## Mini-symposium

Topology and Applications

*dedicated to the memory of Prof. Stoyan Nedev and Acad. Mitrofan Choban*

#### in the frame of the International Conference “Mathematics Days in Sofia”

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he range of talks and presented will be in the field of General Topology, Set-theoretic Topology, Algebraic Topology, Geometric Topology and their application and interplay with other fields as Functional Analysis, Category theory, Dynamical systems and others. The mini-symposium will be dedicated to the memory of the two prominent researchers Stoyan Nedev (IMI, BAS) and Mitrofan Choban (Tiraspol State University, Moldova) devoted strongly to the field of General Topology and its Applications. Their PhD students and co-workers are invited to participate.

#### Organizing Committee

Vladimir Todorov (University of Architecture, Civil Engineering and Geodesy, Bulgaria)

Ekaterina Mihaylova (University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria)

### Participants

**Arkady Leiderman**, Ben-Gurion University of the Negev, Israel**Ekaterina Mihaylova**, University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria**Olivier Otafudu**, North-West University, South Africa**Petar Kenderov**, Institute of Mathematics and Informatics, Bulgaria**Stavros Iliadis**, Moscow State University “M.V. Lomonosov”, Russia**Takamitsu Yamauchi**, Ehime University, Japan**Valdemar Tsanov**, Institute of Mathematics and Informatics, Bulgaria**Vesko Valov**, Nipissing University, Canada**Vladimir Todorov**, University of Architecture, Civil Engineering and Geodesy, Bulgaria**Zlatina Tsolova**, University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria

### Program and Abstracts

A compact space X is called a Corson compactum if it can be homeomorphically embedded into a Σ-product of the real lines. The class of Corson compacta properly contains the class of Gul’ko compacta, which in turn properly contains the class of Eberlein compacta. Every Gul’ko compactum has a dense completely metrizable subspace [1], [2], [3], while there are ZFC examples of Corson compact spaces without dense metrizable subspaces (see [3], [5]). We investigate the following question.

Problem 1. Does there exist a Corson compactum X such that its countable power Xω has no dense metrizable subspace?

We show a few consistent examples of Corson compacta X such that Xω does not contain a dense metrizable subspace. A ZFC example of such a Corson compact space is still unknown.

We also prove that the existence of a ccc counterexample to above Problem 1 is equivalent to the failure of MAω1 for powerfully ccc posets. Then we give a new proof to Kunen and van Mill’s theorem stating that the existence of a non-metrizable Corson compactum with a strictly positive measure is equivalent to the failure of MAω1 for measure algebras.

Our talk is based on a joint work with Santi Spadaro and Stevo Todorcevic [4].

References:

[1] G. Gruenhage, A note on Gul’ko compact spaces, Proc. Amer. Math. Soc. 100 (1987), 371–376. [2] P. S. Kenderov, C(T) is weak Asplund for every Gul’ko compact T, C. R. Acad. Bulgare Sci. 40 (1987), 17–19. [3] A. G. Leiderman, Everywhere dense metrizable subspaces of Corson compacta, Math. Notes of Acad. Science USSR, 38 (1985), 751–755. [4] A. Leiderman, S. Spadaro, S. Todorcevic, Dense metrizable subspaces in powers of Corson compacta, Proc. Amer. Math. Soc. 150 (2022), 3177–3187. [5] S. Todorcevic, Trees and linearly ordered sets, in Handbook of Set-theoretic Topology, (eds. K. Kunen and J. E. Vaughan), North Holland, Amsterdam, 1984, 235–293.The notion of coarse structure is a generalization of metric to describe large-scale uniform properties. By a group coarse structure, we mean a coarse structure compatible with the group operations. A well-known example is the group-compact coarse structure, which is generated by the relatively compact subsets of a topological group. On the other hand, Rosendal (2021) introduced the notion of the left-coarse structure to study Polish groups. In this talk, we discuss these two group coarse structures on locally compact abelian groups.

This talk is based on a joint work with Dmitri Shakhmatov and Nicolo Zava.

The coadjoint orbits of compact Lie groups form an important class of manifolds. Equipped with their Kostant-Kirillov-Sourieau Kähler structures they represent models for all simply connected compact homogeneous Kähler manifolds. The integral orbits admit embeddings as projective algebraic varieties corresponding to the irreducible unitary representations of the group. Several representation theoretic concepts are related to properties of the convex hull of the orbit, and to its projections to subalgebras. I will introduce the notion of partial convex hulls in this context and indicate its relation to representation theory.

Regular topological spaces X containing a dense completely metrizable subset appear rather often in Topology and Functional Analysis. A characterization of these spaces is given which is based on the existence of residually defined continuous single-valued selections for certain set-valued mappings with values in X. These spaces are preserved under continuous and demi-open single-valued mappings. As an application, it is shown that the existence of residually defined continuous selection implies the Closed Graph and the Open Mapping theorems for completely metrizable topological vector spaces.

This is joint work with J. P. Revalski.