$\boldsymbol{\pi_*}$-kernels of Lie groups
Ken-ichi
Maruyama
2335-2351
Abstract: We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of $SU(3)$ and $Sp(2)$ completely. We introduce two natural invariants $lz_p(X)$ and $sz_p(X)$ defined by the filtration, where $p$ is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are $p$-regular or quasi $p$-regular. We apply our results to the groups of self homotopy equivalences.
Associahedra, cellular $W$-construction and products of $A_\infty$-algebras
Martin
Markl;
Steve
Shnider
2353-2372
Abstract: The aim of this paper is to construct a functorial tensor product of $A_\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These constructions in fact already appeared (Saneblidze and Umble, 2000 and 2002); we will try to give a more conceptual presentation. We also prove that there does not exist a coassociative diagonal.
Koszul duality and equivalences of categories
Gunnar
Fløystad
2373-2398
Abstract: Let $A$ and $A^{!}$ be dual Koszul algebras. By Positselski a filtered algebra $U$ with gr$\,U = A$ is Koszul dual to a differential graded algebra $(A^{!},d)$. We relate the module categories of this dual pair by a $\otimes-$Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.
On the contact geometry of nodal sets
R.
Komendarczyk
2399-2413
Abstract: In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a two-dimensional consequence of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3-manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne's conjecture for the free membrane problem. We construct counterexamples to Payne's conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret the conjecture in terms of the tight versus overtwisted dichotomy for contact structures.
Existence and regularity of isometries
Michael
Taylor
2415-2423
Abstract: We use local harmonic coordinates to establish sharp results on the regularity of isometric maps between Riemannian manifolds whose metric tensors have limited regularity (e.g., are Hölder continuous). We also discuss the issue of local flatness and of local isometric embedding with given first and second fundamental form, in the context of limited smoothness.
On the shape of the moduli of spherical minimal immersions
Gabor
Toth
2425-2446
Abstract: The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold $M$ is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of $M$. In this paper we define and study a sequence of metric invariants $\sigma_m$, $m\geq 1$, associated to a compact convex body $\mathcal{L}$ with base point $\mathcal{O}$ in the interior of $\mathcal{L}$. The invariant $\sigma_m$ measures how lopsided $\mathcal{L}$ is in dimension $m$ with respect to $\mathcal{O}$. The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate $\sigma_m$ for convex polytopes.
Global well-posedness in the energy space for a modified KP II equation via the Miura transform
Carlos
E.
Kenig;
Yvan
Martel
2447-2488
Abstract: We prove global well-posedness of the initial value problem for a modified Kadomtsev-Petviashvili II (mKP II) equation in the energy space. The proof proceeds in three main steps and involves several different techniques. In the first step, we make use of several linear estimates to solve a fourth-order parabolic regularization of the mKP II equation by a fixed point argument, for regular initial data (one estimate is similar to the sharp Kato smoothing effect proved for the KdV equation by Kenig, Ponce, and Vega, 1991). Then, compactness arguments (the energy method performed through the Miura transform) give the existence of a local solution of the mKP II equation for regular data. Finally, we approximate any data in the energy space by a sequence of smooth initial data. Using Bourgain's result concerning the global well-posedness of the KP II equation in $L^2$ and the Miura transformation, we obtain convergence of the sequence of smooth solutions to a solution of mKP II in the energy space.
A $(p,q)$ version of Bourgain's theorem
John
J.
Benedetto;
Alexander
M.
Powell
2489-2505
Abstract: Let $1<p,q<\infty$ satisfy $\frac{1}{p} + \frac{1}{q} =1.$ We construct an orthonormal basis $\{ b_n \}$ for $L^2 (\mathbb{R})$ such that $\Delta_p ( b_n )$ and $\Delta_q (\widehat{b_n})$ are both uniformly bounded in $n$. Here $\Delta_{\lambda} (f) \equiv {\rm inf}_{a \in \mathbb{R}} \left( \int \vert x - a\vert^{\lambda} \vert f(x)\vert^2 dx \right)^{\frac{1}{2}}$. This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.
Resolutions for metrizable compacta in extension theory
Leonard
R.
Rubin;
Philip
J.
Schapiro
2507-2536
Abstract: We prove a $K$-resolution theorem for simply connected CW- complexes $K$ in extension theory in the class of metrizable compacta $X$. This means that if $K$ is a connected CW-complex, $G$ is an abelian group, $n\in \mathbb N _{\geq 2}$, $G=\pi _{n}(K)$, $\pi _{k}(K)=0$ for $0\leq k<n$, and $\operatorname{extdim} X\leq K$ (in the sense of extension theory, that is, $K$ is an absolute extensor for $X$), then there exists a metrizable compactum $Z$ and a surjective map $\pi :Z\rightarrow X$ such that: (a) $\pi$ is $G$-acyclic, (b) $\dim Z\leq n+1$, and (c) $\operatorname{extdim} Z\leq K$. This implies the $G$-resolution theorem for arbitrary abelian groups $G$ for cohomological dimension $\dim _{G} X\leq n$ when $n\in \mathbb N_{\geq 2}$. Thus, in case $K$ is an Eilenberg-MacLane complex of type $K(G,n)$, then (c) becomes $\dim _{G} Z\leq n$. If in addition $\pi _{n+1}(K)=0$, then (a) can be replaced by the stronger statement, (aa) $\pi$ is $K$-acyclic. To say that a map $\pi$ is $K$-acyclic means that for each $x\in X$, every map of the fiber $\pi ^{-1}(x)$ to $K$ is nullhomotopic.
Construction of stable equivalences of Morita type for finite-dimensional algebras I
Yuming
Liu;
Changchang
Xi
2537-2560
Abstract: In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a non-trivial example is not known. In this paper, we provide two methods to produce new stable equivalences of Morita type from given ones. The main results are Corollary 1.2 and Theorem 1.3. Here the algebras considered are not necessarily self-injective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension $4$, such that one of them is quasi-hereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasi-heredity of algebras. As another by-product, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras $A$ and $B$. This leads not only to a general formulation of a result by Linckelmann (1996), but also to a nice correspondence of some torsion pairs in $A$-mod with those in $B$-mod if both $A$ and $B$are symmetric algebras. Moreover, under the assumption of symmetric algebras we can get a new stable equivalence of Morita type. Finally, we point out that stable equivalences of Morita type are preserved under separable extensions of ground fields.
A generalization of Marshall's equivalence relation
Ido
Efrat
2561-2577
Abstract: For $p$ prime and for a field $F$ containing a root of unity of order $p$, we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of $F^{\times }$ of index $p$. The equivalence classes then correspond to free pro-$p$ factors of the maximal pro-$p$ Galois group of $F$. We generalize to this setting results of Jacob on the maximal pro-$2$ Galois group of a Pythagorean field.
Bicyclic algebras of prime exponent over function fields
Boris
È.
Kunyavskii;
Louis
H.
Rowen;
Sergey
V.
Tikhonov;
Vyacheslav
I.
Yanchevskii
2579-2610
Abstract: We examine some properties of bicyclic algebras, i.e. the tensor product of two cyclic algebras, defined over a purely transcendental function field in one variable. We focus on the following problem: When does the set of local invariants of such an algebra coincide with the set of local invariants of some cyclic algebra? Although we show this is not always the case, we determine when it happens for the case where all degeneration points are defined over the ground field. Our main tool is Faddeev's theory. We also study a geometric counterpart of this problem (pencils of Severi-Brauer varieties with prescribed degeneration data).
The degree of the variety of rational ruled surfaces and Gromov-Witten invariants
Cristina
Martínez
2611-2624
Abstract: We compute the degree of the variety parametrizing rational ruled surfaces of degree $d$ in $\mathbb{P} ^{3}$ by relating the problem to Gromov-Witten invariants and Quantum cohomology.
Inequalities for eigenvalues of a clamped plate problem
Qing-Ming
Cheng;
Hongcang
Yang
2625-2635
Abstract: Let $D$ be a connected bounded domain in an $n$-dimensional Euclidean space $\mathbb{R}^n$. Assume that $\displaystyle 0 < \lambda_1 <\lambda_2 \le \cdots \le \lambda_k \le \cdots$ are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: $\displaystyle \left \{ \aligned&\Delta^2 u =\lambda u, \text{ in$D$,} &u\ve... ...rac {\partial u}{\partial n}\right \vert _{\partial D}=0. \endaligned \right .$ Then, we give an upper bound of the $(k+1)$-th eigenvalue $\lambda_{k+1}$ in terms of the first $k$ eigenvalues, which is independent of the domain $D$, that is, we prove the following: $\displaystyle \lambda_{k+1} \le \frac 1k\sum_{i=1}^k \lambda_i +\left [\frac {8... ...ac 1k\sum_{i=1}^k \biggl[ \lambda_i(\lambda_{k+1} -\lambda_i) \biggl ]^{1/2}.$ Further, a more explicit inequality of eigenvalues is also obtained.
Teichmüller mapping class group of the universal hyperbolic solenoid
Vladimir
Markovic;
Dragomir
Saric
2637-2650
Abstract: We show that the homotopy class of a quasiconformal self-map of the universal hyperbolic solenoid $H_\infty$ is the same as its isotopy class and that the uniform convergence of quasiconformal self-maps of $H_\infty$ to the identity forces them to be homotopic to conformal maps. We identify a dense subset of $\mathcal{T}(H_\infty )$ such that the orbit under the base leaf preserving mapping class group $MCG_{BLP}(H_\infty)$ of any point in this subset has accumulation points in the Teichmüller space $\mathcal{T}(H_\infty )$. Moreover, we show that finite subgroups of $MCG_{BLP}(H_\infty )$ are necessarily cyclic and that each point of $\mathcal{T}(H_\infty)$ has an infinite isotropy subgroup in $MCG_{BLP}(H_\infty )$.
Bounded Hochschild cohomology of Banach algebras with a matrix-like structure
Niels
Grønbæk
2651-2662
Abstract: Let $\mathfrak{B}$ be a unital Banach algebra. A projection in $\mathfrak{B}$ which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal $\mathfrak{A}$ in $\mathfrak{B}$. In this set-up we prove a theorem to the effect that the bounded cohomology $\mathcal{H}^{n}(\mathfrak{A}, \mathfrak{A}^{*})$ vanishes for all $n\geq 1$. The hypotheses of this theorem involve (i) strong H-unitality of $\mathfrak{A}$, (ii) a growth condition on diagonal matrices in $\mathfrak{A}$, and (iii) an extension of $\mathfrak{A}$ in $\mathfrak{B}$ by an amenable Banach algebra. As a corollary we show that if $X$ is an infinite dimensional Banach space with the bounded approximation property, $L_{1}(\mu ,\Omega )$ is an infinite dimensional $L_{1}$-space, and $\mathfrak{A}$ is the Banach algebra of approximable operators on $L_{p}(X,\mu ,\Omega )\;(1\leq p<\infty )$, then $\mathcal{H}^{n}(\mathfrak{A},\mathfrak{A}^{*})=(0)$ for all $n\geq 0$.
Uniform asymptotics for Jacobi polynomials with varying large negative parameters--- a Riemann-Hilbert approach
R.
Wong;
Wenjun
Zhang
2663-2694
Abstract: An asymptotic expansion is derived for the Jacobi polynomials $P_{n}^{(\alpha_{n},\beta_{n})}(z)$ with varying parameters $\alpha_{n}=-nA+a$ and $\beta_n=-nB+b$, where $A>1, B>1$ and $a,b$ are constants. Our expansion is uniformly valid in the upper half-plane $\overline{\mathbb{C}}^+=\{z:{Im}\; z \geq 0\}$. A corresponding expansion is also given for the lower half-plane $\overline{\mathbb{C}}^-=\{z:{Im}\; z \leq 0\}$. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve $L$, which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of $L$, and tend to $L$ as $n \to \infty$.
Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices
Karlheinz
Gröchenig;
Michael
Leinert
2695-2711
Abstract: We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.
Theta lifting of nilpotent orbits for symmetric pairs
Kyo
Nishiyama;
Hiroyuki
Ochiai;
Chen-bo
Zhu
2713-2734
Abstract: We consider a reductive dual pair $(G, G')$ in the stable range with $G'$ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent $K_{\mathbb{C}}$-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair $( G, K)$. As an application, we prove sphericality and normality of the closure of certain nilpotent $K_{\mathbb{C}}$-orbits obtained in this way. We also give integral formulas for their degrees.
Construction and properties of quasi-linear functionals
Ørjan
Johansen;
Alf
B.
Rustad
2735-2758
Abstract: Quasi-linear functionals are shown to be uniformly continuous and decomposable into a difference of two quasi-integrals. A predual space for the quasi-linear functionals inducing the weak*-topology is given. General constructions of quasi-linear functionals by solid set-functions and q-functions are given.
Block combinatorics
V.
Farmaki;
S.
Negrepontis
2759-2779
Abstract: In this paper we extend the block combinatorics partition theorems of Hindman and Milliken-Taylor in the setting of the recursive system of the block Schreier families $(\mathcal{B}^\xi)$, consisting of families defined for every countable ordinal $\xi$. Results contain (a) a block partition Ramsey theorem for every countable ordinal $\xi$ (Hindman's Theorem corresponding to $\xi=1$, and the Milliken-Taylor Theorem to $\xi$ a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.
Twist points of planar domains
Nicola
Arcozzi;
Enrico
Casadio Tarabusi;
Fausto
Di Biase;
Massimo
A.
Picardello
2781-2798
Abstract: We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.