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This is part one of a series of papers. In this series of papers, we consider problems analogous to the Oppenheim conjecture from the viewpoint of prehomogeneous vector spaces.
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arxiv:math/9605219
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A definition is given of seriate sets as being sets constituted out of structured collections of objects which are recursively internally self- similar. Fundamental (geometrical) objects of Dimension N are conceived to be constituted out of seriate sets of fundamental objects of Dimension N-1, starting with points assigned to Dimension 0. Syntactical rules to enable such objects to be systematically named and combined, are set out. A series of formal proofs of theorems relative to objects of Dimensions 1 and 2 are worked through. A proof that four colours are sufficient to colour any five area map is given in illustration. I know most/all will think all the above impossible, and necessarily the work of some crazy. If it is wrong I believe it is wrong in interesting ways. The underlying idea is easily grasped without working through the theorems given, so I would be grateful to hear from anyone as to why it is wrong - and not simply because it must be so.
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arxiv:math/9606206
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In this paper we explore a connection between descriptive set theory and inner model theory. From descriptive set theory, we will take a countable, definable set of reals, A. We will then show that A is equal to the reals of M, where M is a canonical model from inner model theory. In technical terms, M is a ''mouse''. Consequently, we say that A is a mouse set. For a concrete example of the type of set A we are working with, let OD(n) be the set of reals which are Sigma-n definable over the omega-first level of the model L(R), from an ordinal parameter. In this paper we will show that for all n, OD(n) is a mouse set. Our work extends some similar results due to D.A. Martin, J.R. Steel, and H. Woodin. Several interesting questions in this area remain open.
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arxiv:math/9606207
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The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no rational f,g with this property. In fact, a more general problem is solved. In addition to algebraic methods, some results from local analytic dynamics are used.
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arxiv:math/9606217
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We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities.
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arxiv:math/9607209
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For any Lagrangean K\"ahler submanifold $M \subset T^*{\Bbb C}^n$, there exists a canonical hyper K\"ahler metric on $T^*M$. A K\"ahler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper K\"ahler manifolds with large automorphism group. Using it, a class of pseudo hyper K\"ahler manifolds of complex signature $(2,2n)$ is constructed. For any hyper K\"ahler manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper K\"ahler metric of complex signature $(2,2n)$.
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arxiv:math/9607213
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Let $\Gamma$ be a discrete subgroup of a simply connected, solvable Lie group~$G$, such that $\Ad_G\Gamma$ has the same Zariski closure as $\Ad G$. If $\alpha \colon \Gamma \to \GL_n(\real)$ is any finite-dimensional representation of~$\Gamma $,we show that $\alpha$ virtually extends to a continuous representation~$\sigma $ of~$G$. Furthermore, the image of~$\sigma$ is contained in the Zariski closure of the image of~$\alpha $. When $\Gamma$ is not discrete, the same conclusions are true if we make the additional assumption that the closure of $[\Gamma, \Gamma]$ is a finite-index subgroup of $[G,G] \cap \Gamma$ (and $\Gamma$ is closed and $\alpha$ is continuous).
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arxiv:math/9607221
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Let p be a prime number, and let f, g, and h be three modular forms of weights $\kappa$, $\lambda$, and $\mu$ for $SL(2,\Bbb{Z})$. We suppose $\kappa \geq \lambda + \mu$. In joint work with Kudla, one of the authors obtained a formula for the normalized {\it square root} of the value at $s = {1/2}(\kappa + \lambda + \mu - 2)$ (the {\it central critical value}) of the triple product $L(s,f,g,h)$. We apply this formula, letting $f$ (and thus $\kappa$) vary in a $p$-adic analytic family ${\bold f}$ of ordinary modular forms (a Hida family). By modifying Hida's construction of the $p$-adic Rankin-Selberg convolution, we obtain a generalized $p$-adic measure whose associated analytic function gives a $p$-adic interpolation of the square roots of the central critical values of $L(s,f,g,h)$, normalized by certain universal correction factors. The archimedean correction factor is not determined explicitly. This is an example of what appears to be a very general phenomenon of $p$-adic interpolation of normalized square roots of $L$-functions along the so-called "anti-cyclotomic hyperplane." We note that the $p$-adic triple product itself has not been constructed in the half-space $\kappa \geq \lambda + \mu$.
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arxiv:math/9610228
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Let $\Ga$ be a connected, solvable linear algebraic group over a number field~$K$, let $S$ be a finite set of places of~$K$ that contains all the infinite places, and let $\theints$ be the ring of $S$-integers of~$K$. We define a certain closed subgroup~$\GOS$ of $\Ga_S = \prod_{v \in S} \Ga_{K_v}$ that contains $\Ga_{\theints}$, and prove that $\Ga_{\theints}$ is a superrigid lattice in~$\GOS$, by which we mean that finite-dimensional representations $\alpha\colon \Ga_{\theints} \to \GL_n(\real)$ more-or-less extend to representations of~$\GOS$. The subgroup~$\GOS$ may be a proper subgroup of~$\Ga_S$ for only two reasons. First, it is well known that $\Ga_{\theints}$ is not a lattice in~$\Ga_S$ if $\Ga$ has nontrivial $K$-characters, so one passes to a certain subgroup $\GS$. Second, $\Ga_{\theints}$ may fail to be Zariski dense in $\GS$ in an appropriate sense; in this sense, the subgroup $\GOS$ is the Zariski closure of~$\Ga_{\theints}$ in~$\GS$. Furthermore, we note that a superrigidity theorem for many non-solvable $S$-arithmetic groups can be proved by combining our main theorem with the Margulis Superrigidity Theorem.
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arxiv:math/9611219
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We show that if the lower central series of the fundamental group of a closed oriented $3$-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion $2$-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental groups of closed oriented $3$-manifolds.
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arxiv:math/9612216
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In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between monopoles are identified with holomorphic data on an associated ruled surface, and a dimension formula for such flows is calculated.
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arxiv:math/9612221
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We consider a specialization $Y_M(q,t)$ of the Tutte polynomial of a matroid $M$ which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of $M$. We show that the coefficients of $Y_M(1-p,t)$ are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independent set complexes of the duals of the truncations of $M$. In particular, we obtain a new homological interpretation for the coefficients of the characteristic polynomial of a matroid.
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arxiv:math/9702219
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Let K be an algebraically closed field endowed with a complete non-archimedean norm. Let f:Y -> X be a map of K-affinoid varieties. We prove that for each point x in X, either f is flat at x, or there exists, at least locally around x, a maximal locally closed analytic subvariety Z in X containing x, such that the base change f^{-1}(Z) -> Z is flat at x, and, moreover, g^{-1}(Z) has again this property in any point of the fibre of x after base change over an arbitrary map g:X' -> X of affinoid varieties. If we take the local blowing up \pi:X-tilde -> X with this centre Z, then the fibre with respect to the strict transform f-tilde of f under \pi, of any point of X-tilde lying above x, has grown strictly smaller. Among the corollaries to these results we quote, that flatness in rigid analytic geometry is local in the source; that flatness over a reduced quasi-compact rigid analytic variety can be tested by surjective families; that an inclusion of affinoid domains is flat in a point, if it is unramified in that point.
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arxiv:math/9702230
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We consider the system of N (\ge 2) elastically colliding hard balls with masses m_1,..., m_N, radius r, moving uniformly in the flat torus T_L^{\nu}= R^\nu/L \cdot Z^\nu, \nu \ge 2. It is proved here that the relevant Lyapunov exponents of the flow do not vanish for almost every (N+1)-tuple (m_1,...,m_N;L) of the outer geometric parameters.
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arxiv:math/9704229
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We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases.
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arxiv:math/9705210
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A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system $A_{n-1}$. The proof is based on an explicit formula for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.
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arxiv:math/9705223
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In the Forum section of the November, 1993 Notices of the American Mathematical Society, John Franks discussed the electronic journal of the future. Since then, the New York Journal of Mathematics, the first electronic general mathematics journal, has begun publication. In this article, we explore the issues of electronic journal publishing in the context of this new project. We also discuss future developments.
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arxiv:math/9708203
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The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces $X$ and $Y$, the Kadets distance is defined to be the infimum of the Hausdorff distance $d(B_X,B_Y)$ between the respective closed unit balls over all isometric linear embeddings of $X$ and $Y$ into a common Banach space $Z.$ This is compared with the Gromov-Hausdorff distance which is defined to be the infimum of $d(B_X,B_Y)$ over all isometric embeddings into a common metric space $Z$. We prove continuity type results for the Kadets distance including a result that shows that this notion of distance has applications to the theory of complex interpolation.
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arxiv:math/9709211
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Various versions of club are shown to be different. A question of Soukup, Fuchino and Juhasz, is it consistent to have a stick without club, is answered as a consequence. The more detailed version of the paper, which is coming up, also answers a question of Galvin.
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arxiv:math/9710215
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The aim of this paper is to show that the automorphism and isometry groups of the suspension of $B(H)$, $H$ being a separable infinite dimensional Hilbert space, are algebraically reflexive. This means that every local automorphism, respectively local surjective isometry of $C_0(\mathbb R)\otimes B(H)$ is an automorphism, respectively a surjective isometry.
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arxiv:math/9711208
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We address ZFC inequalities between some cardinal invariants of the continuum, which turned to be true in spite of strong expectations given by [RoSh:470].
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arxiv:math/9711222
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The main result of this article is the decomposition of tensor products of representations of SL(2) in the sum of irreducible representations parametrized by outerplanar graphs. An outerplanar graph is a graph with the vertices 0, 1, 2, ..., m, edges of which can be drawn in the upper half-plane without intersections. I allow for a graph to have multiple edges, but don't allow loops.
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arxiv:math/9712259
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This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. While deformations of complex surfaces from our first class are induced by quasiconformal homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their complex analytic submanifolds) from another class are quasiconformally unstable, but nevertheless their deformations are induced by homeomorphisms.
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arxiv:math/9712281
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We introduce a new class of simple Lie algebras $W(n,m)$ that generalize the Witt algebra by using "exponential" functions, and also a subalgebra $W^*(n,m)$ thereof; and we show each derivation of $W^*(1,0)$ can be written as a sum of an inner derivation and a scalar derivation. The Lie algebra $W(n,m)$ is $Z$-graded and is infinite growth.
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arxiv:math/9712293
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The main new result is the computation of the degeneration of l-adic Eisenstein classes at the cusps. This is done by relating it to the degeneration of the elliptic polylog. These classes come from K-theory and their Hodge regulator can also be computed (see: Dirichlet motives via modula curves, on the K-theory server). This gives a new proof of a comparison conjecture of Bloch and Kato which was used in the proof of their Tamagawa number conjecture for the Riemann zeta-function. The paper contains appendices on the definition of the classical and elliptic polylog, its degeneration and the comparison to Eisenstein classes.
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arxiv:math/9712295
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The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results of Sylvester (1851), Hilbert (1888), Dixon and Stuart (1906) and some more recent results of Mukai (1992) are presented together with new results for the cases (n,d)=(3,8), (4,2), (5,3). In the last case the variety of sums of 8 powers of a general cubic form is a Fano 5-fold of index 1 and degree 660.
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arxiv:math/9801110
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See q-alg/9710003 for the corrected version of this paper.
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arxiv:math/9802023
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An action of the Yangian of the general Lie algebra gl(N) is defined on every irreducible integrable highest weight module of affine gl(N) with level greater than 1. This action is derived, by means of the Drinfeld duality and a subsequent semi-infinite limit, from a certain induced representation of the degenerate double affine Hecke algebra H. Each vacuum module of affine gl(N) is decomposed into irreducible Yangian subrepresentations by means of the intertwiners of H. Components of this decomposition are parameterized by semi-infinite skew Young diagrams.
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arxiv:math/9802048
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In this paper we announce a gluing theorem for conformal structures with anti-self-dual (ASD) Weyl tensor that applies in geometrical situations that are more general than those considered by previous authors. By adapting a method proposed by Floer, sufficient conditions are given for the existence of ASD conformal structures on `generalized connected sums' of non-compact ASD 4-manifolds with cylindrical ends. The gluing theorem applies in particular to give results about connected sums of ASD orbifolds along (isolated) singular points.
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arxiv:math/9802055
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The goal of this paper is to construct quasi-isometrically embedded subgroups of Thompson's group $F$ which are isomorphic to $\fz^n$ for all $n$. A result estimating the norm of an element of Thompson's group is found. As a corollary, Thompson's group is seen to be an example of a finitely presented group which has an infinite-dimensional asymptotic cone.
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arxiv:math/9802095
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For a projective morphism of an smooth algebraic surface $X$ onto a smooth algebraic curve $S$, both given over a perfect field $k$, we construct the direct image morphism in two cases: from $H^i(X,\Omega^2_X)$ to $H^{i-1}(S,\Omega^1_S)$ and when $char k =0$ from $H^i(X,K_2(X))$ to $H^{i-1}(S,K_1(S))$. (If i=2, then the last map is the Gysin map from $CH^2(X)$ to $CH^1(S)$.) To do this in the first case we use the known adelic resolution for sheafs $\Omega^2_X$ and $\Omega^1_S$. In the second case we construct a $K_2$-adelic resolution of a sheaf $K_2(X)$. And thus we reduce the direct image morphism to the construction of some residues and symbols from differentials and symbols of 2-dimensional local fields associated with pairs $x \in C$ ($x$ is a closed point on an irredicuble curve $C \in X$) to 1-dimensional local fields associated with closed points on the curve $S$. We prove reciprocity laws for these maps.
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arxiv:math/9802112
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We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small.
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arxiv:math/9803011
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The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy groups. First, we show how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups \pi_{*-1} X (which are isomorphic to \pi_* Y, if X=\Omega\Y. This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be formulated in the framework of an obstruction theory for realizing \Pi-algebras, which is simplified here by showing that any (suitable) \Delta-simplicial space may be made into a full simplicial space (a result which may be useful in other contexts).
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arxiv:math/9803055
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We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra $\ssll (2)_q$ introduced by Lyubashenko and the second named author. We consider separately the cases of $q$ generic and $q$ at roots of unity. Some of the representations have no classical analog even for generic $q$. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra $U_q(sl(2))$, while in those that do there are different matrix elements.
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arxiv:math/9803095
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We construct modular categories from Hecke algebras at roots of unity. For a special choice of the framing parameter, we recover the Reshetikhin-Turaev invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N) by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We then discuss the choice of the framing parameter. This leads, for any rank N and level K, to a modular category \tilde H^{N,K} and a reduced invariant \tilde\tau_{N,K}. If N and K are coprime, then this invariant coincides with the known PSU(N) invariant at level K. If gcd(N,K)=d>1, then we show that the reduced invariant admits spin or cohomological refinements, with a nice decomposition formula which extends a theorem of H. Murakami.
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arxiv:math/9803114
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If $G$ is an omega-stable group with a normal definable subgroup $H$, then the Sylow-$2$-subgroups of $G/H$ are the images of the Sylow-$2$-subgroups of $G$.
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arxiv:math/9803161
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The known Holstein-Primakoff and Dyson realizations of the Lie algebra $sl(n+1), n=1,2,...$ in terms of Bose operators (Okubo S 1975 J. Math. Phys. 16 528) are generalized to the class of the quantum algebras $U_q[sl(n+1)]$ for any $n$. It is shown how the elements of $U_q[sl(n+1)]$ can be expressed via $n$ pairs of Bose creation and annihilation operators.
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arxiv:math/9804017
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We show that the image of a 2-dimensional set under d-dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure, if and only if the set in question has positive (d/2)-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.
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arxiv:math/9804061
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Generalizing the classical result of Bohr, we show that if an n-variable power series converges in an n-circular bounded complete domain D and its sum has modulus less than 1, then the sum of the maximum of the moduli of the terms is less than 1 in the homothetic domain r*D, where r = 1 - (2/3)^(1/n). This constant is near to the best one for the domain D = {z: |z_1| + ... + |z_n| < 1}.
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arxiv:math/9804102
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The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map $f:{\Bbb C} \to X$ to the surface is constant. In 1970, Kobayashi conjectured more generally that a (very) generic hypersurface of sufficiently high degree in projective space is hyperbolic (here, the terminology "very generic" refers to complements of countable unions of proper algebraic subsets). Our technique follows the stream of ideas initiated by Green and Griffiths in 1979, which consists in considering jet differentials and their associated base loci. However, a key ingredient is the use of a different kind of jet bundles, namely the "Semple jet bundles" previously studied by the first named author (Santa Cruz Summer School, July 1995, Proc. Symposia Pure Math., Vol. 62.2, 1997). The base locus calculation is achieved through a sequence of Riemann-Roch formulas combined with a suitable generic vanishing theorem for order 2-jets. Our method covers the case of surfaces of general type with Picard group ${\Bbb Z}$ and $(13+12\theta_2) c_1^2 - 9 c_2 > 0$, where $\theta_2$ is what we call the "2-jet threshold" (the 2-jet threshold turns out to be bounded below by -1/6 for surfaces in ${\Bbb P}^3$). The final conclusion is obtained by using very recent results of McQuillan on holomorphic foliations.
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arxiv:math/9804129
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We prove a characterization of the support of the law of the solution for a stochastic wave equation with two-dimensional space variable, driven by a noise white in time and correlated in space. The result is a consequence of an approximation theorem, in the convergence of probability, for equations obtained by smoothing the random noise. For some particular classes of coefficients, aproximation in the $L^p$ norm, for $p\geq 1$ is also proved.
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arxiv:math/9804148
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We prove a main gap theorem for e-saturated submodels of a homogeneous structure. We also study the number of e-saturated models, which are not elementarily embeddable to each other
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arxiv:math/9804157
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In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space G'/H'. Our procedure establishes a correspondence between certain unitary representations of G and those of G'. This extends the usual theta--correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E_7 and F_4.
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arxiv:math/9805002
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The $x$-dependence of the symmetries of (1+1)-dimensional scalar translationally invariant evolution equations is described. The sufficient condition of (quasi)polynomiality in time $t$ of the symmetries of evolution equations with constant separant is found. The general form of time dependence of the symmetries of KdV-like non-linearizable evolution equations is presented.
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arxiv:math/9805105
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We prove the consistency with ZFC of ``the length of an ultraproduct of Boolean algebras is smaller than the ultraproduct of the lengths''. Similarly for some other cardinal invariants of Boolean algebras.
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arxiv:math/9805145
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By the classical genus zero Sugawara construction one obtains from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type) representations of the Virasoro algebra. In this lecture first the classical construction is recalled. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in joint work with O.K. Sheinman. We were able to show that also in the higher genus, multi-point situation one obtains from representations of the global algebras of affine type representations of a centrally extended algebra of meromorphic vector fields on Riemann surfaces. The latter algebra is the generalization of the Virasoro algebra to higher genus. Invited lecture at the XVI${}^{th}$ workshop on geometric methods in physics, Bialowieza, Poland, June 30 -- July 6, 1997.
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arxiv:math/9806032
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We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A^e-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A^e-module D to N \otimes M, viewed as a left A^e-module. Some properties of D are described. Finally, we show that when A is a symmetric algebra, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of N \otimes M.
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arxiv:math/9806044
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We consider the action of a subtorus of the big torus on a toric variety. The aim of the paper is to define a natural notion of a quotient for this setting and to give an explicit algorithm for the construction of this quotient from the combinatorial data corresponding to the pair consisting of the subtorus and the toric variety. Moreover, we study the relations of such quotients with good quotients. We construct a good model, i.e. a dominant toric morphism from the given toric variety to some ``maximal'' toric variety having a good quotient by the induced action of the given subtorus.
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arxiv:math/9806049
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We consider vector spaces H(n,l) and F(n,l) spanned by the degree-n coefficients in power series forms of the Homfly and Kauffman polynomials of links with l components. Generalizing previously known formulas, we determine the dimensions of the spaces H(n,l), F(n,l) and H(n,l)+F(n,l) for all values of n and l. Furthermore, we show that for knots the algebra generated by H(n,1)+F(n,1) (n > 0) is a polynomial algebra with dim(H(n,1)+F(n,1))-1=n+[n/2]-4 generators in degree n>3 and one generator in degrees 2 and 3.
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arxiv:math/9806064
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We give a general way of representing the crystal (base) corresponding to the intgrable highest weight modules of quantum Kac-Moody algebras, which is called polyhedral realizations. This is applied to describe explicitly the crystal bases of integrable highest weight modules for arbitrary rank 2 Kac-Moody algebra cases, the classical A_n-case and the affine A^{(1)}_{n-1}-case.
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arxiv:math/9806085
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We investigate, in some details, symplectic equivalence between several conformal classes of Lorentz metrics on the hyperboloid of one sheet $H^{1,1} \cong T \times T - \Delta$ and affine coadjoint orbits of the group $Diff_+(\Delta)$ of orientation preserving diffeomorphisms of $\Delta \cong T$ with its natural projective structure. This will allow for generalizations, namely, to the case of arbitrary projective structures on null infinity.
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arxiv:math/9806135
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We classify completely the surfaces of general type whose canonical map is 3-to-1 onto a surface of minimal degree in projective space. These surfaces fall into 5 distinct classes and we give explicit examples belonging to each of these classes. As far as we know, one of the examples thus constructed was unknown and it is a surface whose canonical system has two infinitely near base points.
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arxiv:math/9807006
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We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
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arxiv:math/9807016
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Generalizations of the classical affine Lelieuvre formula to surfaces in projective three-dimensional space and to hypersurfaces in multi- dimensional projective space are given. A discrete version of the projective Lelieuvre formula is presented too.
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arxiv:math/9807083
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Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of these cases one can define a module of measurable 1-forms and a first-order exterior derivative. We give a general construction which applies to any metric space equipped with a sigma-finite measure and produces the desired result in all of the above cases. It also applies to an important class of Dirichlet spaces, where, however, the known first-order differential calculus in general differs from ours (although the two are related).
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arxiv:math/9807096
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Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds consisting of piecewise geodesic paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian structures $G^i_P$. It is proved that as the mesh of the partition tends to $0$, $$ 1/Z_P^i e^{- 1/2 E(\sigma)} Vol_{G^i_P}(\sigma) \to \rho_i(\sigma)\nu(\sigma) $$ where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma$, and for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is the Riemannian volume form relative $G^i_P$, and $\nu$ is Wiener measure on paths on $M$. Here $\rho_1 = 1$ and $$ \rho_0 (\sigma) = \exp( -1/6 \int_0^1 Scal(\sigma(s))ds ) $$ where $Scal$ is the scalar curvature of $M$. These results are also shown to imply the well know integration by parts formula for the Wiener measure.
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arxiv:math/9807098
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We prove a topological result concerning the kernel of a morphism d : E --> F of holomorphic vector bundles over a complex analytic space. As a consequence, we show that the projectivization P(ker d) is a quasifibration up to some dimension. We give an application to the Abel-Jacobi map of a Riemann surface, and to the space of rational curves in the symmetric product of a Riemann surface.
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arxiv:math/9807103
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In this note, we first derive explicit formulas for Kirby-Melvin's three-manifold invariants $\tau_r^{'}$ for all Lens spaces from our results for another set of invariants $\xi_r$ defined by the first author. Then, as a corollary, we obtain formulas for Othsuki's $\tau$ for all Lens spaces.
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arxiv:math/9807155
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We introduce a 1-cocycle on the group of diffeomorphisms Diff$(M)$ of a smooth manifold $M$ endowed with a projective connection. This cocycle represents a nontrivial cohomology class of $\Diff(M)$ related to the Diff$(M)$-modules of second order linear differential operators on $M$. In the one-dimensional case, this cocycle coincides with the Schwarzian derivative, while, in the multi-dimensional case, it represents its natural and new generalization. This work is a continuation of \cite{bo} where the same problems have been treated in one-dimensional case.
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arxiv:math/9808006
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A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X,L), the cohomologies of L over the GIT quotient X // G equal the invariant part of the cohomologies over X. This generalizes the theorem of [Invent. Math. 67 (1982), 515-538] on global sections, and strengthens its subsequent extensions to Riemann-Roch numbers. Remarkable by-products are the invariance of cohomology of vector bundles over X // G under a small change in the defining polarization or under shift desingularization, as well as a new proof of Boutot's theorem. Also studied are equivariant holomorphic forms and the equivariant Hodge-to-de Rham spectral sequences for X and its strata, whose collapse is shown. One application is a new proof of the Borel-Weil-Bott theorem of [Invent. Math. 134 (1998), 1-57] for the moduli stack of G-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Collapse of the Hodge-to-de Rham sequences for these stacks is also shown.
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arxiv:math/9808029
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The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a lattice over $\bz[q^{-1}]$. This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of $\cl$. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the ``imaginary'' subalgebra which is fixed by the affine braid group. Once the basis is found we construct a PBW-type basis whose $\bz[q^{-1}]$-span reduces to a ``crystal'' basis at $q=\infty,$ with the imaginary component given by the Schur functions.
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arxiv:math/9808060
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A notion of curvature is introduced in multivariable operator theory and an analogue of the Gauss-Bonnet-Chern theorem is established for graded (contractive) Hilbert modules over the complex polynomial algebra in d variables, d=1,2,3,.... The curvature invariant, Euler characteristic, and degree are computed for some explicit examples based on varieties in (multidimensional) complex projective space, and applications are given to the structure of graded ideals in C[z_1,...,z_d] and to the existence of "inner sequences" for closed submodules of the free Hilbert module H^2(C^d).
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arxiv:math/9808100
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The notion of local equivalence relation on a topological space is generalised to that of local subgroupoid. The main result is the construction of the holonomy and monodromy groupoids of certain Lie local subgroupoids, and the formulation of a monodromy principle on the extendibility of local Lie morphisms.
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arxiv:math/9808112
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We show that group algebras kG of polycyclic-by-finite groups G, where k is a field, are catenary: Given prime ideals P and P' of kG, with P contained in P', all saturated chains of primes between P and P' have the same length.
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arxiv:math/9808123
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We construct splittings of some completions of the Z/(p)-Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n-1)'s as a spectrum, where t is shorthand for the fixed points of the Z/(p)-Tate cohomology spectrum (ie Mahowald's inverse limit of P_{-k} smash SE(n)). We also give a multiplicative splitting of tE(n) after a suitable base extension.
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arxiv:math/9808141
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We characterize the subscheme of the moduli space of torsion-free sheaves on an elliptic surface which are stable of relative degree zeero (with respect to a polarization of type aH+bf, H being the section and f the elliptic fibre) which is isomorphic, via the relative Fourier-Mukai transform, with the relative compactified Simpson Jacobian of the family of those curves D in the surface which are flat over the base of the elliptic fibration. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of sheaves whose restriction to each fibre is torsion-free and semistable of rank n and degree zero for higher dimensional elliptic fibrations. The relative Fourier-Mukai transform induces an isomorphic between this relative moduli space and the relative n-th symmetric product of the fibration.
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arxiv:math/9809019
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This paper investigates the Castelnuovo-Mumford regularity of the generic hyperplane section of projective curves in positive characteristic case, and yields an application to a sharp bound on the regularity for nondegenerate projective varieties.
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arxiv:math/9809042
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Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield $2^{43}-43$ independent characteristic numbers mod 2, which generalize the Akbulut-King numbers in dimension three.
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arxiv:math/9809060
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Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.
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arxiv:math/9809071
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An algebra $A$ with multiplication $A\times A \to A, (a,b)\mapsto a\circ b$, is called right-symmetric, if $a\circ(b\circ c)-(a\circ b)\circ a\circ (c\circ b)-(a\circ c)\circ b,$ for any $a,b,c\in A$. The multiplication of right-symmetric Witt algebras $W_n=\{u\der_i: u\in U, U={\cal K}[x_1^{\pm 1},...,x_n^{\pm}$ or $={\cal K}[x_1,...,x_n], i=1,...,n\}, p=0,$ or $W_n({\bf m)}=\{u\der_i: u\in U, U=O_n({\bf m})\}$, are given by $u\der_i\circ v\der_j=v\der_j(u)\der_i.$ An analogue of the Amitsur-Levitzki theorem for right-symmetric Witt algebras is established. Right-symmetric Witt algebras of $ satisfy the standard right-symmetric identity of degree $2n+1:$ $\sum_{\sigma\in Sym_{2n}}sign(\sigma)a_{\sigma(1)}\circ(a_{\sigma(2)}\circ >...(a_{\sigma(2n)}\circ a_{2n+1})...)=0.$ The minimal deg$ left polynomial identities of $W_n^{rsym}, W_n^{+rsym}, p=0,$ i$ The minimal degree of multilinear left polynomial identity of $$ is also $2n+1.$ All left polynomial (also multilinear, if $p>0$) identities of right-symmetric Witt algebras of minimal $ combinations of left polynomials obtained from standard ones by permutations of arguments.
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arxiv:math/9809082
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We review some recent development on the extension problem of pluricanonical forms from a divisor to the ambient space in [Si], [K5] and [N3] with simplified proofs. A section for a correction is added.
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arxiv:math/9809091
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Let K be a number field with ring of integers O, and let G be a finite-index subgroup of SL(n,O). Using a classical construction from the geometry of numbers and the theory of modular symbols, we exhibit a finite spanning set for the highest nonvanishing rational cohomology group of G.
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arxiv:math/9809166
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Let M be a smooth 4-manifold which admits a genus g Lefschetz fibration over D^2 or S^2. We develop a technique to compute the signature of M using the global monodromy of this fibration.
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arxiv:math/9809178
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We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-\kappa^2) =-\kappa - \int_0^a A(\alpha) e^{-2\alpha\kappa} d\alpha +O(e^{-(2a -\epsilon)\kappa}) for all \epsilon > 0. We discuss five issues here. First, we extend the theory to general q in L^1 ((0,a)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure \rho: A(\alpha) = -2\int_{-\infty}^\infty \lambda^{-\frac12} \sin (2\alpha \sqrt{\lambda})\, d\rho(\lambda) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b<\infty. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.
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arxiv:math/9809182
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The work of Chatzidakis and Hrushovski on the model theory of difference fields in characteristic zero showed that groups defined by difference equations have a very restricted structure. Recent work of Chatzidakis, Hrushovski and Peterzil [CHP] extends the class of difference fields for which this sort of result is known to positive characteristic. In this note, we analyze the subgroups of the torsion points of simple commutative algebraic groups over finite fields that can be constructed by such difference equations. Our results are reasonably complete modulo some well-known conjectures in Number Theory. In one case, we need the $p$-adic version of the four exponentials conjecture and in another we need a version of Artin's conjecture on primitive roots. We recover part of a theorem of Boxall on the intersection of varieties with the group of $m$-power torsion points, but in general this theorem does not follow from the model-theoretic analysis, because there may be no field automorphism $\sigma$ so that the $m$-power torsion group is contained in a modular group definable with $\sigma$. On the other hand, some of the groups defined by modular difference equations are much larger than the group of $m$-power torsion points, so our results are stronger in another direction. In some ways, the model theoretic approach extends the approach of Bogomolov and the original one of Lang.
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arxiv:math/9809188
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We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symplectic quotient.
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arxiv:math/9810046
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Let M(j) denote the moduli space of bundles on the blown-up plane which restrict to the exceptional divisor as O(j)+O(-j). We show that there is a topological embedding of M(j) into M(j+1).
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arxiv:math/9810106
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For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on moduli spaces of punctured Riemann surfaces, and then give a more geometric interpretation of our determinant metrics in terms of Selberg zeta functions. We end this paper by proposing an arithmetic factorization for Weil-Petersson metrics, cuspidal metrics and Selberg zeta functions.
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arxiv:math/9810116
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Extensions of the generalized Weierstrass representation to generic surfaces in 4D Euclidean and pseudo-Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey -Stewartson hierarchy. Geometrically these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the total squared mean curvature) is the simplest of them. Various particular classes of surfaces and their integrable deformations are considered.
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arxiv:math/9810138
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The derivation is studied in relation with the space of Schwartz of functions. This paper has been withdrawn by the author.
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arxiv:math/9810141
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We examine the dependence of the deformation obtained by bending quasi-Fuchsian structures on the bending lamination. We show that when we consider bending quasi-Fuchsian structures on a closed surface, the conditions obtained by Epstein and Marden to relate weak convergence of arbitrary laminations to the convergence of bending cocycles are not necessary. Bending may not be continuous on the set of all measured laminations. However we show that if we restrict our attention to laminations with non negative real and imaginary parts then the deformation depends continuously on the lamination.
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arxiv:math/9810195
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Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.
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arxiv:math/9810207
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Yang-Baxter system related to quantum doubles is introduced and large class of both continuous and discrete symmetries of the solution manifold are determined. Strategy for solution of the system based on the symmetries is suggested and accomplished in the dimension two.The complete list of invertible solutions of the system is presented.
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arxiv:math/9811016
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We show how to construct, starting from a quasi-Hopf (super)algebra, central elements or Casimir invariants. We show that these central elements are invariant under quasi-Hopf twistings. As a consequence, the elliptic quantum (super)groups, which arise from twisting the normal quantum (super)groups, have the same Casimir invariants as the corresponding quantum (super)groups.
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arxiv:math/9811052
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We explain the basics of conformal theory using the language of chiral algebras of Beilinson and Drinfeld.
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arxiv:math/9811061
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We survey recent results about asymptotic functions of groups, obtained by the authors in collaboration with J.-C.Birget, V. Guba and E. Rips. We also discuss methods used in the proofs of these results.
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arxiv:math/9811107
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We prove that the Veronese embedding of P^n of degree d with n\ge 2, d\ge 3 does not satisfy property N_p (according to Green and Lazarsfeld) if p\ge 3d-2. We conjecture that the converse holds. This is true for n=2 and for n=d=3.
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arxiv:math/9811131
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We characterize Riesz frames and frames with the subframe property and use this to answer most of the questions from the literature concerning these properties and their relationships to the projection methods etc.
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arxiv:math/9811149
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We prove that a standard realization of the direct image complex via the so-called Douady-Barlet morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi-isomorphism of complexes. This is a more precise form of a special case of the decomposition theorems of Beilinson-Bernstein-Deligne-Gabber and M. Saito. The proof hinges on the special case of the bi-disk in the complex affine plane where we make explicit use of a construction of Nakajima's and of the corresponding representation-theoretic interpretation foreseen by Vafa-Witten. Some consequences of the decomposition theorem: G\"ottsche Formula holds for complex surfaces; interpretation of the rational cohomologies of Douady spaces as a kind of Fock space; new proofs of results of Brian\c{c}on and Ellingsrud-Stromme on punctual Hilbert schemes; computation of the mixed Hodge structure of the Douady spaces in the K\"ahler case. We also derive a natural connection with Equivariant K-Theory for which, in the case of algebraic surfaces, Bezrukavnikov-Ginzburg have proposed a different approach.
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arxiv:math/9811159
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In this paper we prove that any strongly embedded subgroup of a K*-group G of finite Morley rank and odd type that does not interpret any bad field is solvable if its Pruefer 2-rank is at least 2. If the normal 2-rank of G is at least 3 this has two important consequences: If G contains a non-solvable centraliser of an involution, then G does not contain any proper 2-generated core and centralisers of involutions have trivial cores.
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arxiv:math/9811163
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In recently published work Maskit constructs a fundamental domain D_g for the Teichmueller modular group of a closed surface S of genus g>1. Maskit's technique is to demand that a certain set of 2g non-dividing geodesics C_{2g} on S satisfies certain shortness criteria. This gives an a priori infinite set of length inequalities that the geodesics in C_{2g} must satisfy. Maskit shows that this set of inequalities is finite and that for genus g=2 there are at most 45. In this paper we improve this number to 27. Each of these inequalities: compares distances between Weierstrass points in the fundamental domain S-C_4 for S; and is realised (as an equality) on one or other of two special surfaces.
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arxiv:math/9811180
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The subject of this paper is a Jacobian, introduced by F. Lazzeri, (unpublished), associated to every compact oriented riemannian manifold of dimension twice an odd number. We start the investigation of Torelli type problems and Schottky type problem for Lazzeri's Jacobian; in particular we examine the case of tori with flat metrics. Besides we study Lazzeri's Jacobian for Kahler manifolds and its relationship with other Jacobians. Finally we examine Lazzeri's Jacobian of a bundle.
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arxiv:math/9812110
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We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kaehler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. We derive a simple and explicit "differential operator formula" for the necessary quantum products, which applies both to the finite-dimensional and to the infinite-dimensional situations.
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arxiv:math/9812127
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We survey three methods for proving that the characteristic polynomial of a finite lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on Zaslavsky's theory of signed graphs. The second approach is algebraic and employs results of Saito and Terao about free hyperplane arrangements. Finally, we consider a purely combinatorial theorem of Stanley about semimodular supersolvable lattices and its generalizations.
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arxiv:math/9812136
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One of the basic objects in the Morse theory of circle-valued maps is Novikov complex - an analog of the Morse complex of Morse functions. Novikov complex is defined over the ring of Laurent power series with finite negative part. The main aim of this paper is to present a detailed and self-contained exposition of the author's theorem saying that C^0-generically the Novikov complex is defined over the ring of rational functions. The paper contains also a systematic treatment of the topics of the classical Morse theory related to Morse complexes (Chapter 2). We work with a new class of gradient-type vector fields, which includes riemannian gradients. In Ch.3 we suggest a purely Morse-theoretic (not using triangulations) construction of small handle decomposition of manifolds. In the Ch.4 we deal with the gradients of Morse functions on cobordisms. Due to the presence of critical points the descent along the trajectories of such gradient does no define in general a continuous map from the upper component of the boundary to the lower one. We show that for C^0-generic gradients there is an algebraic model of "gradient descent map". This is one of the main tools in the proof of the main theorem (Chapter 5). We give also the generalizations of the result for the versions of Novikov complex defined over completions of group rings (non commutative in general).
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arxiv:math/9812157
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Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group which is determined by the Witt ring WF.
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arxiv:math/9812169
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We present a conjecture about partitions, with a very elementary formulation.
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arxiv:math/9901040
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Rewriting for semigroups is a special case of Groebner basis theory for noncommutative polynomial algebras. The fact is a kind of folklore but is not fully recognised. The aim of this paper is to elucidate this relationship, showing that the noncommutative Buchberger algorithm corresponds step-by-step to the Knuth-Bendix completion procedure.
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arxiv:math/9901044
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This is the fourth installment of a series. The main point of the entire series is the following: given a triangulated category T, it is possible to attach to it a K-theory space.
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arxiv:math/9901091
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In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing.
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arxiv:math/9901141
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