Cartan-decomposition subgroups of $\operatorname{SO}(2,n)$
Hee
Oh;
Dave
Witte
Morris
1-38
Abstract: For $G = \operatorname{SL} (3,\mathord{\mathbb{R} })$ and $G = \operatorname{SO}(2,n)$, we give explicit, practical conditions that determine whether or not a closed, connected subgroup $H$of $G$ has the property that there exists a compact subset $C$ of $G$with $CHC = G$. To do this, we fix a Cartan decomposition $G = K A^+ K$of $G$, and then carry out an approximate calculation of $(KHK) \cap A^+$for each closed, connected subgroup $H$ of $G$.
Cuntz-Krieger algebras of infinite graphs and matrices
Iain
Raeburn;
Wojciech
Szymanski
39-59
Abstract: We show that the Cuntz-Krieger algebras of infinite graphs and infinite $\{0,1\}$-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their $K$-theory. Since the finite approximating graphs have sinks, we have to calculate the $K$-theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.
Semi-linear homology $G$-spheres and their equivariant inertia groups
Zhi
Lü
61-71
Abstract: This paper introduces an abelian group $H\Theta_V^G$ for all semi-linear homology $G$-spheres, which corresponds to a known abelian group $\Theta_V^G$ for all semi-linear homotopy $G$-spheres, where $G$ is a compact Lie group and $V$ is a $G$-representation with $\dim V^G>0$. Then using equivariant surgery techniques, we study the relation between both $H\Theta_V^G$ and $\Theta_V^G$ when $G$ is finite. The main result is that under the conditions that $G$-action is semi-free and $\dim V-\dim V^G\geq 3$ with $\dim V^G >0$, the homomorphism $T: \Theta_V^G\longrightarrow H\Theta_V^G$defined by $T([\Sigma]_G)=\langle \Sigma\rangle_G$ is an isomorphism if $\dim V^G\not=3,4$, and a monomorphism if $\dim V^G=4$. This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology $G$-spheres.
Compact covering mappings between Borel sets and the size of constructible reals
Gabriel
Debs;
Jean
Saint
Raymond
73-117
Abstract: We prove that the topological statement: ``Any compact covering mapping between two Borel sets is inductively perfect" is equivalent to the set-theoretical statement: $\lq\lq \,\forall\alpha\in \omega^\omega,\; \aleph_1^{L(\alpha)}<\aleph_1$".
Backward stability for polynomial maps with locally connected Julia sets
Alexander
Blokh;
Lex
Oversteegen
119-133
Abstract: We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial $f$ with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure $\mu$ this easily implies that one of the following holds: 1. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=J(f)$; 2. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=\omega(c(x))$ for a critical point $c(x)$depending on $x$.
Geometric aspects of Sturm-Liouville problems II. Space of boundary conditions for left-definiteness
Kevin
Haertzen;
Qingkai
Kong;
Hongyou
Wu;
Anton
Zettl
135-157
Abstract: For a given regular Sturm-Liouville equation with an indefinite weight function, we explicitly describe the space of left-definite selfadjoint boundary conditions. The description only uses one value of a fundamental solution of the matrix form of the equation. As a consequence we show that this space has the shape of a solid consisting of two cones sharing a common base.
Closed product formulas for extensions of generalized Verma modules
Riccardo
Biagioli
159-184
Abstract: We give explicit combinatorial product formulas for the polynomials encoding the dimensions of the spaces of extensions of $(g,p)$-generalized Verma modules, in the cases when $(g,p)$corresponds to an indecomposable classic Hermitian symmetric pair. The formulas imply that these dimensions are combinatorial invariants. We also discuss how these polynomials, defined by Shelton, are related to the parabolic $R$-polynomials introduced by Deodhar.
An index for gauge-invariant operators and the Dixmier-Douady invariant
Victor
Nistor;
Evgenij
Troitsky
185-218
Abstract: Let $\mathcal{G}\to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\mathcal{G}^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\mathcal{G}\to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\mathcal{G}\to B$, which, in this approach, is an element of $K_\mathcal{G}^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\mathcal{G}))$, the $K$-theory group of the Banach algebra $C^*(\mathcal{G})$. We prove that $K_0(C^*(\mathcal{G})) \simeq K^0_\mathcal{G}(\mathcal{G})$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\mathcal{G})$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently turned out to be useful in the study of Ramond-Ramond fields.
Vassiliev invariants for braids on surfaces
Juan
González-Meneses;
Luis
Paris
219-243
Abstract: We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.
Ideals of the cohomology rings of Hilbert schemes and their applications
Wei-Ping
Li;
Zhenbo
Qin;
Weiqiang
Wang
245-265
Abstract: We study the ideals of the rational cohomology ring of the Hilbert scheme $X^{[n]}$ of $n$ points on a smooth projective surface $X$. As an application, for a large class of smooth quasi-projective surfaces $X$, we show that every cup product structure constant of $H^*(X^{[n]})$ is independent of $n$; moreover, we obtain two sets of ring generators for the cohomology ring $H^*(X^{[n]})$. Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between $H^*(X^{[n]}; \mathbb{C} )$ and $H^*_{\rm orb}(X^n/S_n; \mathbb{C} )$ for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.
Framings of knots satisfying differential relations
James
J.
Hebda;
Chichen
M.
Tsau
267-281
Abstract: This paper introduces the notion of a differential framing relation for knots in a three-dimensional manifold. There is a canonical map from the space of knots that satisfy a framing relation into the space of framed knots. Under reasonable assumptions this canonical map is a weak homotopy equivalence.
Embedded minimal disks: Proper versus nonproper---global versus local
Tobias
H.
Colding;
William
P.
Minicozzi II
283-289
Abstract: We construct a sequence of compact embedded minimal disks in a ball in $\mathbf{R}^3$ with boundaries in the boundary of the ball and where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper. If instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity, then we have shown previously that any limit must be smooth and proper.
Analysing finite locally $s$-arc transitive graphs
Michael
Giudici;
Cai
Heng
Li;
Cheryl
E.
Praeger
291-317
Abstract: We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms $G$ and are either locally $(G,s)$-arc transitive for $s \geq 2$ or $G$-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of $G$. Given a normal subgroup $N$ which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of $N$ preserves both local primitivity and local $s$-arc transitivity and leads us to study graphs where $G$ acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for $G$ in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.
Eigenvalue and gap estimates for the Laplacian acting on $p$-forms
Pierre
Guerini;
Alessandro
Savo
319-344
Abstract: We study the gap of the first eigenvalue of the Hodge Laplacian acting on $p$-differential forms of a manifold with boundary, for consecutive values of the degree $p$. We first show that the gap may assume any sign. Then we give sufficient conditions on the intrinsic and extrinsic geometry to control it. Finally, we estimate the first Hodge eigenvalue of manifolds whose boundaries have some degree of convexity.
Exponential sums on $\mathbf{A}^n$, II
Alan
Adolphson;
Steven
Sperber
345-369
Abstract: We prove a vanishing theorem for the $p$-adic cohomology of exponential sums on $\mathbf{A}^n$. In particular, we obtain new classes of exponential sums on $\mathbf{A}^n$ that have a single nonvanishing $p$-adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.
Slopes of vector bundles on projective curves and applications to tight closure problems
Holger
Brenner
371-392
Abstract: We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous $R_+$-primary ideal in a two-dimensional normal standard-graded algebra $R$ in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.
Asymptotics of the transition probabilities of the simple random walk on self-similar graphs
Bernhard
Krön;
Elmar
Teufl
393-414
Abstract: It is shown explicitly how self-similar graphs can be obtained as `blow-up' constructions of finite cell graphs $\hat C$. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals. For a class of symmetrically self-similar graphs we study the simple random walk on a cell graph $\hat C$, starting at a vertex $v$ of the boundary of $\hat C$. It is proved that the expected number of returns to $v$before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions, we compute the asymptotic behaviour of the $n$-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpinski graph are generalised to the class of symmetrically self-similar graphs, and at the same time the error term of the asymptotic expression is improved. Finally, we present a criterion for the occurrence of oscillating phenomena of the $n$-step transition probabilities.
The classical problem of the calculus of variations in the autonomous case: Relaxation and Lipschitzianity of solutions
Arrigo
Cellina
415-426
Abstract: We consider the problem of minimizing \begin{displaymath}\int _{a}^{b} L(x(t),x^{\prime }(t)) \, dt, \qquad x(a)=A, x(b)=B.\end{displaymath} Under the assumption that the Lagrangian $L$is continuous and satisfies a growth assumption that does not imply superlinear growth, we provide a result on the relaxation of the functional and show that a solution to the minimum problem is Lipschitzian.