On the complexity of the integral closure
Bernd
Ulrich;
Wolmer
V.
Vasconcelos
425-442
Abstract: The computation of the integral closure of an affine ring has been the focus of several modern algorithms. We will treat here one related problem: the number of generators the integral closure of an affine ring may require. This number, and the degrees of the generators in the graded case, are major measures of cost of the computation. We prove several polynomial type bounds for various kinds of algebras, and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus.
Quantum cohomology of partial flag manifolds
Anders
Skovsted
Buch
443-458
Abstract: We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties $\operatorname{SL}(n)/P$, including the quantum Pieri and quantum Giambelli formulas and the presentation.
A tracial quantum central limit theorem
Greg
Kuperberg
459-471
Abstract: We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.
On the behavior of the algebraic transfer
Robert
R.
Bruner;
Lê
M.
Hà;
Nguyên
H. V.
Hung
473-487
Abstract: Let $Tr_k:\mathbb{F}_2\underset{GL_k}{\otimes} PH_i(B\mathbb{V}_k)\to Ext_{\mathcal{A}}^{k,k+i}(\mathbb{F}_2, \mathbb{F}_2)$ be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\mathbb{V} _k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$. However, Singer showed that $Tr_5$ is not an epimorphism. In this paper, we prove that $Tr_4$does not detect the nonzero element $g_s\in Ext_{\mathcal{A}}^{4,12\cdot 2^s}(\mathbb{F}_2, \mathbb{F}_2)$ for every $s\geq 1$. As a consequence, the localized $(Sq^0)^{-1}Tr_4$ given by inverting the squaring operation $Sq^0$ is not an epimorphism. This gives a negative answer to a prediction by Minami.
Moduli of suspension spectra
John
R.
Klein
489-507
Abstract: For a $1$-connected spectrum $E$, we study the moduli space of suspension spectra which come equipped with a weak equivalence to $E$. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie's calculus of homotopy functors.
Discrete Morse functions from lexicographic orders
Eric
Babson;
Patricia
Hersh
509-534
Abstract: This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with $\hat{0}$ and $\hat{1}$ from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Möbius function may be computed as the alternating sum of Morse numbers. The above construction enables us to prove that the poset $\Pi_n/S_{\lambda }$ of partitions of a set $\{ 1^{\lambda_1 },\dots ,k^{\lambda_k }\}$ with repetition is homotopy equivalent to a wedge of spheres of top dimension when $\lambda$ is a hook-shaped partition; it is likely that the proof may be extended to a larger class of $\lambda$ and perhaps to all $\lambda$, despite a result of Ziegler (1986) which shows that $\Pi_n/S_{\lambda }$ is not always Cohen-Macaulay.
Knot theory for self-indexed graphs
Matías
Graña;
Vladimir
Turaev
535-553
Abstract: We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.
Stein's method and Plancherel measure of the symmetric group
Jason
Fulman
555-570
Abstract: We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.
Analysis on products of fractals
Robert
S.
Strichartz
571-615
Abstract: For a class of post-critically finite (p.c.f.) fractals, which includes the Sierpinski gasket (SG), there is a satisfactory theory of analysis due to Kigami, including energy, harmonic functions and Laplacians. In particular, the Laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods. The probabilistic method is also available for non-p.c.f. fractals such as the Sierpinski carpet. In this paper we show how to extend Kigami's construction to products of p.c.f. fractals. Since the products are not themselves p.c.f., this gives the first glimpse of what the analytic theory could accomplish in the non-p.c.f. setting. There are some important differences that arise in this setting. It is no longer true that points have positive capacity, so functions of finite energy are not necessarily continuous. Also the boundary of the fractal is no longer finite, so boundary conditions need to be dealt with in a more involved manner. All in all, the theory resembles PDE theory while in the p.c.f. case it is much closer to ODE theory.
On structurally stable diffeomorphisms with codimension one expanding attractors
V.
Grines;
E.
Zhuzhoma
617-667
Abstract: We show that if a closed $n$-manifold $M^n$ $(n\ge 3)$ admits a structurally stable diffeomorphism $f$ with an orientable expanding attractor $\Omega$ of codimension one, then $M^n$ is homotopy equivalent to the $n$-torus $T^n$ and is homeomorphic to $T^n$ for $n\ne 4$. Moreover, there are no nontrivial basic sets of $f$ different from $\Omega$. This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on $T^n$, $n\ge 3$.
Dynamical systems disjoint from any minimal system
Wen
Huang;
Xiangdong
Ye
669-694
Abstract: Furstenberg showed that if two topological systems $(X,T)$ and $(Y,S)$ are disjoint, then one of them, say $(Y,S)$, is minimal. When $(Y,S)$ is nontrivial, we prove that $(X,T)$ must have dense recurrent points, and there are countably many maximal transitive subsystems of $(X,T)$ such that their union is dense and each of them is disjoint from $(Y,S)$. Showing that a weakly mixing system with dense periodic points is in ${\mathcal{M}}^{\perp }$, the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in ${\mathcal{M}}^{\perp }$. We show that a weakly mixing system with dense regular minimal points is in ${\mathcal{M}}^{\perp }$, and each system in ${\mathcal{M}}^{\perp }$ has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in ${\mathcal{M}}^{\perp }$ and having no periodic points are constructed. Moreover, we show that there is a distal system in ${\mathcal{M}}^{\perp }$. Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing.
Finite time blow-up for a dyadic model of the Euler equations
Nets
Hawk
Katz;
Natasa
Pavlovic
695-708
Abstract: We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.
The relationship between homological properties and representation theoretic realization of artin algebras
Osamu
Iyama
709-734
Abstract: We will study the relationship of quite different objects in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, $\tau$-categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers.
On the Cohen-Macaulay modules of graded subrings
Douglas
Hanes
735-756
Abstract: We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings.
The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature
Bent
Fuglede
757-792
Abstract: This is a continuation of the Cambridge Tract ``Harmonic maps between Riemannian polyhedra'', by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.
A note on the hyperbolic 4--orbifold of minimal volume
Ruth
Kellerhals
793-793
Abstract: Paper withdrawn by author after original posting date of July 16, 2004 and prior to preparation of the printed issue.
The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics
Thomas
Bieske;
Luca
Capogna
795-823
Abstract: We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the $L^{\infty}$variational problem \begin{displaymath}\begin{cases} \inf \vert\vert\nabla_0 u\vert\vert _{L^{\infty... ...g\in Lip(\partial\Omega) \text{ on }\partial\Omega, \end{cases}\end{displaymath} where $\Omega\subset \mathbf{G}$ is an open subset of a Carnot group, $\nabla_0 u$ denotes the horizontal gradient of $u:\Omega\to \mathbb{R}$, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends previous work of Jensen and of Crandall, Evans and Gariepy. We also derive the Aronsson-Euler equation for more ``regular" absolutely minimizing Lipschitz extensions corresponding to those Carnot-Carathéodory metrics which are associated to ``free" systems of vector fields.
A theta function identity and its implications
Zhi-Guo
Liu
825-835
Abstract: In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for $\prod_{n=1}^\infty(1-q^n)^{10}$is given. The proofs are self-contained and elementary.
Harnack inequalities for non-local operators of variable order
Richard
F.
Bass;
Moritz
Kassmann
837-850
Abstract: We consider harmonic functions with respect to the operator \begin{displaymath}\mathcal{L} u(x)=\int [u(x+h)-u(x)-1_{(\vert h\vert\leq 1)} h\cdot \nabla u(x)] n(x,h) \, dh. \end{displaymath} Under suitable conditions on $n(x,h)$ we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator $\mathcal{L}$ is allowed to be anisotropic and of variable order.