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Let A denote the ring of arithmetical functions with unitary convolution, and let V be a finite subset of the positive integers having the property that for every v in V, all unitary divisors of v lie in V. We study the truncation A_V, an artinian monomial quotient of a polynomial ring in finitely many indeterminates, isomorphic to the ``Artinified'' Stanley-Reisner ring C[\bar{\Delta(V)}] of a certain simplicial complex \Delta(V).
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arxiv:math/0205242
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It is demonstrated that hypersurfaces with a flat centroaffine metric are governed by a system of nonlinear PDEs known as the equations of associativity of 2-dimensional topological field theory.
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arxiv:math/0205248
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We generalize a theorem of Kapranov by showing that the Hall algebra of the category of coherent sheaves on a weighted projective line (over a finite field) provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the correponding Kac-Moody algebra. In particular this yieds a geometric realization of the quantized enveloping algebra of 2-toroidal (or elliptic) algebras of types D_4, E_6, E_7 or E_8 in terms of weighted projective lines of genus one.
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arxiv:math/0205267
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Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are called {\it $D$-equivalent} if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while {\it $K$-equivalent} if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide at least for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.
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arxiv:math/0205287
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It is proved that there exists a separable reflexive Banach space W that contains an isomorphic image of every separable superreflexive Banach space. This gives the affirmative answer on one J. Bourgain's question
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arxiv:math/0206010
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We prove a quantized version of a theorem by M. V. Sheinberg: A uniform algebra equipped with its canonical, i.e. minimal, operator space structure is operator amenable if and only if it is a commutative $C^\ast$-algebra.
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arxiv:math/0206042
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We show that every orientable 3-manifold is a classifying space B\Gamma where \Gamma is a groupoid of germs of homeomorphisms of R. This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. The F we construct can be taken to be C^1 but not C^2. The existence of such an F answers positively a question posed by Tsuboi [Classifying spaces for groupoid structures, notes from minicourse at PUC, Rio de Janeiro (2001)], but leaves open the question of whether M = B\Gamma for some C^\infty groupoid \Gamma.
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arxiv:math/0206066
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Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum at a prime p. They discussed May's notions of nuclear complexes and of cores of spaces, spectra, and commutative S-algebras. Their most striking conclusions, due to Hu and Kriz, were negative: cores are not unique up to equivalence, and BP is not a core of MU considered as a commutative S-algebra, although it is a core of MU considered as a p-local spectrum. We investigate these ideas further, obtaining much more positive conclusions. We show that nuclear complexes have several non-obviously equivalent characterizations. Up to equivalence, they are precisely the irreducible complexes, the minimal atomic complexes, and the Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz dimension, which we call complexes with no mod p detectable homotopy. Unlike the notion of a nuclear complex, these other notions are all invariant under equivalence. This simple and conceptual criterion for a complex to be minimal atomic allows us to prove that many familiar spectra, such as ko, eo_2, and BoP at the prime 2, all BP<n> at any prime p, and the indecomposable wedge summands of the suspension spectra of $CP^\infty$ and $HP^\infty$ at any prime p are minimal atomic.
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arxiv:math/0206067
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Many Properties of a category X, as for instance the existence of an adjoint or a factorization system, are a consequence of the cowellpoweredness of X. In the absence of cowellpoweredness, for general results, fairly strong assumption on the category are needed. This paper provides a number of novel and useful observations to tackle the cowellpoweredness problem of subcategories by means of regular closure operators. Our exposition focusses on the question when two subcategories A and B induce the same regular closure operators, then information about (non)-cowellpoweredness of A may be gained from corresponding property of B, and vice versa.
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arxiv:math/0206124
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We construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player andyet the new combined game is favourable -- this is a counter-intuitive phenomenoknown as Parrondo's paradox. We observe that all of the games in question are random walks in periodic environments (RWPE) when viewed on the proper time scale. Consequently, we use RWPE techniques to derive conditions under which Parrondo's paradox occurs.
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arxiv:math/0206151
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We prove a conjecture of Denef on parameterized $p$-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.
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arxiv:math/0206161
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A set of all symmetric Banach function spaces defined on [0,1] is equipped with the partial order by the relation of continuous inclusion. Properties of symmetric spaces, which do not depend of their position in the ordered structure, are studied. With the help of the J. Peetre's interpolation scheme it is shown that for any pair of symmetric spaces E, F such that F is absolutely continuously included in E there exists an intermediate (so called, Peetre's) space K(E,F;W), where W is a space with an unconditional basis, that is reflexive or, respectively, weakly sequentially complete provided W also is reflexive or, respectively, weakly sequentially complete.
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arxiv:math/0206183
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We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Caratheodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.
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arxiv:math/0206230
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We describe a natural way to plant cherry- and plumtrees at prescribed generic locations in an orchard.
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arxiv:math/0206266
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A minor error in the necessary conditions for the algebraic form of the Lam\'e equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lam\'e's differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lam\'e equation. [See R. C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric, Riemann, and Lam\'e equations", J. Symbolic Computation 28 (1999), 521-545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.
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arxiv:math/0206285
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In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists are roots of unity, and square of the braiding has finite order. We also discuss the notion of the quasi-exponent of a finite rigid tensor category, which is motivated by results 1 and 2 and the paper math/0109196 of S.Gelaki and the author.
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arxiv:math/0207007
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We study equivariant projective compactifications of reductive groups obtained by closing the image of a group in the space of operators of a projective representation. We describe the structure and the mutual position of their orbits under the action of the doubled group by left/right multiplications, the local structure in a neighborhood of a closed orbit, and obtain some conditions of normality and smoothness of a compactification. Our methods of research use the theory of equivariant embeddings of spherical homogeneous spaces and of reductive algebraic semigroups.
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arxiv:math/0207034
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Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p, t^2 + kappa t + 1), where p is a prime number and t^2 + kappa t + 1 is irreducible modulo p. For each such quandle, there is an invariant with values in the group ring Z[C_p] of a cyclic group of order p. We shall show that the values of this invariant all have the form Gamma_p^r p^{2s} for a fixed element Gamma_p of Z[C_p] and integers r >= 0 and s > 0. We also describe some machine computations, which lead us to conjecture that the invariant is determined by the Alexander module of the link. This conjecture is verified for all torus and two-bridge knots.
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arxiv:math/0207099
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Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.
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arxiv:math/0207172
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A construction for sphere packings is introduced that is parallel to the ``anticode'' construction for codes. This provides a simple way to view Vardy's recent 20-dimensional sphere packing, and also produces packings in dimensions 22, 44--47 that are denser than any previously known.
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arxiv:math/0207182
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The crosscap number of a knot in the 3-sphere is the minimal genus of non-orientable surface bounded by the knot. We determine the crosscap numbers of torus knots.
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arxiv:math/0207203
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We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that relates the total vorticity to the gradient of the back-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusive analogue for viscous flows). The results include a nonlinear version of the Taylor-Proudman theorem: in a steady solution of the rotating Euler equations, two fluid material points which were initially on a vertical vortex line, will perpetually maintain their vertical separation unchanged. For more general situations, including unsteady flows, we obtain bounds for the vertical gradients of the Lagrangian displacement that vanish linearly with the maximal local Rossby number.
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arxiv:math/0207220
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We give a geometric proof of Conn's linearization theorem for analytic Poisson structures, without using the fast convergence method.
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arxiv:math/0207263
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The exceptional log Del Pezzo surfaces with delta=1 are classified.
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arxiv:math/0207269
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In a previous paper [Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243--270, arXiv:math.GT/0010247], a group H_g of homology cylinders over the oriented surface of genus g is defined. A filtration of H_g is defined, using the Goussarov-Habiro notion of finite-type. It is erroneously claimed that this filtration essentially coincides with the relative weight filtration. The present note corrects this error and studies the actual relation between the two filtrations.
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arxiv:math/0207290
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Various distribution free goodness-of-fit test procedures have been extracted from literature. We present two new binning free tests, the univariate three-region-test and the multivariate energy test. The power of the selected tests with respect to different slowly varying distortions of experimental distributions are investigated. None of the tests is optimum for all distortions. The energy test has high power in many applications and is superior to the chi^2 test.
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arxiv:math/0207300
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We consider the problem of recovering l-adic representations from a knowledge of the character values at the Frobenius elements associated to l-adic representations constructed algebraically out of the original representations. These results generalize earlier results in of the author concerning refinements of strong multiplicity one for $l$-adic represntations, and a result of Ramakrishnan recovering modular forms from a knowledge of the squares of the Hecke eigenvalues. For example, we show that if the characters of some tensor or symmetric powers of two absolutely irreducible l-adic representation with the algebraic envelope of the image being connected, agree at the Frobenius elements corresponding to a set of places of positive upper density, then the representations are twists of each other by a finite order character.
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arxiv:math/0207308
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We prove that for any field k of characteristic p>0, any separated scheme X of finite type over k, and any overconvergent F-isocrystal E over X, the rigid cohomology H^i(X, E) and rigid cohomology with compact supports H^i_c(X,E) are finite dimensional vector spaces. We also establish Poincare duality and the Kunneth formula with coefficients. The arguments use a pushforward construction in relative dimension 1, based on a relative version of Crew's conjecture on the quasi-unipotence of certain p-adic differential equations.
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arxiv:math/0208027
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Let $(M,F)$ be a Finsler manifold. We construct a 1-cocycle on $\Diff(M)$ with values in the space of differential operators acting on sections of some bundles, by means of the Finsler function $F.$ As an operator, it has several expressions: in terms of the Chern, Berwald, Cartan or Hashiguchi connection, although its cohomology class does not depend on them. This cocycle is closely related to the conformal Schwarzian derivatives introduced in our previous work. The second main result of this paper is to discuss some properties of the conformally invariant quantization map by means of a Sazaki (type) metric on the slit bundle $TM\backslash 0$ induced by $F.$
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arxiv:math/0208030
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Using the Palm measure notion, we prove the existence of the diffraction measure of all stationary and ergodic point processes. We get precise expressions of those measures in the case of specific processes : stochastic subsets of Z^d, sets obtained by the ``cut-and-project'' method.
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arxiv:math/0208064
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We define united K-theory for real C*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors -- real K-theory, complex K-theory, and self-conjugate K-theory -- and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Kunneth-type, non-splitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C*-algebras A and B which holds as long as the complexification of A is in the bootstrap category. Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product O_{k+1} otimes O_{l+1} is not determined solely by the greatest common divisor of k and l. Hence we have examples of non-isomorphic, simple, purely infinite, real C*-algebras whose complexifications are isomorphic.
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arxiv:math/0208068
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We consider multimodal C^3 interval maps f satisfying a summability condition on the derivatives D_n along the critical orbits which implies the existence of an absolutely continuous f -invariant probability measure mu. If f is non-renormalizable, mu is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence D_n as n tends to infinity . We also give sufficient conditions for mu to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.
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arxiv:math/0208114
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Nous avons obtenu des formules explicites representant les fonctions E(z) apparaissant dans la theorie des ``Espaces de Sonine'' associes par de Branges a la transformation de Fourier.
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arxiv:math/0208121
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A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in a previous paper, is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT. By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r^4 up to polylogarithmic factors, confirming heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r^2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon converges in distribution to an asymmetric stable random variable of type 1/2. By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability p_c=1/2 and that at p_c percolation does not occur.
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arxiv:math/0208123
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Certain types of bilinearly defined sets in $\mathbb{R}^n$ exhibit a higher degree of linearity than what is apparent by inspection.
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arxiv:math/0208131
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In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations on chain complexes of graphs', Mathematische Annalen, 327 (2003) 545-573] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [James Conant, `Fusion and fission in graph complexes', Pac. J. 209 (2003), 219-230] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.
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arxiv:math/0208169
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We construct the Fock space representations of classical quantum affine algebras using combinatorics of Young walls. We also show that the crystal graphs of the Fock space representations can be realized as the abstract crystal consisting of proper Young walls. Finally, we give a generalized version of Lascoux-Leclerc-Thibon algorithm for computing the global bases of the basic representations of classical quantum affine algebras.
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arxiv:math/0208204
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We recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the firdt nontrivial case of curves in RP^3. Namely, we show that i) the tangent developable surface of any convex curve in RP^3 has 'degree' 4 and ii) construct an example of 4 tangent lines to a convex curve in RP^3 such that no real line intersects all four of them.
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arxiv:math/0208218
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We show that the roots of any smooth curve of polynomials with only real roots can be parametrized twice differentiably (but not better).
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arxiv:math/0208228
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A scheme $X\subset \PP^{n+c}$ of codimension $c$ is called {\em standard determinantal} if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous $t \times (t+c-1)$ matrix and $X$ is said to be {\em good determinantal} if it is standard determinantal and a generic complete intersection. Given integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$ we denote by $W(\underline{b};\underline{a})\subset \Hi ^p(\PP^{n+c})$ (resp. $W_s(\underline{b};\underline{a})$) the locus of good (resp. standard) determinantal schemes $X\subset \PP^{n+c}$ of codimension $c$ defined by the maximal minors of a $t\times (t+c-1)$ matrix $(f_{ij})^{i=1,...,t}_{j=0,...,t+c-2}$ where $f_{ij}\in k[x_0,x_1,...,x_{n+c}]$ is a homogeneous polynomial of degree $a_j-b_i$. In this paper we address the following three fundamental problems : To determine (1) the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) in terms of $a_j$ and $b_i$, (2) whether the closure of $W(\underline{b};\underline{a})$ is an irreducible component of $\Hi ^p(\PP^{n+c})$, and (3) when $\Hi ^p(\PP^{n+c})$ is generically smooth along $W(\underline{b};\underline{a})$. Concerning question (1) we give an upper bound for the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) which works for all integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$, and we conjecture that this bound is sharp. The conjecture is proved for $2\le c\le 5$, and for $c\ge 6$ under some restriction on $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$. For questions (2) and (3) we have an affirmative answer for $2\le c \le 4$ and $n\ge 2$, and for $c\ge 5$ under certain numerical assumptions.
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arxiv:math/0209011
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We give another proof for a result of Brick stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of $\p1i$ for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact lattices in nilpotent and semi-simple Lie groups, and in particular for fundamental groups of geometric 3-manifolds.
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arxiv:math/0209014
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This paper grew out of an attempt to find a suitable finite sheeted covering of an aspherical 3-manifold so that the cover either has infinite or trivial first homology group. With this motivation we define a new class of groups. These groups are in some sense eventually perfect. We prove results giving several classes of examples of groups which do (not) belong to this class. Also we prove some elementary results on these groups and state two conjectures. A direct application of one of the conjectures to the virtual Betti number conjecture of Thurston is mentioned.
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arxiv:math/0209121
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We give a formula for the Dixmier-Douady class of a continuous-trace groupoid crossed product that arises from an action of a locally trivial, proper, principal groupoid on a bundle of elementary $C^*$-algebras that satisfies Fell's condition.
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arxiv:math/0209125
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This paper gives a complete primary decomposition of the first, that is, the smallest, Mayr-Meyer ideal, its radical, and the intersection of its minimal components. The particular membership problem which makes the Mayr-Meyer ideals' complexity doubly exponential in the number of variables is here examined also for the radical and the intersection of the minimal components. It is proved that for the first Mayr-Meyer ideal the complexity of this membership problem is the same as for its radical. This problem was motivated by a question of Bayer, Huneke and Stillman.
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arxiv:math/0209154
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The standard Faddeev quantization of the simple groups is modified in such a way that the quantum analogs of the nonsemisimple groups are obtained by contractions. The contracted quantum groups are regarded as the algebras of noncommutative functions generated by elements $J_{ik}t_{ik},$ where $J_{ik}$ are some products of generators of the algebra ${\bf D}(\iota)$ and $t_{ik}$ are the noncommutative generators of guantum group. Possible contractions of quantum orthogonal groups essentially depend on the choice of primitive elements of the Hopf algebra. All such choices are considered for quantum group $SO_{q}(N;C)$ and all allowed contractions in Cayley--Klein scheme are described. The quantum deformations of the complex kinematical groups have been investigated as a contractions of $SO_q(5;C).$ The quantum Euclead $E_q(4;C)$ and Newton $N_q(4;C)$ groups with unchanged deformation parameter as well as Newton group $N_v(4;C)$ with transformed deformation parameter are obtained. But there is no quantum analog of the (complex) Galilei group $G(1,3).$ According to correspondence principle a new physical theory must include an old one as a particular case. For space-time symmetries this principle is realized as the chain of contractions of the kinematical groups: $$ S^{\pm}(1,3)\stackrel{K \to 0}{\longrightarrow} P(1,3)\stackrel{c \to \infty}{\longrightarrow}G(1,3). $$ As it was mentioned above there is no quantum deformation of the complex Galilei group in the standard Cayley--Klein scheme, therefore it is not possible to construct the quantum analog of the full chain of contractions of the (1+3) kinematical groups even at the level of a complex groups.
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arxiv:math/0209158
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We prove that the existence of an automorphism of finite order on a (defined over a number field) variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Gamma-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne. Our proof relies on the arithmetic fixed point formula (equivariant arithmetic Riemann-Roch theorem) proved by K. Koehler and the second author, and the vanishing of the equivariant analytic torsion for the Dolbeault complex.
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arxiv:math/0209177
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We study limits of quasifuchsian groups for which the bending measures on the convex hull boundary tend to zero, giving necessary and sufficient conditions for the limit group to exist and be Fuchsian. As an application we complete the proof of a conjecture made in \cite{S1}, that the closure of pleating varieties for quasifuchsian groups meet Fuchsian space exactly in Kerckhoff's lines of minima of length functions. Doubling our examples gives rise to a large class of cone manifolds which degenerate to hyperbolic surfaces as the cone angles approach $2\pi$.
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arxiv:math/0209190
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Spectral transformation is known to set up a birational morphism between the Hitchin and Beauville-Mukai integrable systems. The corresponding phase spaces are: (a) the cotangent bundle of the moduli space of bundles over a curve C, and (b) a symmetric power of the cotangent surface T^*(C). We conjecture that this morphism can be quantized, and we check this conjecture in the case where C is a rational curve with marked points and rank 2 bundles. We discuss the relation of the resulting isomorphism of quantized algebras with Sklyanin's separation of variables.
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arxiv:math/0209294
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We formulate problems of tight closure theory in terms of projective bundles and subbundles. This provides a geometric interpretation of such problems and allows us to apply intersection theory to them. This yields new results concerning the tight closure of a primary ideal in a two-dimensional graded domain.
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arxiv:math/0209300
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Let S be a bordered Riemann surface with genus g and m boundary components. For a smooth family of smooth Jordan curves in the complex plane parametrized by the boundary of S and such that all curves contain 0 in their interior we show that there exists a holomorphic solution of the corresponding Riemann-Hilbert problem with at most 2g+m-1 zeros on S.
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arxiv:math/0209320
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This first part of the paper describes the support of top graded local cohomology modules. As a corrolary one obtains a simple criteria for the vanishing of these modules and also the fact that they have finitely many minimal primes. The second part of this paper constructs examples of cohomological Hilbert functions which are not of polynomial type.
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arxiv:math/0209350
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The linear span P_n of the sums of all permutations in the symmetric group S_n with a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra D_n; and the direct sum P of all P_n is a Hopf sub-algebra of the direct sum D of all D_n, dual to the Stembridge algebra of peak functions. In our self-contained approach, peak counterparts of several results on the descent algebra are established, including a simple combinatorial characterization of the algebra P_n; an algebraic characterization of P_n based on the action on the Poincar'e-Birkhoff-Witt basis of the free associative algebra; the display of peak variants of the classical Lie idempotents; an Eulerian-type sub-algebra of P_n; a description of the Jacobson radical of P_n and its nil-potency index, of the principal indecomposable and irreducible P_n-modules, and of the Cartan matrix of P_n. Furthermore, it is shown that the primitive Lie algebra of P is free, and that P is its enveloping algebra.
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arxiv:math/0209376
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In the paper the notion of truncating twisting function $\tau :X\to Q$ from a simplicial set $X$ to a cubical set $Q$ and the corresponding notion of twisted Cartesian product of these sets $X\times_{\tau}Q$ are introduced. The latter becomes a cubical set whose chain complex coincides with the standard twisted tensor product $C_*(X)\otimes_{\tau_*}C_*(Q)$. This construction together with the theory of twisted tensor products for homotopy G-algebras allows to obtain multiplicative models for fibrations.
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arxiv:math/0210006
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In this paper we introduce a generalisation of the notion of holonomy for connections over a bundle map on a principal fibre bundle. We prove that, as in the standard theory on principal connections, the holonomy groups are Lie subgroups of the structure group of the principle fibre bundle and we also derive a straightforward generalisation of the Reduction Theorem.
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arxiv:math/0210020
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In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas underlying the theory. This article is going to appear in the collected works of Erich K\"ahler.
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arxiv:math/0210060
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A classification of normal affine surfaces admitting a $\bf C^*$-action was given in the work of Bia{\l}ynicki-Birula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of such surfaces in terms of their graded rings as well as by defining equations. This is based on a generalization of the Dolgachev-Pinkham-Demazure construction in the case of a hyperbolic grading. As an apllication we determine the structure of singularities, of the orbits and the divisor class groups for such surfaces.
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arxiv:math/0210153
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A vanishing theorem for a convex cocompact hyperbolic manifold is established, which relates the L2 cohomology to the Hausdorff dimension of the limit set. The borderline case is shown to characterize the manifold completely.
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arxiv:math/0210163
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To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category Aff(C)_\kappa of smooth modules (in the sense of Kazhdan and Lusztig [KL1]) of finite length over the corresponding affine Kac-Moody algebra in the case of central charge less than the critical level. Equivalent characterizations of these categories are obtained in the spirit of the works of Kazhdan-Lusztig [KL1] and Lian-Zuckerman [LZ1]. In the main part of this paper we establish a finiteness result for the Kazhdan-Lusztig tensor product which can be considered as an affine version of a theorem of Kostant [K]. It contains as special cases the finiteness results of Kazhdan, Lusztig [KL] and Finkelberg [F], and states that for any subalgebra f of g which is reductive in g the "affinization" of the category of finite length admissible (g, f) modules is stable under Kazhdan-Lusztig's tensoring with the "affinization" of the category of finite dimensional g modules (which is O_\kappa in the notation of [KL1, KL2, KL3]).
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arxiv:math/0210180
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There are several ways to associate a complex structure to a ribbon graph. In the construction of dessins d'enfants a single riemann surface is associated to each graph. We call it the Grothendieck model of a ribbon graph. The goal of the present paper is to discuss one more such construction, depending on some parameters - that of Chekhov-Fock. We prove that putting all parametres equal to 0, we obtain the Grothendieck model of this graph.
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arxiv:math/0210183
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We consider a class of homogeneous manifolds including all semisimple coadjoint orbits. We describe manifolds of that class admitting deformation q uantizations equivariant under the action of $G$ and the corresponding quantum group. We also classify Poisson brackets relating to such quantizations.
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arxiv:math/0210188
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The Hopf algebra generated by the l-functionals on the quantum double C_q[G] \bowtie C_q[G] is considered, where C_q[G] is the coordinate algebra of a standard quantum group and q is not a root of unity. It is shown to be isomorphic to C_q[G]^op \bowtie U_q(g). This was conjectured by T. Hodges in [Ho]. As an algebra it can be embedded into U_q(g) \otimes U_q(g). Here it is proven that there is no bialgebra structure on U_q(g) \otimes U_q(g), for which this embedding becomes a homomorphism of bialgebras. In particular, it is not an isomorphism. As a preliminary a lemma of [Ho] concerning the structure of l-functionals on C_q[G] is generalized. For the classical groups a certain choice of root vectors is expressed in terms of l-functionals. A formula for their coproduct is derived.
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arxiv:math/0210203
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Given a family $X/B$ of nodal curves, we construct canonically and compatibly with base-change, via an explicit blow-up of the Cartesian product $X^r/B$, a family $W^r(X/B)$ parametrizing length-$r$ subschemes of fibres of $X/B$ (plus some additional data). Though $W^r(X/B)$ is singular, the important sheaves on it are locally free, which allows us to study intersection theory on it and deduce enumerative applications, including some relative multiple point formulae, enumerating the length-$r$ schemes contained simultaneously in some fibre of $X/B$ and some fibre of a given map from $X$ to a smooth variety.
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arxiv:math/0210209
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Many of the known complemented subspaces of L_p have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well-known complemented subspaces of L_p. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of L_p. Using this we define a space Y_n with norm given by partitions and weights with distance to any subspace of L_p growing with n. This allows us to construct an example of a Banach space with norm given by partitions and weights which is not isomorphic to a subspace of L_p.
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arxiv:math/0210228
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In this note we discuss a class of hyperelliptic curves introduced by Abel in a 1826 paper. After some indications of the context in which he introduced them and a description of his main result we give some results on the moduli space of such curves. In particular we compute the dimension of it at each of its points as well as giving a combinatorial formula for the number of components.
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arxiv:math/0210273
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We extend some of the results obtained for subvarieties of the moduli stack of canonically polarized manifolds in "Base spaces of non-isotrivial families of smooth minimal models" (math.AG/0103122) to moduli of polarized minimal models of Kodaira dimension zero. To this aim we first show that for those manifolds there are no non isotrivial families of minimal models, which are birationally isotrivial. In the last chapter, we discuss the rigidity of curves in the moduli stack of canonically polarized manifolds and of polarized minimal models of Kodaira dimension zero.
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arxiv:math/0210310
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Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomials ?
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arxiv:math/0210321
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In the paper I study properties of random polynomials with respect to a general system of functions. Some lower bounds for the mathematical expectation of the uniform and recently introduced integral-uniform norms of random polynomials are established. {\sc Key words and phrases:} Random polynomial, estimates for maximum of random process, integral-uniform norm.
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arxiv:math/0210341
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In this note we will show how to get consistency for first order classical logic, in a purely syntactic way, without going through cut elimination. The procedure is very simple and it uses the calculus of structures in an essential way. It also shows how finitaryness (in the sense of finite choice of premises for each rule) is actually a triviality (contrarily to what one would guess from textbooks).
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arxiv:math/0210387
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We study affine immersions as introduced by Nomizu and Pinkall. We classify those affine immersions of a surface in 4-space which are degenerate and have vanishing cubic form (i.e. parallel second fundamental form). This completes the classification of parallel surfaces of which the first results were obtained in the beginning of this century by Blaschke and his collaborators.
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arxiv:math/0210406
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We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yau's relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsager's reciprocity relations must hold. We check validity of these identities for a wide class of models. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. The Onsager relations arising in this context and its consequences seem to be novel. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena.
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arxiv:math/0210426
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In the abelian case (the subject of several beautiful books) fixing some combinatorial structure (so called theta structure of level k) one obtains a special basis in the space of sections of canonical polarization powers over the jacobians. These sections can be presented as holomorphic functions on the "abelian Schottky space". This fact provides various applications of these concrete analytic formulas to the integrable systems, classical mechanics and PDE's. Our practical goal is to do the same in the non abelian case that is to give an answer to the Beauville's question. In future we hope to extend this digest to a mathematical mohograph with title "VBAC".
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arxiv:math/0210466
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It is shown that every algebra over the chain operad of the little disks operad gives naturally rise to a Hertling-Manin's F-manifold, that is a smooth manifold equipped with an integrable graded commutative associative product on the tangent sheaf. In particular, moduli spaces of extended deformations of complex/symplectic structures are shown to have a canonical structure of F-manifold. With the help of the $G_\infty$-operad a strong homotopy version of the notion of F-manifold is constructed. Among natural examples of $F_\infty$-manifolds one finds formal manifolds associated with the Hochschild cohomology of an associative algebra and with the singular cohomology of an arbitrary compact topological space.
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arxiv:math/0210478
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We construct an A_infinity-category D(C|B) from a given A_infinity-category C and its full subcategory B. The construction is similar to a particular case of Drinfeld's quotient of differential graded categories. We use D(C|B) to construct an A_infinity-functor of K-injective resolutions of a complex. The conventional derived category is obtained as the 0-th cohomology of the quotient of differential graded category of complexes over acyclic complexes.
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arxiv:math/0211037
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We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $ \exp (C |s|^\delta) $ where $ \delta $ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound $ \exp (C |s|^{n+1}) $, and it gives new bounds on the number of resonances (scattering poles) of $ \Gamma \backslash {\mathbb H}^{n+1} $. The proof of this result is based on the application of holomorphic $ L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider $ \Gamma \backslash {\mathbb H}^{n+1} $ as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets.
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arxiv:math/0211041
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Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Cech-de Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bundles. We reprove the relation with the determinant of cohomology construction. Furthermore, if we specialize the covering to the one provided by a Kleinian uniformization (thereby allowing possibly disconnected surfaces) the same class can be reinterpreted as the transgression of the regulator class expressed by the Bloch-Wigner dilogarithm.
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arxiv:math/0211055
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Let a be a positive integer greater than 1, and Q_a(x;k,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo k. In this paper, the natural densities of Q_a(x;4,j) (j=0,1,2,3) are considered. We assume a is square-free and a is congruent to 1 (mod 4). Then, for j=0, 2, we can prove unconditionally that their natural densities are equal to 1/3. On the contrary, for j=1, 3, we assume Generalized Riemann Hypothesis, then we can prove that their densities are equal to 1/6.
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arxiv:math/0211077
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This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from `noncommutative functional analysis', more specifically the new field of {\em operator spaces.} In our illustration we show how the classical characterization of (possibly non-surjective) isometries between function algebras generalizes to operator algebras. We give some variants of this characterization, and a new proof which has some advantages.
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arxiv:math/0211098
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We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics; this requires no nondegeneracy hypothesis. We also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join.
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arxiv:math/0211099
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Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau [LPY] on spectral gap and logarithmic Sobolev inequalities for unbounded conservative spin systems, we study uniform bounds in these inequalities for Glauber dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n), (x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or small perturbation of strictly convex) and, following [LPY], the case V is a bounded perturbation of a quadratic potential. By a simple path counting argument for the standard random walk, uniform bounds for the Glauber dynamics yields, in a transparent way, the classical L^{-2} decay for the Kawasaki dynamics on d-dimensional cubes of length L. The arguments of proofs however closely follow and make heavy use of the conservative approach and estimates of [LPY], relying in particular on the Lu-Yau martingale decomposition and clever partitionings of the conditional measure.
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arxiv:math/0211108
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In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid monodromy type of algebraic surfaces and symplectic 4-manifolds.
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arxiv:math/0211197
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A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has cardinality exactly n+1 for each positive integer n. Our main result is that the volume of the simplex whose n+1 vertices are the n+1 points in F_n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation $\pi$ (depending on an irrational x and a positive integer n) that orders the fractional parts {1 x}, {2 x}, ..., {n x}, i.e., 0 < {\pi(1) x} < {\pi(2) x} < ... < {\pi(n) x} < 1. We give a formula for the sign of $\pi$, and prove that for every irrational x there are infinitely many n such that the order of $\pi$ (as an element of the symmetric group S_n) is less than n.
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arxiv:math/0211200
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We characterize the pairs of sup-lattices which occur as pairs of Morita equivalence bimodules between quantales in terms of the mutual relation between the sup-lattices.
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arxiv:math/0211236
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We prove a new patchworking theorem for singular algebraic curves, which states the following. Given a complex toric threefold $Y$ which fibers over ${\mathbb C}$ with a reduced reducible zero fiber $Y_0$ and other fibers $Y_t$ smooth, and given a reduced curve $C_0\subset Y_0$, the theorem provides a sufficient condition for the existence of a one-parametric family of curves $C_t\subset Y_t$, which induces an equisingular deformation for some singular points of $C_0$ and certain prescribed deformations for the other singularities. As application we give a comment on a recent theorem by G. Mikhalkin on enumeration of nodal curves on toric surfaces via non-Archimedean amoebas [arXiv:math.AG/0209253]. Namely, using our patchworking theorem, we establish link between nodal curves over the field of complex Puiseux series and their non-Archimedean amoebas, what has been done by Mikhalkin in a different way. We discuss also the case of curves with a cusp as well as real nodal curves.
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arxiv:math/0211278
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Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3-manifolds of classical results by Turaev and Goldman concerning intersection and skein theory on oriented surfaces.
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arxiv:math/0211392
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Let $f(x,y), g(x,y)$ denote either a pair of holomorphic function germs, or a pair of monic polynomials in $x$ whose coefficients are Laurent series in $y$. A relative polar arc is a Newton-Puiseux root, $x=\gamma(y)$, of the Jacobian $J=f_yg_x-f_xg_y$. We define the tree-model, $T(f,g)$, for the pair, using the contact orders of the Newton-Puiseux roots of $f$ and $g$. We then describe how the $\gamma$'s climb, and where they leave, the tree. We shall also show by two examples that the way the $\gamma$'s leave the tree is not an invariant of the tree; this phenomenon is in sharp contrast to that in the one function case where the tree completely determines how the polar roots split away. Our result yield a factorisation of the Jacobian determinant.
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arxiv:math/0211408
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The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The classifications for these give isolated examples that are isotropy irreducible spaces, and known families that are the nearly K\"ahler manifolds in dimension 6 and Gray's weak holonomy G$_2$ structures in dimension 7.
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arxiv:math/0211446
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This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the algebraic moment map. In particular, we explain the relation between linear precision and the algebraic moment map. This builds on the introduction to toric varieties by David Cox: What is a Toric Variety? at http://www.cs.amherst.edu/~dac/lectures/tutorial.ps
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arxiv:math/0212044
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Let M be an orientable and irreducible 3-manifold whose boundary is an incompressible torus. Suppose that M does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in M with the 4-plane property can realize only finitely many boundary slopes. Moreover, we will show that only finitely many Dehn fillings of M can yield 3-manifolds with nonpositive cubings. This gives the first examples of hyperbolic 3-manifolds that cannot admit any nonpositive cubings.
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arxiv:math/0212111
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We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process.
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arxiv:math/0212116
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We propose a simple formula for the coordinates of the vertices of the Stasheff polytope (associahedron) and we compare it to the permutohedron.
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arxiv:math/0212126
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We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite union of discs of arbitrarily small but fixed radii. These exceptional discs are the neighborhoods of the so-called band edges of the associated equilibrium measure. As applications, we prove universality results for correlation functions of a general class of discrete orthogonal polynomial ensembles, and in particular we deduce asymptotic formulae with error bound for certain statistics relevant in the random tiling of a hexagon with rhombus-shaped tiles. The discrete orthogonal polynomials are characterized in terms of a a Riemann-Hilbert problem formulated for a meromorphic matrix with certain pole conditions. By extending the methods of [17, 22], we suggest a general and unifying approach to handle Riemann-Hilbert problems in the situation when poles of the unknown matrix are accumulating on some set in the asymptotic limit of interest.
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arxiv:math/0212149
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For the hypoelliptic differential operators $P={\partial^2_ x}+(x^k\partial_ y -x^l{\partial_t})^2$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be {\em{analytic}} hypoelliptic (except for $(k,l)=(0,1)$), in accordance with Treves' conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.
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arxiv:math/0212167
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Let F denote a homogeneous degree 4 polynomial in 3 variables, and let s be an integer between 1 and 5. We would like to know if F can be written as a sum of fourth powers of s linear forms (or a degeneration). We determine necessary and sufficient conditions for this to be possible. These conditions are expressed as the vanishing of certain concomitants of F for the natural action of SL_3.
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arxiv:math/0212169
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Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.
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arxiv:math/0212196
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We construct three compatible quadratic Poisson structures such that generic linear combination of them is associated with Elliptic Sklyanin algebra in n generators. Symplectic leaves of this elliptic Poisson structure is studied. Explicit formulas for Casimir elements are obtained.
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arxiv:math/0212210
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We introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for our theory of differential graded schemes, as they give rise to affine differential graded schemes. We also introduce etale morphisms. The purpose for studying these, is that they will be used to glue differential graded schemes from affine ones with respect to an etale topology.
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arxiv:math/0212225
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The main motivation for this work was to find an explicit formula for a "Szego-regularized" determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold. The idea of the Szego-regularization was suggested by V. Guillemin and K. Okikiolu. They have computed the second term in a Szego type expansion on a Zoll manifold of an arbitrary dimension. In the present work we compute the third asymptotic term in any dimension. In the case of dimension 2, our formula gives the above mentioned expression for the Szego-redularized determinant of a zeroth order PsDO. The proof uses a new combinatorial identity, which generalizes a formula due to G.A. Hunt and F.J. Dyson. This identity is related to the distribution of the maximum of a random walk with i.i.d. steps on the real line. The full version of this paper is also available, math.FA/0212275.
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arxiv:math/0212273
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We utilize the obstruction theory of Galewski-Matumoto-Stern to derive equivalent formulations of the Triangulation Conjecture. For example, every closed topological manifold M^n with n > 4 can be simplicially triangulated if and only if the two distinct combinatorial triangulations of RP^5 are simplicially concordant.
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arxiv:math/0212297
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Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several counter-examples. We consider the subadditivity theorem for monomial ideals on toric rings, and construct a counter-example on a three-dimensional toric ring.
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arxiv:math/0212340
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The graded algebra Lambda defined by Pierre Vogel is of general interest in the theory of finite-type invariants of knots and of 3-manifolds because it acts on the corresponding spaces of connected graphs subject to relations called IHX and AS. We examine a subalgebra Lambda_0 that is generated by certain elements called t and x_n with n >= 3. Two families of relations in Lambda_0 are derived and it is shown that the dimension of Lambda_0 grows at most quadratically with respect to degree. Under the assumption that t is not a zero divisor in Lambda_0, a basis of Lambda_0 and an isomorphism from Lambda_0 to a sub-ring of Z[t,u,v] is given.
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arxiv:math/0301019
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