On loop spaces of configuration spaces
F.
R.
Cohen;
S.
Gitler
1705-1748
Abstract: This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as ``twistings" between the factors. The main homological results are given in terms of extensions of the ``infinitesimal braid relations" or ``universal Yang-Baxter Lie relations".
On the structure of $P(n)_\ast P((n))$ for $p=2$
Christian
Nassau
1749-1757
Abstract: We show that $P(n)_\ast(P(n))$ for $p=2$ with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation $\epsilon$ nor the coproduct $\Delta$are multiplicative. As a consequence the algebra structure of $P(n)_\ast(P(n))$ is slightly different from what was supposed to be the case. We give formulas for $\epsilon(xy)$ and $\Delta(xy)$ and show that the inversion of the formal group of $P(n)$is induced by an antimultiplicative involution $\Xi:P(n)\rightarrow P(n)$. Some consequences for multiplicative and antimultiplicative automorphisms of $K(n)$ for $p=2$ are also discussed.
Polar and coisotropic actions on Kähler manifolds
Fabio
Podestà;
Gudlaugur
Thorbergsson
1759-1781
Abstract: The main result of the paper is that a polar action on a compact irreducible homogeneous Kähler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.
Katetov's problem
Paul
Larson;
Stevo
Todorcevic
1783-1791
Abstract: In 1948 Miroslav Katetov showed that if the cube $X^{3}$ of a compact space $X$ satisfies the separation axiom T$_{5}$ then $X$ must be metrizable. He asked whether $X^{3}$ can be replaced by $X^{2}$ in this metrization result. In this note we prove the consistency of this implication.
Asymptotic linear bounds for the Castelnuovo-Mumford regularity
Jürgen
Herzog;
Lê
Tuân
Hoa;
Ngô
Viêt
Trung
1793-1809
Abstract: We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not be Noetherian.
Monoidal extensions of a Cohen-Macaulay unique factorization domain
William
J.
Heinzer;
Aihua
Li;
Louis
J.
Ratliff Jr.;
David
E.
Rush
1811-1835
Abstract: Let $A$ be a Noetherian Cohen-Macaulay domain, $b$, $c_1$, $\dots$, $c_g$an $A$-sequence, $J$ = $(b,c_1,\dots,c_g)A$, and $B$ = $A[J/b]$. Then $B$ is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets $\mbox{{Ass}}_B(B/bB)$ and $\mbox{{Ass}}_A(A/J)$, and each $q$ $\in$ $\mbox{{Ass}}_A(A/J)$ has height $g+1$. If $B$ does not have unique factorization, then some height-one prime ideals $P$ of $B$ are not principal. These primes are identified in terms of $J$ and $P \cap A$, and we consider the question of how far from principal they can be. If $A$ is integrally closed, necessary and sufficient conditions are given for $B$ to be integrally closed, and sufficient conditions are given for $B$ to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if $P$is a height-one prime ideal of $B$, then $P \cap A$ also has height one if and only if $b$ $\notin$ $P$ and thus $P \cap A$ has height one for all but finitely many of the height-one primes $P$ of $B$. If $A$ has unique factorization, a description is given of whether or not such a prime $P$ is a principal prime ideal, or has a principal primary ideal, in terms of properties of $P \cap A$. A similar description is also given for the height-one prime ideals $P$ of $B$with $P \cap A$ of height greater than one, if the prime factors of $b$ satisfy a mild condition. If $A$ is a UFD and $b$ is a power of a prime element, then $B$ is a Krull domain with torsion class group if and only if $J$ is primary and integrally closed, and if this holds, then $B$ has finite cyclic class group. Also, if $J$ is not primary, then for each height-one prime ideal $p$ contained in at least one, but not all, prime divisors of $J$, it holds that the height-one prime $pA[1/b] \cap B$ has no principal primary ideals. This applies in particular to the Rees ring ${\mathbf R}$ $=$ $A[1/t, tJ]$. As an application of these results, it is shown how to construct for any finitely generated abelian group $G$, a monoidal transform $B$ = $A[J/b]$ such that $A$ is a UFD, $B$ is Cohen-Macaulay and integrally closed, and $G$ $\cong$ $\mbox{{Cl}}(B)$, the divisor class group of $B$.
Existence of curves with prescribed topological singularities
Thomas
Keilen;
Ilya
Tyomkin
1837-1860
Abstract: Throughout this paper we study the existence of irreducible curves $C$ on smooth projective surfaces $\Sigma$ with singular points of prescribed topological types $\mathcal S_1,\ldots,\mathcal S_r$. There are necessary conditions for the existence of the type $\sum_{i=1}^r \mu(\mathcal S_i)\leq \alpha C^2+\beta C.K+\gamma$ for some fixed divisor $K$on $\Sigma$ and suitable coefficients $\alpha$, $\beta$ and $\gamma$, and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case $\Sigma=\mathbb P_{\mathbb C}^2$. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up $\Sigma$ of the form $\mathcal O_{\widetilde{\Sigma}}(\pi^*D-\sum_{i=1}^rm_iE_i)$, deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in $\mathbb P_{\mathbb C}^3$, and K3-surfaces.
On crepant resolution of some hypersurface singularities and a criterion for UFD
Hui-Wen
Lin
1861-1868
Abstract: In this article, we find some diagonal hypersurfaces that admit crepant resolutions. We also give a criterion for unique factorization domains.
Verlinde bundles and generalized theta linear series
Mihnea
Popa
1869-1898
Abstract: In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties. We study a naturally defined class of vector bundles on a Jacobian, called Verlinde bundles, in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.
Principal bundles over a projective scheme
Donghoon
Hyeon
1899-1908
Abstract: We prove the existence of a quasi-projective moduli scheme for principal bundles over an arbitrary projective scheme.
Sums of squares in real analytic rings
José
F.
Fernando
1909-1919
Abstract: Let $A$ be an analytic ring. We show: (1) $A$ has finite Pythagoras number if and only if its real dimension is $\leq 2$, and (2) if every positive semidefinite element of $A$ is a sum of squares, then $A$ is real and has real dimension $2$.
Small rational model of subspace complement
Sergey
Yuzvinsky
1921-1945
Abstract: This paper concerns the rational cohomology ring of the complement $M$ of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for $M$. Inside it we find a much smaller subalgebra $D$ quasi-isomorphic to the whole algebra. $D$ is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice $L$whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of $H^*(M)$. The algebra $D$ has a natural integral version that is a good candidate for an integral model of $M$. If the rational local homology of $L$ can be computed explicitly we obtain an explicit presentation of the ring $H^*(M,{\mathbf Q})$. For example, this is done for the cases where $L$ is a geometric lattice and where $M$ is a $k$-equal manifold.
On the maximal Bochner-Riesz conjecture in the plane for $p<2$
Terence
Tao
1947-1959
Abstract: We give a new estimate on the maximal Bochner-Riesz operator in the plane, for $p<2$; as a corollary we obtain an almost everywhere convergence result for certain Bochner-Riesz means. This work was inspired by discussions with Michael Christ and Chris Sogge.
Nonlinear Cauchy-Riemann operators in $\mathbb{R}^{n}$
Tadeusz
Iwaniec
1961-1995
Abstract: This paper has arisen from an effort to provide a comprehensive and unifying development of the $L^{p}$-theory of quasiconformal mappings in $\mathbb{R}^{n}$. The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by $\mathcal{D}^{-}$and $\mathcal{D}^{+}$, which act on weakly differentiable deformations $f:\Omega \to \mathbb{R}^{n}$ of a domain $\Omega \subset \mathbb{R}^{n}$. Solutions to the so-called Cauchy-Riemann equations $\mathcal{D}^{-}f=0$ and $\mathcal{D}^{+}f=0$ are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental $L^{p}$-estimate \begin{displaymath}\Vert\mathcal{D}^{+}f\Vert _{p} \le A_{p}(n)\Vert\mathcal{D}^{-}f\Vert _{p}. \end{displaymath} In quest of the best constant $A_{p}(n)$, we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.
Trudinger inequalities without derivatives
Paul
MacManus;
Carlos
Pérez
1997-2012
Abstract: We prove that the Trudinger inequality holds on connected homogeneous spaces for functions satisfying a very weak type of Poincaré inequality. We also illustrate the connection between this result and the John-Nirenberg theorem for BMO.
$A_p$ weights for nondoubling measures in $R^n$ and applications
Joan
Orobitg;
Carlos
Pérez
2013-2033
Abstract: We study an analogue of the classical theory of $A_p(\mu)$weights in $\mathbb{R} ^n$ without assuming that the underlying measure $\mu$is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions ($BMO$), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if $f$ is a locally integrable function satisfying $\frac{1}{\mu(Q)}\int_{Q} \vert f-f_{Q}\vert d\mu \le a(Q)$ for all cubes $Q$, then it is possible to deduce a higher $L^p$ integrability result for $f$, assuming a certain simple geometric condition on the functional $a$.
A semigroup of operators in convexity theory
Christer
O.
Kiselman
2035-2053
Abstract: We consider three operators which appear naturally in convexity theory and determine completely the structure of the semigroup generated by them. RESUMO. Duongrupo de operatoroj en la teorio pri konvekseco. Ni konsideras tri operatorojn kiuj aperas nature en la teorio pri konvekseco kaj plene determinas la strukturon de la duongrupo generita de ili.
The super order dual of an ordered vector space and the Riesz--Kantorovich formula
Charalambos
D.
Aliprantis;
Rabee
Tourky
2055-2077
Abstract: A classical theorem of F. Riesz and L. V. Kantorovich asserts that if $L$ is a vector lattice and $f$ and $g$are order bounded linear functionals on $L$, then their supremum (least upper bound) $f\lor g$ exists in $L^\sim$ and for each $x\in L_+$ it satisfies the so-called Riesz-Kantorovich formula: \begin{displaymath}\bigl[f\lor g\bigr](x)=\sup\bigl\{f(y)+g(z)\colon y,z\in L_+ \,\hbox{and} \, y+z=x\bigr\}\,. \end{displaymath} Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals $f$ and $g$ on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the above-mentioned problem and to the properties of the Riesz-Kantorovich formula.
Constructing division rings as module-theoretic direct limits
George
M.
Bergman
2079-2114
Abstract: If $R$ is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring $D$ generated by a homomorphic image of $R$ is a rule putting on all free $R$-modules of finite rank matroid structures (closure operators satisfying the exchange axiom) subject to certain functoriality conditions. This note gives a new description of how $D$ may be constructed from this data. (A classical precursor of this is the construction of $\mathbf Q$ as a field with additive group a direct limit of copies of $\mathbf Z$.) The division rings of fractions of right and left Ore rings, the universal division ring of a free ideal ring, and the concept of a specialization of division rings are then interpreted in terms of this construction.
Certain imprimitive reflection groups and their generic versions
Jian-yi
Shi
2115-2129
Abstract: The present paper is concerned with the connection between the imprimitive reflection groups $G(m,m,n)$, $m\in \mathbb{N}$, and the affine Weyl group $\widetilde {A}_{n-1}$. We show that $\widetilde {A}_{n-1}$ is a generic version of the groups $G(m,m,n)$, $m\in \mathbb{N}$. We introduce some new presentations of these groups which are shown to have some group-theoretic advantages. Then we define the Hecke algebras of these groups and of their braid versions, each in two ways according to two presentations. Finally we give a new description for the affine root system $\overline{\Phi }$ of $\widetilde {A}_{n-1}$ such that the action of $\widetilde {A}_{n-1}$ on $\overline{\Phi }$ is compatible with that of $G(m,m,n)$ on its root system in some sense.