemannian manifold, it has characteristics similar to those of CartanHadamard manifolds.

Ricci flow and nonnegativity of sectional curvature

Abstract: In this paper, we extend the general maximum principle in (NT3) to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional cur- vature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow.

Module categories over representations of $SL_q(2)$ and graphs

Abstract: We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same category.

The Chow motive of semismall resolutions

Abstract: We consider proper, algebraic semismall maps f from a complex algebraic manifold X. We show that the topological Decomposition Theorem implies a "motivic" decomposition theorem for the rational algebraic cycles of X and, in the case X is compact, for the Chow motive of X.The result is a Chow-theoretic analogue of Borho-MacPherson's observation concerning the cohomology of the fibers and their relation to the relevant strata for f. Under suitable assumptions on the stratification, we prove an explicit version of the motivic decomposition theorem. The assumptions are fulfilled in many cases of interest, e.g. in connection with resolutions of orbifolds and of some configuration spaces. We compute the Chow motives and groups in some of these cases, e.g. the nested Hilbert schemes of points of a surface. In an appendix with T. Mochizuki, we do the same for the parabolic Hilbert scheme of points on a surface. The results above hold for mixed Hodge structures and explain, in some cases, the equality between orbifold Betti/Hodge numbers and ordinary Betti/Hodge numbers for the crepant semismall resolutions in terms of the existence of a natural map of mixed Hodge structures. Most results hold over an algebraically closed field and in the Kaehler context.

Sharp estimates for green potentials on non-smooth domains

Abstract: Given an open, bounded, connected domain Ω ⊂ R n , let GD, GN be the solution operators for the Poisson equation for the Laplacian in Ω with homogeneous Dirichlet and Neumann boundary conditions, respectively.The aim of this note is to describe the sharp ranges of indices for which GD, GN are smoothing operators of order two, on Besov and Triebel-Lizorkin scales in arbitrary Lipschitz domains.This builds on the work of many people who have dealt with homogeneous/inhomogenous problems for the Laplacian with Dirichlet/Neumann boundary conditions in Lipschitz domains; an excellent account can be found in [23].Earlier results emphasized homogeneous problems with boundary data exhibiting an integer amount of smoothness.Such estimates underpin the entire work here and play the role of end-point/limiting cases in our theory. The main theorems we prove extend the work of D.Jerison and C.Kenig [20], whose methods and results are largely restricted to the case p ≥ 1, and answer the open problem # 3.2.21 on p.121 in C. Kenig’s book [23] in the most complete fashion.When specialized to Hardy spaces (viewed as a subclass of Triebel-Lizorkin scale), our results provide a solution of a (strengthened form of a) conjecture made by D.-C.Chang, S.Krantz and E.Stein regarding the regularity of the Green potentials on Hardy spaces in Lipschitz domains.Cf. p.130 of [5] where the authors write: “ For some applications it would be desirable to find minimal smoothness conditions on ∂Ω in order for our analysis of the Dirichlet and Neumann problems to remain valid. We do not know whether C 1+e boundary is sufficient in order to obtain [Hardy space] estimates for the Dirichlet problem when p is near 1. [...]

A new integrable system on the sphere

Abstract: This pre-print has been submitted, and accepted, to the journal, Mathematical Research Letters. The definitive version: DULLIN, H.R. and MATVEEV, V.S., 2004. A new integrable system on the sphere. Mathematical Research Letters, 11(5-6),pp. 715-722.

A note on the Fourier transform of fractal measures

Abstract: We are interested in the behavior of the Fourier transform of measures with finite energy. It is easy to see that Iα(μ) < ∞ does not imply any pointwise decay of |μ̂(ξ)| as |ξ| → ∞. However, in general, averages of μ̂(ξ) behave much better. Let Γ be a smooth submanifold of R and let νΓ be a smooth surface measure on Γ. One may ask the following general question: Fix α ∈ (0, d) and assume that Iα(μ) = 1. For which β > 0 ∫

A lower bound for the canonical height on abelian varieties over abelian extensions

Abstract: Let A be an abelian variety defined over a number field K and let ĥ be the canonical height function on A(K) attached to a symmetric ample line bundle L. We prove that there exists a constant C = C(A,K,L) > 0 such that ĥ(P ) ≥ C for all nontorsion points P ∈ A(K), where K is the maximal abelian extension of K.

Polynomial values, the linking form and unknotting numbers

Abstract: We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 8_{16}, 9_{49} and 6 further new entries in Kawauchi's tables. Another method is developed by applying and extending the linking form criterion of Lickorish. This leads to several conjectured relations between the Jones value of Q and the linking form.

Hilbert-Kunz Functions for Normal Rings

Abstract: Let (R,m,k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R=d. Let M be a finitely generated R-module. We show that there exists a real number beta(M) such that lambda(M/I^[q]M) = e_{HK}(M) q^d + beta(M) q^{d-1} + O(q^{d-2}).

A morera type theorem in the strip

Abstract: We prove the following result. Let f be a continuous function in the closed infinite strip in complex plane. Suppose the restriction of f to every circle inscribed in the strip extends holomorphically inside the circle. Then f is holomorphic in the open strip.

Almost All Palindromes Are Composite

Abstract: First published in Mathematical Research Letters 11 (2004) nos.5-6, pp.853-868, published by International Press. ©International Press.

An example of Fourier–Mukai partners of minimal elliptic surfaces

Abstract: Let $X$ and $Y$ be smooth projective varieties over $\C$. We say that $X$ and $Y$ are \emph{D-equivalent} (or, $X$ is a \emph{Fourier--Mukai partner} of $Y$) if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories. The aim of this short note is to find an example of mutually D-equivalent but not isomorphic relatively minimal elliptic surfaces.

Highest Weight Irreducible Representations of Rank 2 Quantum Tori

Abstract: For any nonzero q ∈ C (the complex numbers), the rank 2 quan- tum torus Cq is the skew Laurent polynomial algebra C(t ±1 1 ,t ±1 2 ) with defining relations: t2t1 = qt1t2 and tit −1 i = t −1 i ti = 1. Here we consider Cq as the naturally associated Lie algebra. We add the one dimensional center Cc1 and the outer derivation d1 to Cq to get the extended torus Lie algebra � Cq (and � Cq, in a different manner), where we assume q is a primitive m-th root of unity for � Cq. Before this paper, there appeared highest weight representations for � Cq and � Cq with only positive integral levels. In this paper, we define the highest weight irreducible (Z Z-graded) module V (φ) over � Cq and � Cq for any linear map φ : C(t ±1 2 )+ Cc1 + Cd1 → C, thus the central charge (level) can be any complex numbers. We obtain the necessary and sufficient conditions for V (φ) to have finite dimensional weight spaces, thus obtaining a lot of new irreducible weight repre- sentations for these Lie algebras. The corresponding irreducible Z Z × Z Z-graded modules with finite dimensional weight spaces over � Cq are also constructed.

Euclidean scissor congruence groups and mixed Tate motives over dual numbers

Abstract: We define Euclidean scissor congruence groups for an arbitrary algebraically closed field F and propose their conjectural description. We suggest how they should be related to mixed Tate motives over dual numbers for F.

Arithmetic Properties of Periodic Maps

Abstract: Let 1,... , k be periodic maps from Z to a field of char- acteristic p (where p is zero or a prime). Assume that positive integers n1,... ,nk not divisible by p are their periods respectively. We show that 1 + ··· + k is constant if 1(x) + ··· + k(x) equals a constant for |S| consecutive integers x where S = Sk=1{r/ns : r = 0,... ,ns 1}. We also present some new results on finite systems of arithmetic sequences.

The asymptotic behaviour of Heegaard genus

Abstract: Heegaard splittings have recently been shown to be related to a number of important conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b1 conjecture and the virtually fibred conjecture [3]. Of particular importance is the rate at which the Heegaard genus of finite-sheeted covering spaces grows as a function of their degree. This was encoded in the following definitions.

Between Maharam's and von Neumann's problems

Abstract: In the context of definable algebras Maharam’s and von Neumann’s problems essentially coincide. Consequently, random forcing is the only definable ccc forcing adding a single real that does not make the ground model reals null, and the only pairs of definable ccc -ideals with the Fubini property are (meager,meager) and (null,null). In Scottish Book, von Neumann asked whether every ccc, weakly distributive complete Boolean algebra carries a strictly positive probability measure. Von Neumann’s problem naturally splits into two: (a) whether all such algebras carry a strictly positive continuous submeasure, and (b) whether every algebra that carries a strictly positive continuous submeasure carries a strictly positive measure. The latter problem is known under the names of Maharam’s Problem and Control Measure Problem (see [16], [9], [5, §393]). While von Neumann’s problem has a consistently negative answer ([16]), Maharam’s problem can be stated as a 1 statement and is therefore, by Shoenfield’s theorem, absolute between transitive models of set theory containing all countable ordinals. Theorem 0.1. Let I be a c.c.c. -ideal on Borel subsets of 2 ! that is analytic on G . The following are equivalent:

On Lens Spaces and Their Symplectic Fillings

Abstract: The standard contact structure on the three-sphere is invariant under the action of the cyclic group of order p yielding the lens space L(p,q). Therefore, every lens space carries a natural quotient contact structure Q. A theorem of Eliashberg and McDuff classifies the symplectic fillings of (L(p,1), Q) up to diffeomorphism. We announce a generalization of that result to every lens space. In particular, we give an explicit handlebody decomposition of every symplectic filling of (L(p,q), Q) for every p and q. Our results imply that: (a) there exist infinitely many lens spaces L(p,q) with q>1 such that (L(p,q), Q) admits only one symplectic filling up to blowup and diffeomorphism; (b) for any natural number N, there exist infinitely many lens spaces L(p,q) such that (L(p,q), Q) admits more than N symplectic fillings up to blowup and diffeomorphism.