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Dimension and rank for mapping class groups
"... Dedicated to the memory of Candida Silveira. Abstract. We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An appl ..."
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Cited by 20 (4 self)
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Dedicated to the memory of Candida Silveira. Abstract. We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of BrockFarb’s Rank Conjecture which asserts that MCG has quasiflats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasiflats in Teichmuller space with the WeilPetersson metric. The coarse geometric structure of a finitely generated group can be studied by passage to its asymptotic cone, which is a space obtained by a limiting process from sequences of rescalings of the group. This has played an important role in the quasiisometric rigidity results of [DS], [KL1] [KL2], and others. In this paper we study the asymptotic cone M ω (S) of the mapping
Duality beyond Sober Spaces: Topological Spaces and Observation Frames
 and Completion in Semantics
, 1995
"... We introduce observation frames as an extension of ordinary frames. The aim is to give an abstract representation of a mapping from observable predicates to all predicates of a specific system. A full subcategory of the category of observation frames is shown to be dual to the category of T 0 topolo ..."
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Cited by 10 (4 self)
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We introduce observation frames as an extension of ordinary frames. The aim is to give an abstract representation of a mapping from observable predicates to all predicates of a specific system. A full subcategory of the category of observation frames is shown to be dual to the category of T 0 topological spaces. The notions we use generalize those in the adjunction between frames and topological spaces in the sense that we generalize finite meets to infinite ones. We also give a predicate logic of observation frames with both infinite conjunctions and disjunctions, just like there is a geometric logic for (ordinary) frames with infinite disjunctions but only finite conjunctions. This theory is then applied to two situations: firstly to upper power spaces, and secondly we restrict the adjunction between the categories of topological spaces and of observation frames in order to obtain dualities for various subcategories of T 0 spaces. These involve non sober spaces. Contents 1 Introduct...
Vizio, Continuity of the radius of convergence of padic differential equations on berkovich spaces
, 2007
"... analytic spaces ..."
Online regression competitive with reproducing kernel Hilbert spaces
, 2005
"... We consider the problem of online prediction of realvalued labels of new objects. The prediction algorithm’s performance is measured by the squared deviation of the predictions from the actual labels. No probabilistic assumptions are made about the way the labels and objects are generated. Instead ..."
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Cited by 6 (3 self)
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We consider the problem of online prediction of realvalued labels of new objects. The prediction algorithm’s performance is measured by the squared deviation of the predictions from the actual labels. No probabilistic assumptions are made about the way the labels and objects are generated. Instead, we are given a benchmark class of prediction rules some of which are hoped to produce good predictions. We show that for a wide range of infinitedimensional benchmark classes one can construct a prediction algorithm whose cumulative loss over the first N examples does not exceed the cumulative loss of any prediction rule in the class plus O ( √ N). Our proof technique is based on the recently developed method of defensive forecasting. 1
Diagrammatic Representations in DomainSpecific Languages
, 2000
"... One emerging approach to reducing the labour and costs of software development favours the specialisation of techniques to particular application domains. The rationale is that programs within a given domain often share enough common features and assumptions to enable the incorporation of substantia ..."
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Cited by 1 (1 self)
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One emerging approach to reducing the labour and costs of software development favours the specialisation of techniques to particular application domains. The rationale is that programs within a given domain often share enough common features and assumptions to enable the incorporation of substantial support mechanisms into domainspecific programming languages and associated tools. Instead of being machineoriented, algorithmic implementations, programs in many domainspecific languages (DSLs) are rather userlevel, problemoriented specifications of solutions. Taken further, this view suggests that the most appropriate representation of programs in many domains is diagrammatic, in a way which derives from existing design notations in the domain. This thesis conducts an investigation, using mathematical techniques and supported by case studies, of issues arising from the use of diagrammatic representations in DSLs. Its structure is conceptually divided into two parts: the first is co...
On sequentially hcomplete groups
 Proceedings of the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories
, 2002
"... A topological group G is hcomplete if every continuous homomorphic image of G is (Raĭkov)complete; we say that G is hereditarily hcomplete if every closed subgroup of G is hcomplete. In this paper, we establish openmap properties of hereditarily hcomplete groups with respect to large classes o ..."
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Cited by 1 (1 self)
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A topological group G is hcomplete if every continuous homomorphic image of G is (Raĭkov)complete; we say that G is hereditarily hcomplete if every closed subgroup of G is hcomplete. In this paper, we establish openmap properties of hereditarily hcomplete groups with respect to large classes of groups, and prove a theorem on the (total) minimality of subdirectly represented groups. Numerous applications are presented, among them: 1. Every hereditarily hcomplete group with quasiinvariant basis is the projective limit of its metrizable quotients; 2. If every countable discrete hereditarily hcomplete group is finite, then every locally compact hereditarily hcomplete group that has small invariant neighborhoods is compact. In the sequel, several open problems are formulated.
Computable Separation in Topology, from T0 to T3
"... Abstract. This article continues the study of computable elementary topology started in [7]. We introduce a number of computable versions of the topological T0 to T3 separation axioms and solve their logical relation completely. In particular, it turns out that computable T1 is equivalent to computa ..."
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Abstract. This article continues the study of computable elementary topology started in [7]. We introduce a number of computable versions of the topological T0 to T3 separation axioms and solve their logical relation completely. In particular, it turns out that computable T1 is equivalent to computable T2. The strongest axiom SCT3 is used in [2] to construct a computable metric. 1
University that enabled me to do this research. Contents
, 2008
"... I gratefully acknowledge the generous financial support received from the Killam Trusts and Dalhousie ..."
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I gratefully acknowledge the generous financial support received from the Killam Trusts and Dalhousie
On zerodimensionality and the connected component of locally pseudocompact groups *
, 2009
"... A topological group is locally pseudocompact if it contains a nonempty open set with pseudocompact closure. In this note, we study connectedness and disconnectedness properties of groups G with the property that every closed subgroup of G is locally pseudocompact. We show that the completion of the ..."
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A topological group is locally pseudocompact if it contains a nonempty open set with pseudocompact closure. In this note, we study connectedness and disconnectedness properties of groups G with the property that every closed subgroup of G is locally pseudocompact. We show that the completion of the component G0 of G contains every connected compact subgroup of the completion of G. We also prove that the question of whether G/G0 is zerodimensional (or equivalently, whether G0 is dense in the component of the completion of G) can be reduced to the case where G is a dense subgroup of a group of the form N ×R, where N is zerodimensional and compact. 1.