Imperial College
London
United Kingdom
Title. A domain-theoretic generalisation of Henstock-Kurzweil integral for compact metric spaces
Download
Abstract. Given a compact metric space and a normalised Borel measure on the space, we introduce the notion of a gauge and a subordinate tagged partition induced by crescents1 of the space, extending the corresponding notions in Henstock-Kurzweil integration of real-valuedfunctions with respect to the Lebesgue measure on the unit interval.We then consider the integration of any real-valued function on the compact metric space with respect to the normalised Borel measure.
A partition of the space by crescents induces a simple valuation on the upper space of the metric space, while a tagged partition subordinate to a gauge induces a simple point valuation on the metric
space. We introduce a partial order on pairs of a tagged partition and a gauge, to which the tagged partition is subordinate, which makes the set of such pairs into a directed set, emulating the partial order of simple valuations way-below a continuous valuation in the probabilistic power domain.
We show that for the integration of continuous functions with respect to a normalised Borel measure, the tagged partition/gauge pairs are equivalent to a directed set of normalized simple valuations way-below the given Borel measure in the probabilistic power domain of the upper space of the metric space.
For general, possibly unbounded functions, we define the D-integral of a real-valued function with respect to a Borel measure using the limit of the net of the integrals of the simple valuations induced by pairs of tagged partitions and gauges for the function. The D-integral of functions on a compact metric space with respect to a normalised Borel measure satisfies the basic properties of an integral and generalizes the Henstock-Kurzweil integral. We show that when the Lebesgue integral of the function exists then the D-integral also exists and they have the same value. We provide a family of real-valued unbounded functions on the Cantor space that are D-integrable but not Lebesgue integrable with respect to a self-similar Borel measure.
ENS Cachan
France
Title.On completeness for Kantorovich-Rubinstein quasi-metrics
Download
View complete package
Abstract. A famous 1956 theorem due to Prohorov states that the space of probability measures, with the weak topology, over a Polish space is itself Polish. One way of proving this is through the so-called Kantorovich-Rubinstein metric. I will explain how the latter generalizes naturally to quasi-metric spaces. A quasi-metric space is like a metric space, without symmetry, and the notion generalizes both metric spaces and posets. The goal of this talk is to show an analogue of Prohorov’s theorem in the quasi-metric world: given any continuous complete quasi-metric space, its space of (sub)probability continuous valuations is continuous complete under the Kantorovich-Rubinstein quasi-metric, and the induced topology is the weak topology. Rather curiously, we will find ourselves at least twice in situations where we have the impression that the proof is almost complete, but where overcoming the remaining, apparently minor, difficulty will require pretty elaborate techniques. As a central tool, we will extensively rely on the poset of formal balls of a quasi-metric space, and its many properties.
Louisiana State University
United States of America
Title.Chains in Partially Ordered Spaces
Abstract. In this paper we study chains in partially ordered spaces, topological spaces equipped with a partial order, which we assume to be semiclosed, i.e., $\downarrow x$ and $\uparrow x$ are closed for each $x$. The strong order-theoretic property of being a chain allows us to develop some significant theory about chains in the weak setting of a semiclosed relation. We obtain condition for chains to be compact, to be connected, and to have the relative topology agree with the order topology. We illustrate applications of these results in the setting of semitopological semilattices.
Tulane University
United States of America
Title.Completions, K-categories and Commutative Probability Monads
Download
Abstract. The standard approach to probability over domains is via valuations and the probabilistic power domain. With one exception, this works well, the exception being the lack of the Cartesian closed category of domains that is invariant under this power domain. (This is the Jung-Tix Problem.) On the other hand, the probabilistic power domain also defines a monad on the larger category DCPO of directed complete posets and Scott-continuous maps, but it is unknown whether this monad is commutative – i.e., whether the Fubini Theorem holds. In a related invited talk, Xiaodong Jia will describe some commutative probability monads over DCPO we, together with Andre Kornell, Bert Lindenhovius and Vladimir Zamdzhiev discovered. In this talk, I will outline the mathematical theory that underpins the construction of these monads. The basic idea is to start with the simple valuations, and apply a topological completion, such as the d-completion due to Zhao and Fan. The remaining ingredients are the K-categories of Keimel and Lawson and monotone convergence spaces.
Hunan University
China
Title.D-completion, well-filterification and sobrification
Abstract. In this speech, we will discuss the D-completion, well-filterification and sobrification of a $T_0$ space. First, we supply an example of a tapered closed set which is neither the closure of a directed set nor a closed KF-set. This also gives a negative answer to the problem proposed by Xu in 2020, since each tapered closed set is a closed WD-set. Second, we provide a direct characterization for the D-completion of a poset by using the notion of pre-C-compact elements. Finally, for a given $T_0$ space, we give some sufficient conditions which guarantee that each pair of its standard D-completion, standard well-filterification and standard sobrification to agree.
Institute of Mathematics of the Czech Academy of Sciences
Czech Republic
Title.Lattice-Free and Point-Free - Vickers Duality for Subbases of Stably Locally Compact Spaces (joint work with Wiesław Kubiś)
Download
Abstract. The past 80+ years have seen a large proliferation of extensions of the classic Stone duality. While most research has focused on lattice theoretic extensions, our contention is that entailment relations provide a more flexible alternative, at least when it comes to stably locally compact spaces. Not only do entailments retain the finitary character of the original Stone duality (in contrast to frames, for example) but they also provide a common framework in which to unify many of the various extensions already established. In our talk, we will outline their general theory, based largely on previous work of Vickers and Jung-Kegelmann-Moshier, and show how they encompass a number of motivating examples from topology and order theory.
Hunan University
China
Title.Probabilistic computation and valuations monads on DCPO
Download
Abstract. I will try to explain our newly found commutative valuations monads on the category of dcpo’s and Scott-continuous functions. The findings of these monads are driven by the semantic purposes and indeed, we will see that all of them, together with the category of dcpo’s, are suitable for modelling higher-order programming languages with probabilistic effects. This allows us to bypass the Jung-Tix Problem. Some questions about our new monads are left open in this talk.
Minnan Normal University
China
Title.Scott power spaces
Download
Abstract. In this talk, we mainly discuss some basic properties of Scott power spaces. For a $T_0$ space $X$, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of $X$ endowed with the Smyth order. It is proved that if a $T_0$ space $X$ is well-filtered, then its Scott power space $\Sigma \mathsf{K}(X)$ is well-filtered, and $X$ is well-filtered iff the upper Vietoris topology is course that the Scott topology on $\mathsf{K}(X)$ and $\Sigma \mathsf{K}(X)$ is well-filtered. A sober space is constructed for which its Scott power space is not sober. Some sufficient conditions are given under which a Scott power spaces is sober. Several other properties, such as local compactness, first-countability and Rudin property, of Scott power spaces are also investigated.
Shimane University
Japan
Title. The dimension Dind of finite topological $T_0$-spaces
Download
Abstract. The talk is based on the paper by D. N. Georgiou, A. C. Megaritis, F. Sereti and myself. A.V. Arhangelskii introduced the dimension Dind and some properties of this dimension have been studied. In this talk, we study this dimension for finite $T_0$-spaces. Especially, we prove that in the realm of finite $T_0$-spaces, Dind is less than or equal to the small inductive dimension ind, the large inductive dimension Ind and the covering dimension dim. We also study the ``gaps" between Dind and the dimensions ind, Ind and dim, presenting various examples which shows these ``gaps". Moreover, in this field of spaces, we give characterizations of Dind, inserting the meaning of the maximal family of pairwise disjoint open sets, and study properties of this dimension.