Past Analytic Topology in Mathematics and Computer Science

1 June 2011
16:00
to
17:30
Abstract
<p>It is known for long that the set of possible compactifications of a topological space (up to homeomorphism) is in order-preserving bijection to "strong inclusion" relations on the lattice of open sets. Since these relations do not refer to points explicitly, this bijection has been generalised to point-free topology (a.k.a. locales). The strong inclusion relations involved are typically "witnessed" relations. For example, the Stone-Cech compactification has a strong inclusion witnessed by real-valued functions. This makes it natural to think of compactification in terms of d-frames, a category invented by Jung and Moshier for bitopological Stone duality. Here, a witnessed strong inclusion is inherent to every object and plays a central role. </p> <p>We present natural analogues of the topological concepts regularity, normality, complete regularity and compactness in d-frames. Compactification is then a coreflection into the category of d-frames dually equivalent to compact Hausdorff spaces. The category of d-frames has a few surprising features. Among them are: </p> <ul> <li>The real line with the bitopology of upper and lower semicontinuity admits precisely one compactification, the extended reals.</li> <li>Unlike in the category of topological spaces (or locales), there is a coreflection into the subcategory of normal d-frames, and every compactification can be factored as "normalisation" followed by Stone-Cech compactification.</li></ul>
  • Analytic Topology in Mathematics and Computer Science
4 May 2011
16:00
to
17:30
Dr Secil Tokgoz
Abstract
A topological space $(X,\tau)$ is submaximal if $\tau$ is the maximal element of $[{\tau}_{s}]$. Submaximality was first defined and characterized by Bourbaki. Since then, some mathematicians presented several characterizations of submaximal spaces. In this paper, we will attempt to develop the concept of submaximality and offer some new results. Furthermore, some results concerning $\alpha$-scattered space will be obtained.
  • Analytic Topology in Mathematics and Computer Science
9 March 2011
16:00
Mehrnoosh Sadrzadeh
Abstract
<p>Words are building blocks of sentences, yet the meaning of a sentence goes well beyond meanings of its words. Formalizing the process of meaning assignment is proven a challenge for computational and mathematical linguistics; with the two most successful approaches each missing on a key aspect: the 'algebraic' one misses on the meanings of words, the vector space one on the grammar.</p> <p>I will present a theoretical setting where we can have both! This is based on recent advances in ordered structures by Lambek, referred to as pregroups and the categorical/diagrammatic approach used to model vector spaces by Abramsky and Coecke. Surprisingly. both of these structures form a compact category! If time permits, I will also work through a concrete example, where for the first time in the field we are able to compute and compare meanings of sentences compositionally. This is collaborative work with E. Greffenstete, C. Clark, B. Coecke, S. Pulman.</p>
  • Analytic Topology in Mathematics and Computer Science
24 September 2010
16:00
Dr Alessandra Palmigiano
Abstract

Abstract: Quantales are ordered algebras which can be thought of as pointfree noncommutative topologies. In recent years, their connections have been studied with fundamental notions in noncommutative geometry such as groupoids and C*-algebras. In particular, the setting of quantales corresponding to étale groupoids has been very well understood: a bijective correspondence has been defined between localic étale groupoids and inverse quantale frames. We present an equivalent but independent way of defining this correspondence for topological étale groupoids and we extend this correspondence to a non-étale setting.

  • Analytic Topology in Mathematics and Computer Science
26 May 2010
14:00
Drew Moshier
Abstract
<p>Consider the following simple question:</p> <p>Is there a subcategory of Top that is dually equivalent to Lat?</p> <p>where Top is the category of topological spaces and continuous maps and Lat is the category</p> <p>of bounded lattices and bounded lattice homomorphisms.</p> <p>Of course, the question has been answered positively by specializing Lat, and (less</p> <p>well-known) by generalizing Top.</p> <p>The earliest examples are of the former sort: Tarski showed that every complete atomic</p> <p>Boolean lattice is represented by a powerset (discrete topological space); Birkhoff showed</p> <p>that every finite distributive lattice is represented by the lower sets of a finite partial order</p> <p>(finite T0 space); Stone generalized Tarski and then Birkhoff, for arbitrary Boolean and</p> <p>arbitrary bounded distributive lattices respectively. All of these results specialize Lat,</p> <p>obtaining a (not necessarily full) subcategory of Top.</p> <p>As a conceptual bridge, Priestley showed that distributive lattices can also be dually</p> <p>represented in a category of certain topological spaces augmented with a partial order.</p> <p>This is an example of the latter sort of result, namely, a duality between a category of</p> <p>lattices and a subcategory of a generalization of Top.</p> <p>Urquhart, Hartung and Hartonas developed dualities for arbitrary bounded lattices in</p> <p>the spirit of Priestley duality, in that the duals are certain topological spaces equipped with</p> <p>additional structure.</p> <p>We take a different path via purely topological considerations. At the end, we obtain</p> <p>an affirmative answer to the original question, plus a bit more, with no riders: the dual</p> <p>categories to Lat and SLat (semilattices) are certain easily described subcategories of Top</p> <p>simpliciter. This leads directly to a very natural topological characterization of canonical</p> <p>extensions for arbitrary bounded lattices.</p> <p>Building on the topological foundation, we consider lattices expanded with quasioperators,</p> <p>i.e., operations that suitably generalize normal modal operatos, residuals, orthocomplements</p> <p>and the like. This hinges on both the duality for lattices and for semilattices</p> <p>in a natural way.</p> <p>This talk is based on joint work with Peter Jipsen.</p> <p>Date: May 2010.</p> <p>1</p>
  • Analytic Topology in Mathematics and Computer Science

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