Analytic geometry of complex superspaces
H.
Flenner;
D.
Sundararaman
1-40
Abstract: A detailed account of the analytic geometry of complex superspaces is given in this paper. Several representability criteria and representability theorems are proved. In particular, the existence of a versal family of deformations (parametrized by a complex superspace) for any compact complex superspace is proved.
The conormal derivative problem for equations of variational type in nonsmooth domains
Gary M.
Lieberman
41-67
Abstract: It is well known that elliptic boundary value problems in smooth domains have smooth solutions, but if the domain is, say, $ {C^1}$, the solutions need not be Lipschitz. Recently Korevaar has identified a class of Lipschitz domains, in which solutions of the capillary problem are Lipschitz assuming the contact angle relates correctly to the geometry of the domain. Lipschitz bounds for more general boundary value problems in the same class of domains are proved. Applications to variational inequalities are also considered.
Earthquakes on Riemann surfaces and on measured geodesic laminations
Francis
Bonahon
69-95
Abstract: Let $S$ be a closed orientable surface of genus at least $2$. We study properties of its Teichmüller space $ \mathcal{T}(S)$, namely of the space of isotopy classes of conformal structures on $S$. W. P. Thurston introduced a certain compactification of $ \mathcal{T}(S)$ by what he called the space of projective measured geodesic laminations. He also introduced some transformations of Teichmüller space, called earthquakes, which are intimately related to the geometry of $\mathcal{T}(S)$. A general problem is to understand which geometric properties of Teichmüller space subsist at infinity, on Thurston's boundary. In particular, it is natural to ask whether earthquakes continuously extend at certain points of Thurston's boundary, and at precisely which points they do so. This is the principal question addressed in this paper.
A local Weyl's law, the angular distribution and multiplicity of cusp forms on product spaces
Jonathan
Huntley;
David
Tepper
97-110
Abstract: Let $\Gamma /\mathcal{H}$ be a finite volume symmetric space with $ \mathcal{H}$ the product of half planes. Let $ {\Delta _i}$ be the Laplacian on the $i$th half plane, and assume that we have a cusp form $\phi$, so we have ${\Delta _i}\phi = {\lambda _i}\phi$ for $i = 1,2, \ldots,n$. Let $\vec \lambda = ({\lambda _1}, \ldots,{\lambda _n})$ and let $\displaystyle R = \sqrt {r_1^2 + \cdots + r_n^2} $ with $ r_i^2 + \frac{1} {4} = {\lambda _i}$. Letting $\vec r = ({r_1}, \ldots,{r_n})$, we let $ M(\vec r)$ denote the dimension of the space of cusp forms with eigenvalue $\vec \lambda$. More generally, let $M(\vec r,a)$ denote the number of independent eigenfunctions such that the $\vec r$ associated to an eigenfunction is inside the ball of radius $a$, centered at $\vec r$. We will define a function $f(\vec r)$, which is generally equal to a linear sum of products of the ${r_i}$. We prove the following theorems. Theorem 1. $\displaystyle M(\vec r) = O\left(\frac{f(\vec r)} {(\log R)^n} \right).$ Theorem 2. $\displaystyle M (\vec{r}, A) = 2^n f(\vec{r})+O\left(\frac{f(\vec r)}{\log R} \right).$
On the symmetric square: definitions and lemmas
Yuval Z.
Flicker
111-124
Abstract: We define the symmetric square lifting for admissible and automorphic representations, from the group $H = {H_0} = {\text{SL}}(2)$, to the group $ G = {\text{PGL}}(3)$, and derive its basic properties. This lifting is defined by means of Shintani character relations. The definition is suggested by the computation of orbital integrals (stable and unstable) in our On the symmetric square: Orbital integrals, Math. Ann. 279 (1987), 173-193. It is compatible with dual group homomorphisms ${\lambda _0}:\widehat{H} \to \widehat{G}$ and ${\lambda _1}:{\widehat{H}_1} \to \widehat{G}$, where ${H_1} = {\text{PGL}}(2)$. The lifting is proven for induced, trivial and special representations, and both spherical functions and orthogonality relations of characters are studied.
On the symmetric square: applications of a trace formula
Yuval Z.
Flicker
125-152
Abstract: In this paper we prove the existence of the symmetric-square lifting of admissible and of automorphic representations from the group $ {\text{SL}}(2)$ to the group $ {\text{PGL}}(3)$. Complete local results are obtained, relating the character of an $ {\text{SL}}(2)$-packet with the twisted character of self-contragredient $ {\text{PGL}}(3)$-modules. Our global results relate packets of cuspidal representations of $ {\text{SL}}(2)$ with a square-integrable component, and self-contragredient automorphic $ {\text{PGL}}(3)$-modules with a component coming from a square-integrable one. The sharp results, which concern ${\text{SL}}(2)$ rather than ${\text{GL}}(2)$, are afforded by the usage of the trace formula. The surjectivity and injectivity of the correspondence implies that any self-contragredient automorphic $ {\text{PGL}}(3)$-module as above is a lift, and that the space of cuspidal ${\text{SL}}(2)$-modules with a square-integrable component admits multiplicity one theorem and rigidity ("strong multiplicity one") theorem for packets (and not for individual representations). The techniques of this paper, based on the usage of regular functions to simplify the trace formula, are pursued in the sequel [VI] to extend our results to all cuspidal ${\text{SL}}(2)$-modules and self-contragredient $ {\text{PGL}}(3)$-modules
Homotopie de l'espace des \'equivalences d'homotopie
Geneviève
Didierjean
153-163
Abstract: The spectral sequence of the self-fiber-homotopy-equivalences of a fibration provides a method to compute the homotopy groups of the space of self-equivalences of a space.
On Auslander-Reiten components of blocks and self-injective biserial algebras
Karin
Erdmann;
Andrzej
Skowroński
165-189
Abstract: We investigate the existence of Auslander-Reiten components of Euclidean type for special biserial self-injective algebras and for blocks of group algebras. In particular we obtain a complete description of stable Auslander-Reiten quivers for the tame self-injective algebras considered here.
Positive solutions of semilinear equations in cones
Henrik
Egnell
191-201
Abstract: In this paper we consider the problem of finding a positive solution of the equation $\Delta u + \vert x{\vert^\nu }{u^{(n + 2 + 2\nu)/(n - 2)}} = 0$ in a cone $\mathcal{C}$, with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that $ u(x) = o(\vert x{\vert^{2 - n}})$, as $\mathcal{C} \ni x \to \infty$). We will always assume that $\nu > - 2$. We show that the existence of a solution depends on the sign of $\nu$ and also on the shape of the cone $\mathcal{C}$.
Characterization for the solvability of nonlinear partial differential equations
Elemer E.
Rosinger
203-225
Abstract: Within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been obtained. One of them has been the first global version of the Cauchy-Kovalevskaia theorem, which proves the existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear PDEs. These generalized solutions are analytic everywhere, except for closed, nowhere dense subsets which can be chosen to have zero Lebesgue measure. This paper gives a certain extension of that result by establishing an algebraic necessary and sufficient condition for the existence of generalized solutions for arbitrary polynomial nonlinear PDEs with continuous coefficients. This algebraic characterization, given by the so-called neutrix or off diagonal condition, is proved to be equivalent to certain densely vanishing conditions, useful in the study of the solutions of general nonlinear PDEs.
${\rm II}\sb 1$ factors, their bimodules and hypergroups
V. S.
Sunder
227-256
Abstract: In this paper, we introduce a notion that we call a hypergroup; this notion captures the natural algebraic structure possessed by the set of equivalence classes of irreducible bifinite bimodules over a II$ _{1}$ factor. After developing some basic facts concerning bimodules over II$_{1}$ factors, we discuss abstract hypergroups. To make contact with the problem of what numbers can arise as index-values of subfactors of a given II$_{1}$ factor with trivial relative commutant, we define the notion of a dimension function on a hypergroup, and prove that every finite hypergroup admits a unique dimension function, we then give some nontrivial examples of hypergroups, some of which are related to the Jones subfactors of index $4{\cos ^2}\pi /(2n + 1)$. In the last section, we study the hypergroup invariant corresponding to a bifinite module, which is used, among other things, to obtain a transparent proof of a strengthened version of what Ocneanu terms 'the crossed-product remembering the group.'
Th\'eorie de Sullivan pour la cohomologie \`a coefficients locaux
Antonio
Gómez-Tato
235-305
Abstract: The classical moment map of symplectic geometry is used to canonically associate to a unitary representation of a Lie group $G$ a $G$-invariant subset of the dual of the Lie algebra. This correspondence is in some sense dual to geometric quantization. The nature and convexity of this subset is investigated for $G$ compact semisimple.
The moment map of a Lie group representation
N. J.
Wildberger
257-268
Abstract: Given an $m \times m$ Hadamard matrix one can extract $ {m^2}$ symmetric designs on $m - 1$ points each of which extends uniquely to a $ 3$-design. Further, when $ m$ is a square, certain Hadamard matrices yield symmetric designs on $ m$ points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary association of linear codes with designs. This leads naturally to the notion, defined for any prime $p$, of $p$-equivalence for Hadamard matrices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty $24 \times 24$ matrices fall into only six $ 2$-equivalence classes. In the $16 \times 16$ case, $ 2$-equivalence is identical to the standard equivalence, but our results illuminate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian $2$-group, why two of the matrices cannot be obtained from a symmetric design on $16$ points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.
Hadamard matrices and their designs: a coding-theoretic approach
E. F.
Assmus;
J. D.
Key
269-293
Abstract: To every finite dimensional algebraic coefficient system (defined below) $ (\Theta,V)$ over the De Rham algebra $\Omega (M)$ of a manifold $M$, Sullivan builds a local system $ {\rho _\Theta }:{\pi _1}(M) \to V$, in the topological sense, such that the two cohomologies $H_{{\rho _\Theta }}^{\ast}(M;V)$ and $H_\Theta ^{\ast}(\Omega (M);V)$ are isomorphic. In this paper, if ${\mathbf{K}}$ is a simplicial set and $(\Theta,V)$ an algebraic system over the ${C^\infty }$ forms ${A_\infty }({\mathbf{K}})$, we prove a similar result. We use it to extend the Hirsch lemma to the case of fibration whose fiber is an Eilenberg-Mac Lane space with certain non nilpotent action of the fundamental group of the basis. We apply this to a model of the hyperbolic torus; different from the nilpotent one, this new model is a better mirror of the topology.
Equivalence of families of functions on the natural numbers
Claude
Laflamme
307-319
Abstract: We present some consequences of the inequality $\mathfrak{u} < \mathfrak{g}$ among cardinal invariants of the continuum, which has previously been shown to be consistent relative to ZFC. We are interested in its effect on two orderings of families of functions on the natural numbers; in particular we show that, under $\mathfrak{u} < \mathfrak{g}$, there are exactly five equivalence classes for both orderings (excluding the families bounded by a fixed constant function). This implies, under the same hypothesis, the existence of exactly four classes of rarefaction of measure zero sets.
Motion of level sets by mean curvature. II
L. C.
Evans;
J.
Spruck
321-332
Abstract: We give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface. Our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface
Examples of pseudo-Anosov homeomorphisms
Max
Bauer
333-359
Abstract: We generalize a construction in knot theory to construct a large family $\mathcal{G}\mathcal{R} = \cup \,GR(\mathcal{P})$ of mapping classes of a surface of genus $ g$ and one boundary component, where $ \mathcal{P}$ runs over some finite index set. We exhibit explicitly the set $ \mathcal{G}{\mathcal{R}^{\ast}} \subset \mathcal{G}\mathcal{R}$ that consists of pseudo-Anosov maps, find the map that realizes the smallest dilatation in $ \mathcal{G}{\mathcal{R}^{\ast}}$, and for every $ \mathcal{P}$, we give a set of defining relations for $GR(\mathcal{P})$.
An upper bound for the least dilatation
Max
Bauer
361-370
Abstract: We given an upper bound for the least dilatation arising from a pseudo-Anosov map of a closed surface of genus greater or equal to three.
Hyperfinite transversal theory
Boško
Živaljević
371-399
Abstract: A measure theoretic version of a well-known P. Hall's theorem, about the existence of a system of distinct representatives of a finite family of finite sets, has been proved for the case of the Loeb space of an internal, uniformly distributed, hyperfinite counting space. We first prove Hall's theorem for $ \Pi _1^0(\kappa)$ graphs after which we develop the version of discrete Transversal Theory. We then prove a new version of Hall's theorem in the case of $ \Sigma _1^0(\kappa)$ monotone graphs and give an example of a $\Sigma _1^0$ graph which satisfies Hall's condition and which does not possess an internal a.e. matching.
Pettis integrability
Gunnar F.
Stefánsson
401-418
Abstract: A weakly measurable function $ f:\Omega \to X$ is said to be determined by a subspace $D$ of $X$ if for each ${x^{\ast} } \in {X^{\ast} }$, ${x^{\ast}}{\vert _D} = 0$ implies that ${x^{\ast}}\;f= 0$ a.e. For a given Dunford integrable function $ f:\Omega \to X$ with a countably additive indefinite integral we show that $ f$ is Pettis integrable if and only if $f$ is determined by a weakly compactly generated subspace of $X$ if and only if $f$ is determined by a subspace which has Mazur's property. We show that if $f:\Omega \to X$ is Pettis integrable then there exists a sequence ( $ {\varphi _n}$) of $ X$ valued simple functions such that for all ${x^{\ast}} \in {X^{\ast}}$, ${x^{\ast}}f= {\lim _n}{x^{\ast}}\,{\varphi _n}$ a.e. if and only if $f$ is determined by a separable subspace of $ X$. For a bounded weakly measurable function $f:\Omega \to {X^{\ast} }$ into a dual of a weakly compactly generated space, we show that $f$ is Pettis integrable if and only if $ f$ is determined by a separable subspace of $ {X^{\ast}}$ if and only if $ f$ is weakly equivalent to a Pettis integrable function that takes its range in ${\text{cor}}_f^{\ast} (\Omega)$.
A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three
J. W.
Cannon;
Daryl
Cooper
419-431
Abstract: We show that a cocompact hyperbolic group in dimension $3$ is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finite-volume hyperbolic groups in dimension $3$.
Ramsey theory in noncommutative semigroups
Vitaly
Bergelson;
Neil
Hindman
433-446
Abstract: By utilizing ultrafilters we give a general version of the Central Sets Theorem [$6$, Proposition 8.21]. This enables us to derive noncommutative versions of van der Waerden's Theorem and several of its generalizations. We also derive some standard results, including the Hales-Jewett Theorem.
The Gaussian map for rational ruled surfaces
Jeanne
Duflot;
Rick
Miranda
447-459
Abstract: In this paper the Gaussian map $\Phi :{ \wedge ^2}{H^0}(C,K) \to {H^0}(C,3K)$ of a smooth curve $C$ lying on a minimal rational ruled surface is computed. It is shown that the corank of $\Phi$ is determined for almost all such curves by the rational surface in which it lies. Hence, except for some special cases, a curve cannot lie on two nonisomorphic minimal rational ruled surfaces.