TOPOLOGY SEMINARS NOTE: Seminars in this series prior to Spring 2004 are listed on a separate archive page. Visit http://www.cs.bham.ac.uk/research/events/topology-seminar/topology.html for more information. -------------------------------- Date and time: Friday 8th November 2002 at 16:00 Location: UG40, School of Computer Science Title: Topologies on spaces of continuous functions Speaker: Martin Escardo (http://www.cs.bham.ac.uk/~mhe) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: For various applications of topology (e.g. to homotopy theory and to the lambda-calculus), one needs to consider function spaces, that is, suitably topologized sets of continuous functions. A space X is called exponentiable if such a suitable topology exists on the set continuous maps from X to Y for every space Y. It is well-known that a Hausdorff space is exponentiable if and only if it is locally compact, and that in this case the exponential topology is the compact-open topology. However, the spaces that occur in the applications to the lambda-calculus (domains under the Scott topology) are highly non-Hausdorff. In general, the exponentiable spaces are known to be precisely the core-compact spaces. The available approaches to this general characterization are based on either category theory or domain theory, or even both. In this talk we provide a self-contained, elementary and brief development of general function spaces. The only prerequisite to this development is a basic knowledge of general topology (continuous functions, product topology and compactness). However, we'll take the opportunity to make some excursions in domain-theory land (but without assuming familiarity with the subject). (This is based on joint work [http://www.cs.bham.ac.uk/~mhe/papers/newyork.ps] with Reinhold Heckmann. Slides [http://www.cs.bham.ac.uk/~mhe/papers/functionspaces-talk.ps] are available.) -------------------------------- Date and time: Friday 15th November 2002 at 16:00 Location: UG40, School of Computer Science Title: Topological digital topology Speaker: Ralph Kopperman (http://math0.sci.ccny.cuny.edu/faculty/kopperman/kopperman.html) Abstract: Digital topology apparently asks questions about the topology of digital images, but most often answers them in terms of "adjacencies". This is natural, since such relations allow easy algorithm construction, but the intuition associated with such relations can be misleading, and yield incorrect answers. We discuss some of these issues, giving the strengths and weaknesses of a purely topological approach. There remain important open problems, two of which will be presented. (A reformatted version of the slides [http://www.cs.bham.ac.uk/~mhe/topology/kopperman.ps] is available.) -------------------------------- Date and time: Friday 22nd November 2002 at 16:00 Location: UG40, School of Computer Science Title: Defining a topology on a set so that a given selfmap is continuous Speaker: Chris Good (http://web.mat.bham.ac.uk/C.Good/) Institution: School of Mathematics, The University of Birmingham (http://www.mat.bham.ac.uk) Abstract: Joint with Sina Greenwood, Dave MacIntyre (Auckland), Robin Knight (Oxford), Steve Watson (York, CA). A contraction mapping on a complete metric space has a fixed point. De Groot asked whether a converse of the following form holds: Given a map T: X -> X of a set to itself with a fixed point, is there a compact metric topology on X with respect to which T is continuous. For good reasons, 'compact metric' is too hard a question. 'Metric' alone and 'compact' alone on the other hand are trivial questions, the discrete and indiscrete topologies work, so the question became whether there is a compact, Hausdorff topology on X with respect to which T is continuous. Smith et al showed that the answer is no, in general. But there is a more general question here. Given a map T:X -> X (not necessarily with a fixed point), when is there a compact Hausdorff topology on X with respect to which T is continuous? Given the generality of the question, the answer is surprisingly elegant, and you don't really need to know anything more than the meaning of compact and Hausdorff to follow the talk. ( Slides are available.) -------------------------------- Date and time: Friday 29th November 2002 at 16:00 Location: UG40, School of Computer Science Title: A tutorial introduction to the partial metric Speaker: Steve Matthews (http://www.dcs.warwick.ac.uk/people/academic/Steve.Matthews) Institution: The University of Warwick (http://www.dcs.warwick.ac.uk) Abstract: The purpose of this talk is to offer a motivation for considering the use of generalized metrics in both topology and applications to computer science. The key idea is to drop the reflexive axiom of metrics that self distance d(x,x) is necessarily zero. We will consider the application of such functions for proving absence of deadlock in Kahn data flow networks. This work leads to an unexpected interest in studying the relation of a T2 topology to a T0 sub topology. -------------------------------- Date and time: Friday 6th December 2002 at 16:00 Location: UG40, School of Computer Science Title: Finitely approximated spaces Speaker: Achim Jung (http://www.cs.bham.ac.uk/~axj) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: For some time now, I have been concerned with a logical approach to stably compact spaces (also known as "skew compact spaces"). This work grew out of my inability to extend Samson Abramsky's "Domain Theory in Logical Form" to continuous domains, especially, FS-domains. In this talk I will try to explain some of the ingredients of this work, and bring back some of the original motivation. This will involve making use of an old paper which I wrote with Philipp Suenderhauf in 1995, called "Uniform approximation of spaces" (available from my web-site). PS: I have been motivated to give this talk by Steve Matthews' talk last week and I will be curious to hear his comments on my approach. Nonetheless, I aim to make the talk as self-contained as I can manage. -------------------------------- Date and time: Friday 24th January 2003 at 14:00 Location: UG40, School of Computer Science Title: Ask what can computer science do for topology? Speaker: Ralph Kopperman (http://math0.sci.ccny.cuny.edu/faculty/kopperman/kopperman.html) Abstract: Joint talk with Theoretical Computer Science Seminar [http://www.cs.bham.ac.uk/events/seminars/all_seminars.htmlseries_id=theory] . Theoretical computer science has required from general topology a reformulation of the most basic concepts of separation and compactness, initially developed 70-100 years ago. It has also offered the guidance needed to think these issues through. (A reformatted version of the slides [http://www.cs.bham.ac.uk/~mhe/topology/bhamcs12.ps] is available.) -------------------------------- Date and time: Friday 24th January 2003 at 16:00 Location: UG40, School of Computer Science Title: A universal characterization of the double powerlocale Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv/) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: The opens of a topological space X are the continuous maps from X to the Sierpinski space $, and the same works if spaces are replaced by locales. It follows that the exponential $^X in the category of spaces or locales is important, since it represents the topology on X - the points of $^X are the opens on X. It has long been known that the locale exponential $^X exists if and only if X is exponentiable (i.e. all exponentials Y^X exist), and that holds iff X is locally compact. In that case, $^X is also locally compact, and I proved in 2001 that the double exponential $^($^X) could also be obtained as the double powerlocale PP(X). This is got by applying (in either order) two powerlocale construction, upper and lower, the localic analogues of the Smyth and Hoare powerdomains. This gives a categorical characterization of PP, independently of its frame construction, in the locally compact case. The double powerlocale PP(X) is defined for all locales X, not necessarily locally compact, and so one might ask whether there is a more general such characterization. We show that there is a sense in which PP(X) is $^($^X) for arbitrary X, by embedding the category Loc of locales in the presheaf category Set^(Loc^op). Thus within this presheaf category, if X is a representable presheaf (i.e. the image of an actual locale under the Yoneda embedding) then $^X need not be representable but $^($^X) is. An interesting lemma shows a relationship between Scott continuity and naturality. If X and Y are locales, then the natural transformations from $^X to $^Y are in bijection with the Scott continuous functions from the frame of X to the frame of Y, which were already known to be in bijection with the maps from Y to PP(X). This is joint work with Christopher Townsend. -------------------------------- Date and time: Friday 31st January 2003 at 16:00 Location: UG40, School of Computer Science Title: What is and what is not point-free topology? Speaker: Harold Simmons (http://www.cs.man.ac.uk/~hsimmons/) Institution: University of Manchester (http://www.cs.man.ac.uk) Abstract: In this context point-sensitive topology is what is usually called point-set topology (as developed in Kelly's book). This is part of a larger subject 'topology' which includes algebraic topology, homotopy theory, homological algebra, perhaps some algebraic geometry, and other things. Of course, it impinges on many other topics, and it is not clear where 'topology' ends and other subjects begin. Along side this there is another area of mathematics which is even less well defined. It impinges on such topics as ring theory, algebraic number theory, module theory, algebraic geometry, and some parts of 'topology'. For instance, the study and use of quantales is a part of this area. This area of mathematics uses entirely algebraic methods. Point-free topology is a part of this area, and consist of the study of certain lattices called frames. There is a direct relationship between point-free and point-sensitive, but the information flow goes entirely one way. I will show, by an example, how the entirely algebraic view of the point-free subject leads to a much deeper understanding to that part of the point-sensitive subject which instigated, and made necessary, its development. The secret is to remember a part of the larger algebraic area which seems to have nothing to do with topology. ( Slides [http://www.cs.bham.ac.uk/~mhe/topology/pointfree.ps] are available.) -------------------------------- Date and time: Friday 7th February 2003 at 16:00 Location: UG40, School of Computer Science Title: Universal p-metrizability Speaker: Homeira Pajoohesh (http://www.cs.bham.ac.uk/people/person.php?group=3&name=hxp) Abstract: Steve Matthews defined pmetrics because of a need to have distances between partially defined objects.We will show that pmetrics can  be generalized so that for each continuous dcpo, the Scott and lower topologies arise from a generalized pmetric, and also, each topology arises from a generalized pmetric. (A reformatted version of the slides [http://www.cs.bham.ac.uk/~mhe/topology/homeira.ps] is available.) -------------------------------- Date and time: Friday 14th February 2003 at 16:00 Location: UG40, School of Computer Science Title: Partial metrization of domains Speaker: Pawel Waszkiewicz (http://www.cs.bham.ac.uk/~pqw) Abstract: We will present various representation theorems for partial metrics including a complete characterization of partial metrizability of algebraic posets. We will discuss existence of measurements and partial metrics for diverse classes of domains. -------------------------------- Date and time: Friday 21st February 2003 at 16:00 Location: UG40, School of Computer Science Title: Compact topological spaces and boolean algebras coming from functional analysis Speaker: Mirna Dzamonja (http://www.mth.uea.ac.uk/people/md.html) Abstract: Our interest will be in general in compact spaces coming from functional analysis- for example Corson compacta, Valdivia compacta and in particular in one such class of spaces namely uniform Eberlein compacta. We shall specifically address the question of the existence of universal objects in these classes and show how this question has been approached from the point of view of boolean algebras. Note that the spaces in question are not necessarily zero-dimensional so one cannot simply use the duality theory. -------------------------------- Date and time: Friday 28th February 2003 at 16:00 Location: UG40, School of Computer Science Title: Domain Theoretic Models of Topological Spaces Speaker: Mike Reed (http://web.comlab.ox.ac.uk/oucl/people/mike.reed.html) -------------------------------- Date and time: Friday 7th March 2003 at 16:00 Location: UG40, School of Computer Science Title: Constructive Maximal Point Spaces Speaker: Michael B. Smyth (http://www.doc.ic.ac.uk/directory/peopleS.html) -------------------------------- Date and time: Friday 14th March 2003 at 16:00 Location: UG40, School of Computer Science Title: Axioms for a Category of Spaces Speaker: Christopher Townsend (http://mcs.open.ac.uk/cft36/) Abstract: A recent observation (with Steve Vickers) states that directed join preserving (i.e. Scott continuous/dcpo) maps between frames can be described as natural transformations in the functor category [Loc^op,Set]. This observation is taken as the basis for an axiomatization of a category of spaces. The axioms essentially relate continuous maps between spaces to dcpo maps (that is, natural transformations) in the same manner that locale maps are related to dcpo maps between frames. In this setting the theory of compact Hausdorff spaces is order-dual to the theory of discrete spaces and so the theories of these two classes have equal status. -------------------------------- Date and time: Friday 2nd May 2003 at 16:00 Location: UG40, School of Computer Science Title: Flatness in the enriched case and metric spaces Speaker: Vincent Schmitt Institution: University of Leicester Abstract: Flatness in the enriched case and metric spaces I shall present a notion of flatness that works for enriched categories. From this one may define a "free flat-cocompletion" of enrichments. I could apply this result using Lawvere's equivalence between metric spaces and enrichments. I showed that flat presheaves on A viewed as a category correspond exactly to particular filters on A viewed as a metric space. - I called these filters "flat". It happens that any Cauchy filter is flat. When the space A is symmetric, flat filters on A are exactly the Cauchy ones thus in this special case, the free flat-completion of A is exactly the (Lawvere-)Cauchy one. Another result is that flat filters are exactly the directed cocompletions of forward Cauchy sequences (with the right notion of morphisms though). I will also give a translation the categorical completion of in topological terms of the flat completion. -------------------------------- Date and time: Friday 9th May 2003 at 16:00 Location: UG40, School of Computer Science Title: Noncommutative algebraic discretizations of smooth manifolds Speaker: Ioannis Raptis Institution: Department of Physics, Imperial College Abstract: Sorkin's finitary (:locally finite) poset substitutes of topological manifolds will be revisited. Then we build the Gel'fand dual algebraic picture in terms of finite-dimensional noncommutative incidence Rota algebras. Casting Sorkin's posets as simplicial complexes a la Cech-Alexandrov, we present the associated Rota algebras as `discrete differential manifolds'---`differential functional algebras' with well mannered Gel'fand spatialization behaviour. Based on the recently developed Calculus and manifold-free sheaf-theoretic Abstract Differential Geometry (ADG) theory of Mallios, we first define finitary spacetime sheaves of incidence algebras as discrete differential (not just topological!) substitutes of sheaves of modules of smooth differential forms over differential manifolds, and we heuristically suggest a way of recovering the smooth spacetime continuum by suitable direct (inductive) and inverse (projective) limit constructions, with a concomitant new and significantly generalized notion of 'emergent smoothness'. At the end, a physical interpretation, as Bohr's correspondence principle, will be given to these classical continuum limit procedures, and, time permitting, we will discuss the relevance of it all to current structural issues in both classical and quantum gravity research. _Keywords_ : finitary topological spaces, differential incidence algebras of locally finite posets, abstract differential geometry, sheaf theory, category theory, classical and quantum gravity. -------------------------------- Date and time: Friday 23rd May 2003 at 16:00 Location: UG40, School of Computer Science Title: Subbases that give a canonical representation of the elements in a space Speaker: Hideki Tsuiki Institution: Kyoto University -------------------------------- Date and time: Friday 20th June 2003 at 16:00 Location: UG40, School of Computer Science Title: A domain-theoretic model for quantum computation Speaker: Abbas Edalat Institution: Imperial College Abstract: SD has now provided both computable and topos-based theories of locally compact spaces. In the computable theory, spaces are specified by the containment relation of basic compact subspaces in basic open ones, for example using closed and open real intervals. By adding an "underlying set" axiom, the standard theory of locally compact locales over an elementary topos is recovered. Unlike other current reformulations of topology, this work does not consist of altering an existing category of spaces. It begins from an categorical identification of properties of the Sierpinski space that first appeared in the context of synthetic domain theory. The categorical axioms are elementary, and a corresponding type theory has been described. A strikingly original feature that has emerged from this is the dual treatment of compact Hausdorff and overt discrete objects. The latter play the role of sets, forming an arithmetic universe (pretopos with lists) with general recursion. ASD therefore provides a theory of general topology with such a computable "set theory" in place of a topos. Attention is now shifting to applications. One is to develop a type theory for inductive and coinductive types that would be interpreted respectively as overt discrete and compact Hausdorff objects in ASD and so translate into a very simple lambda calculus. -------------------------------- Date and time: Friday 25th July 2003 at 14:00 Location: UG40, School of Computer Science Title: Abstract Stone Duality Speaker: Paul Taylor (http://www.di.unito.it/~pt/ASD) Abstract: Joint talk with Theoretical Computer Science Seminar [http://www.cs.bham.ac.uk/events/seminars/all_seminars.html?series_id=theory] . ASD has now provided both computable and topos-based theories of locally compact spaces. In the computable theory, spaces are specified by the containment relation of basic compact subspaces in basic open ones, for example using closed and open real intervals. By adding an "underlying set" axiom, the standard theory of locally compact locales over an elementary topos is recovered. Unlike other current reformulations of topology, this work does not consist of altering an existing category of spaces. It begins from an categorical identification of properties of the Sierpinski space that first appeared in the context of synthetic domain theory. The categorical axioms are elementary, and a corresponding type theory has been described. A strikingly original feature that has emerged from this is the dual treatment of compact Hausdorff and overt discrete objects. The latter play the role of sets, forming an arithmetic universe (pretopos with lists) with general recursion. ASD therefore provides a theory of general topology with such a computable "set theory" in place of a topos. Attention is now shifting to applications. One is to develop a type theory for inductive and coinductive types that would be interpreted respectively as overt discrete and compact Hausdorff objects in ASD and so translate into a very simple lambda calculus. -------------------------------- Date and time: Friday 25th July 2003 at 16:00 Location: UG40, School of Computer Science Title: A gentle introduction to locales for topologists Speaker: Martin Escardo (http://www.cs.bham.ac.uk/~mhe/) Institution: School of Computer Science (http://www.cs.bham.ac.uk) -------------------------------- Date and time: Wednesday 14th January 2004 at 16:30 Location: 217, School of Computer Science Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv/) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the first talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. Copies of slides are available. OUTLINE: * VIA LOGIC: In my book, "via Logic" referred to the way it presents topologies as Lindenbaum algebras for theories in a particular non-classical logic, the so called geometric logic. This lies behind an alternative approach to topology known as locale theory, which has been used in the semantics of programming languages - notably in Abramsky's thesis. * OBSERVABILITY: In my book I motivate the use of geometric logic by an informal discussion of how certain logical connective have observational content (this is also present in the work of Smyth and Abramsky). Then the basic deal in semantics is that a domain, with its Scott topology, embodies a logical theory of how one can observe some type of program by watching it run (without access to the source code). * GOING CONSTRUCTIVE: The book is entirely classical in its mathematics. However, since it was written I have come to understand much better the key role of constructive mathematics in topology and have exploited it in my papers such as "Topical categories of domains". This is what I want to explain in the lectures. * BETTER CONSTRUCTIVE TOPOLOGY: One aspect of this is that if you are already committed to constructivism (maybe in order to take better account of computational issues), then the localic approach is known to give better results in constructive topology than topology itself. * CONTINUITY FOR FREE. But there are deeper reasons topological reasons for being constructive. Associated with the geometric logic, there is a "geometric mathematics". It is constructive, and also has an intrinsic continuity. If you adhere to the constructive constraints of geometric reasoning then you do not have to give explicit continuity proofs - continuity is automatic. In effect, the usual continuity proofs in classical reasoning are bureaucratic certifications that you haven't transgressed the geometric constraints. So there are quite practical reasons for accepting those constraints. * TOPOSES A major spinoff is that the same approach adapts quite naturally to a generalization of topological space, namely toposes. This aspect of toposes is normally concealed, but it is fundamental to their importance. Roughly speaking, from this point of view a (Grothendieck) topos is a topological space for which there are "not enough opens" and instead the topological structure has to be defined through what are known as its sheaves. -------------------------------- Date and time: Wednesday 21st January 2004 at 16:30 Location: 217, School of Computer Science Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the second talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 23rd January 2004 at 16:00 Location: LG34, Learning Centre Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the third talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 30th January 2004 at 16:00 Location: LG34, Learning Centre Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the fourth talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 6th February 2004 at 16:00 Location: LG34, Learning Centre Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the fifth talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 20th February 2004 at 16:00 Location: LG34, Learning Centre Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the sixth talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 5th March 2004 at 16:00 Location: LG34, Learning Centre Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the eighth talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 19th March 2004 at 16:00 Location: LG34, Learning Centre Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the tenth talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 26th March 2004 at 16:00 Location: LG34, Learning Centre Title: Topology via Logic Speaker: Steve Vickers (http://www.cs.bham.ac.uk/~sjv) Institution: School of Computer Science (http://www.cs.bham.ac.uk) Abstract: This is the eleventh talk in Steve Vickers' _Topology via Logic_ series: I am giving a series of talks this term on topology from a computer science perspective. It will in many ways follow my book "Topology via Logic" (CUP 1989), but will also go beyond that in bringing out the deep relationship with constructive logic. My lectures will be new, and in fact part of the aim is to put together the shape of a constructive 2 nd edition of Topology via Logic. See the abstract of the first talk [http://www.cs.bham.ac.uk/events/seminars/seminar_details.html?seminar_id=26] for a detiled outline of the series. -------------------------------- Date and time: Friday 19th November 2004 at 16:00 Location: UG40, School of Computer Science Title: Clash with other school activities Speaker: Slot not available -------------------------------- Date and time: Friday 26th November 2004 at 14:00 Location: UG40, School of Computer Science Title: Towards a Geometry of Realizability Toposes? Speaker: Prof. Peter T. Johnstone (http://www.dpmms.cam.ac.uk/site2002/People/johnstone_pt.html) Institution: Cambridge University, Department of Pure Mathematics and Mathematical Statistics (http://www.dpmms.cam.ac.uk/) Host: Achim Jung Abstract: This is a joint theory and topology seminar. Please refer to the theory seminar page. -------------------------------- Date and time: Friday 17th September 2010 at 12:30 Location: 124, School of Computer Science Title: A bitopological point-free approach to compactifications Speaker: Olaf Klinke (http://www.cs.bham.ac.uk/~okk/) Institution: School of Computer Science, University of Birmingham (http://www.cs.bham.ac.uk/) Host: topology lab lunch Abstract: We study structures called 'd-frames' which were developed by Jung and Moshier for a bitopological treatment of Stone duality. These structures consist of a pair of frames thought of as the opens of two topologies, together with two relations which serve as abstractions of disjointness and covering of the space. With these relations, the topological separation axioms regularity and normality have natural analogues in d-frames. We develop a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames. Given that normal topological spaces do not behave well with respect to products and subspaces, probably the most surprising result is this: The category of d-frames has a normal coreflection. Moreover, any compactification can be obtained by first producing a regular normal d-frame and then applying the Stone-Cech compactification to it. Our bitopological compactification subsumes all classical compactifications of frames as well as Smyth's stable compactifications.