The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature
Frank
Morgan;
Michael
Hutchings;
Hugh
Howards
4889-4909
Abstract: We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.
Traces on algebras of parameter dependent pseudodifferential operators and the eta--invariant
Matthias
Lesch;
Markus
J.
Pflaum
4911-4936
Abstract: We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\mathbb{R}$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $\Gamma\subset\mathbb{R} ^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This construction is in the spirit of a trace due to Kontsevich and Vishik in the nonparametric case. Our trace allows us to construct various trace functionals in a systematic way. Furthermore, we study the higher-dimensional eta-invariants on algebras with parameter space $\mathbb{R} ^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\mathbb{R} ^{2k-1}$. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.
Stability theory, permutations of indiscernibles, and embedded finite models
John
Baldwin;
Michael
Benedikt
4937-4969
Abstract: We show that the expressive power of first-order logic over finite models embedded in a model $M$ is determined by stability-theoretic properties of $M$. In particular, we show that if $M$ is stable, then every class of finite structures that can be defined by embedding the structures in $M$, can be defined in pure first-order logic. We also show that if $M$ does not have the independence property, then any class of finite structures that can be defined by embedding the structures in $M$, can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let $I$ be a set of indiscernibles in a model $M$and suppose $(M,I)$ is elementarily equivalent to $(M_1,I_1)$ where $M_1$ is $\vert I_1\vert^+$-saturated. If $M$ is stable and $(M,I)$ is saturated, then every permutation of $I$extends to an automorphism of $M$ and the theory of $(M,I)$ is stable. Let $I$ be a sequence of $<$-indiscernibles in a model $M$, which does not have the independence property, and suppose $(M,I)$ is elementarily equivalent to $(M_1,I_1)$ where $(I_1,<)$ is a complete dense linear order and $M_1$ is $\vert I_1\vert^+$-saturated. Then $(M,I)$-types over $I$are order-definable and if $(M,I)$ is $\aleph_1$-saturated, every order preserving permutation of $I$ can be extended to a back-and-forth system.
Strongly almost disjoint sets and weakly uniform bases
Z.
T.
Balogh;
S.
W.
Davis;
W.
Just;
S.
Shelah;
P.
J.
Szeptycki
4971-4987
Abstract: A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of $\Box_\lambda$ for singular $\lambda$ is proved. CECA is used to show that certain ``almost point-$<\tau$'' families can be refined to point-$< \tau$ families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable $T_1$-space with a weakly uniform base has a point-countable base.''
Effectively dense Boolean algebras and their applications
André
Nies
4989-5012
Abstract: A computably enumerable Boolean algebra ${\mathcal{B}}$ is effectively dense if for each $x \in{\mathcal{B}}$ we can effectively determine an $F(x)\le x$ such that $x \neq 0$ implies $0 < F(x) < x$. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a Boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of ${\mathcal{E}}$ (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals $[{\mathbf{0}},{\mathbf{a}}]$ of subrecursive degree structures, where ${\mathbf{a}}$is the degree of a set of relatively small complexity, for instance a set in exponential time.
A compactification of a family of determinantal Godeaux surfaces
Yongnam
Lee
5013-5023
Abstract: In this paper, we present a geometric description of the compactification of the family of determinantal Godeaux surfaces, via the study of the bicanonical pencil and using classical Prym theory. In particular, we reduce the problem of compactifying the space of bicanonical pencils of determinantal Godeaux surfaces to the compactification of the family of twisted cubic curves in $\mathbb{P}^{3}$ with certain given tangent conditions.
Unipotent groups associated to reduced curves
David
Penniston
5025-5043
Abstract: Let $X$ be a curve defined over an algebraically closed field $k$ with $\operatorname{char}(k)=p>0$. Assume that $X/k$ is reduced. In this paper we study the unipotent part $U$ of the Jacobian $J_{X/k}$. In particular, we prove that if $p$ is large in terms of the dimension of $U$, then $U$ is isomorphic to a product of additive groups $\mathbb{G} _a$.
The density of rational lines on cubic hypersurfaces
Scott
T.
Parsell
5045-5062
Abstract: We provide a lower bound for the density of rational lines on the hypersurface defined by an additive cubic equation in at least 57 variables. In the process, we obtain a result on the paucity of non-trivial solutions to an associated system of Diophantine equations.
Euclidean weights of codes from elliptic curves over rings
José
Felipe
Voloch;
Judy
L.
Walker
5063-5076
Abstract: We construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools, notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest.
Infinitely Renormalizable Quadratic Polynomials
Yunping
Jiang
5077-5091
Abstract: We prove that the Julia set of a quadratic polynomial which admits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles.
Homology manifold bordism
Heather
Johnston;
Andrew
Ranicki
5093-5137
Abstract: The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact $ANR$ homology manifolds of dimension $\geq 6$ is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston. First, we establish homology manifold transversality for submanifolds of dimension $\geq 7$: if $f:M \to P$ is a map from an $m$-dimensional homology manifold $M$ to a space $P$, and $Q \subset P$ is a subspace with a topological $q$-block bundle neighborhood, and $m-q \geq 7$, then $f$ is homology manifold $s$-cobordant to a map which is transverse to $Q$, with $f^{-1}(Q) \subset M$ an $(m-q)$-dimensional homology submanifold. Second, we obtain a codimension $q$ splitting obstruction $s_Q(f) \in LS_{m-q}(\Phi)$ in the Wall $LS$-group for a simple homotopy equivalence $f:M \to P$ from an $m$-dimensional homology manifold $M$ to an $m$-dimensional Poincaré space $P$ with a codimension $q$ Poincaré subspace $Q \subset P$ with a topological normal bundle, such that $s_Q(f)=0$ if (and for $m-q \geq 7$ only if) $f$ splits at $Q$ up to homology manifold $s$-cobordism. Third, we obtain the multiplicative structure of the homology manifold bordism groups $\Omega^H_*\cong\Omega^{TOP}_*[L_0(\mathbb Z)]$.
Spectra of $\text{BP}$-linear relations, $v_n$-series, and $\text{BP}$ cohomology of Eilenberg-Mac Lane spaces
Hirotaka
Tamanoi
5139-5178
Abstract: On Brown-Peterson cohomology groups of a space, we introduce a natural inherent topology, BP topology, which is always complete Hausdorff for any space. We then construct a spectra map which calculates infinite BP-linear sums convergent with respect to the BP topology, and a spectrum which describes infinite sum BP-linear relations in BP cohomology. The mod $p$ cohomology of this spectrum is a cyclic module over the Steenrod algebra with relations generated by products of exactly two Milnor primitives. We show a close relationship between BP-linear relations in BP cohomology and the action of the Milnor primitives on mod $p$ cohomology. We prove main relations in the BP cohomology of Eilenberg-Mac Lane spaces. These are infinite sum BP-linear relations convergent with respect to the BP topology. Using BP fundamental classes, we define $v_{n}$-series which are $v_{n}$-analogues of the $p$-series. Finally, we show that the above main relations come from the $v_{n}$-series.
Center manifolds for smooth invariant manifolds
Shui-Nee
Chow;
Weishi
Liu;
Yingfei
Yi
5179-5211
Abstract: We study dynamics of flows generated by smooth vector fields in ${\mathbb{R} }^n$ in the vicinity of an invariant and closed smooth manifold $Y$. By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of $Y$) based on the information of the linearization along $Y$, which contains every locally bounded solution and is persistent under small perturbations.
$C^1$ Connecting Lemmas
Lan
Wen;
Zhihong
Xia
5213-5230
Abstract: Like the closing lemma, the connecting lemma is of fundamental importance in dynamical systems. Hayashi recently proved the $C^1$ connecting lemma for stable and unstable manifolds of a hyperbolic invariant set. In this paper, we prove several very general $C^1$ connecting lemmas. We simplify Hayashi's proof and extend the results to more general cases.
Local differentiability of distance functions
R.
A.
Poliquin;
R.
T.
Rockafellar;
L.
Thibault
5231-5249
Abstract: Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets $C$ for which the distance function $d_{C}$ is continuously differentiable everywhere on an open ``tube'' of uniform thickness around $C$. Here a corresponding local theory is developed for the property of $d_{C}$ being continuously differentiable outside of $C$ on some neighborhood of a point $x\in C$. This is shown to be equivalent to the prox-regularity of $C$ at $x$, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of $d_{C}^{2}$ being locally of class $C^{1+}$ or such that $d_{C}^{2}+\sigma \vert\cdot \vert^{2}$ is convex around $x$ for some $\sigma >0$. Prox-regularity of $C$ at $x$ corresponds further to the normal cone mapping $N_{C}$ having a hypomonotone truncation around $x$, and leads to a formula for $P_{C}$ by way of $N_{C}$. The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.
Optimal factorization of Muckenhoupt weights
Michael
Brian
Korey
5251-5262
Abstract: Peter Jones' theorem on the factorization of $A_p$ weights is sharpened for weights with bounds near $1$, allowing the factorization to be performed continuously near the limiting, unweighted case. When $1<p<\infty$ and $w$ is an $A_p$ weight with bound $A_p(w)=1+\varepsilon$, it is shown that there exist $A_1$ weights $u,v$ such that both the formula $w=uv^{1-p}$ and the estimates $A_1(u), A_1(v)=1+\mathcal O(\sqrt\varepsilon)$ hold. The square root in these estimates is also proven to be the correct asymptotic power as $\varepsilon\to 0$.
A classification of one dimensional almost periodic tilings arising from the projection method
James
A.
Mingo
5263-5277
Abstract: For each irrational number $\alpha$, with continued fraction expansion $[0; a_1,\allowbreak a_2,a_3, \dots ]$, we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope $\alpha$. The invariant is a sequence of integers in the space $X_\alpha = \{(x_i)_{i=1}^\infty \mid x_i \in \{0,1,2, \dots ,a_i\}$ and $x_{i+1} = 0$ whenever $x_i = a_i\}$ modulo the equivalence relation generated by tail equivalence and $(a_1, 0, a_3, 0, \dots ) \sim (0, a_2, 0, a_4, \dots ) \sim (a_1 -1, a_2 - 1, a_3 - 1, \dots )$. Each tile in a tiling $\textsf{T}$, of slope $\alpha$, is coded by an integer $0 \leq x \leq [\alpha]$. Using a composition operation, we produce a sequence of tilings $\textsf{T}_1 = \textsf{T}{}, \textsf{T}_2, \textsf{T}_3, \dots$. Each tile in $\textsf{T}_i$ gets absorbed into a tile in $\textsf{T}_{i+1}$. A choice of a starting tile in $\textsf{T}_1$ will thus produce a sequence in $X_\alpha$. This is the invariant.
Path stability and nonlinear weak ergodic theorems
Yong-Zhuo
Chen
5279-5292
Abstract: Let $\{f_{n} \}$ be a sequence of nonlinear operators. We discuss the asymptotic properties of their inhomogeneous iterates $f_{n} \circ f_{n-1} \circ \cdots \circ f_{1}\,$ in metric spaces, then apply the results to the ordered Banach spaces through projective metrics. Theorems on path stability and nonlinear weak ergodicity are obtained in this paper.
Hypercyclic operators that commute with the Bergman backward shift
Paul
S.
Bourdon;
Joel
H.
Shapiro
5293-5316
Abstract: The backward shift $B$ on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: ``Which operators that commute with $B$ inherit its hypercyclicity?'' We show that the problem reduces to the study of operators of the form $\varphi(B)$ where $\varphi$ is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely $\varphi(z)$ is allowed to approach the unit circle as $\vert z\vert\to 1-$.
Semi-classical limit for random walks
Ursula
Porod;
Steve
Zelditch
5317-5355
Abstract: Let $(G, \mu)$ be a discrete symmetric random walk on a compact Lie group $G$ with step distribution $\mu$ and let $T_{\mu}$ be the associated transition operator on $L^2(G)$. The irreducibles $V_{\rho}$ of the left regular representation of $G$ on $L^2(G)$ are finite dimensional invariant subspaces for $T_{\mu}$ and the spectrum of $T_{\mu}$ is the union of the sub-spectra $\sigma(T_{\mu}\upharpoonleft_{V_{\rho}})$ on the irreducibles, which consist of real eigenvalues $\{ \lambda_{\rho 1},...,\lambda_{\rho \dim V_{\rho}}\}$. Our main result is an asymptotic expansion for the spectral measures \begin{displaymath}m_{\rho}^{\mu}(\lambda) := \frac{1}{\dim V_{\rho}} \sum_{j=1}^{\dim V_{\rho}} \delta(\lambda - \lambda_{\rho j})\end{displaymath} along rays of representations in a positive Weyl chamber $\mathbf{t}^*_+$, i.e. for sequences of representations $k \rho$, $k\in \mathbb{N}$ with $k\rightarrow \infty$. As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on $U(n)$ (for which $T_{\mu}$ is essentially a direct sum of Harper operators).
Cotorsion theories and splitters
Rüdiger
Göbel;
Saharon
Shelah
5357-5379
Abstract: Let $R$ be a subring of the rationals. We want to investigate self splitting $R$-modules $G$ (that is $\operatorname{Ext}_R(G,G) = 0)$. Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism $R$-algebras with a free $R$-module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.
The Jantzen sum formula for cyclotomic $q$--Schur algebras
Gordon
James;
Andrew
Mathas
5381-5404
Abstract: The cyclotomic $q$-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen's sum formula for the cyclotomic $q$-Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.