• The example of a pseudocompact paratopological group which is not topological
  • We present a functionally Hausdorff second countable paratopological group G such that each power of G is countably pracompact but G is not a topological group. We discuss some questions from descriptive topology related with the cardinal invariants of the constructed examples.

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  • The active participants of the seminar were discuss new results and posed some open problems.

• Large free subgroups of automorphisms groups of ultrahomogeneous spaces
  • We consider the following largeness notion of subgroups of the symmetric group S_∞. A group G is large if it contains a free subgroup with c generators. We give a necessary condition for a countable structure A to have the large automorphism group Aut(A). We also prove that under this condition any non-meager subgroup of Aut(A) with the Baire property is also large. It turns out that any countable free subgroup of S_∞ is contained in a large free subgroup of S_∞, and under Martin's Axiom any free subgroup of S_∞ of cardinality less than continuum is contained in a large free subgroup of S_∞. Finally, we show that the countable product of finitely generated groups either is large or else contains no uncountable free subgroup.

• Multigeometric sequences, Cantors and Cantorvals
  • For an absolute summing sequence x=(x_n) by E(x) we denote the set of all sumsums of the series Σ_nx_n. It is known that the set E(x) is one of the following forms: a finite union of closed intervals, a copy of the Cantor set, or so-called Cantorvals. We describe families of sequences which contain, according to our knowledge, all known examples of sequences x with Cantorval sumset E(x). Also, we give a sufficient condition for E(x) being Cantor.

• On generators and automorphizms of the the monoid of cofinite monotone partial bijections of L_nx_lexZ
  • The reporter discusses on the generators and automorphisms of the the monoid of cofinite monotone partial bijections of L_nx_lexZ. In particular this semigroup has four generators and all its automorphisms are inner.

• The structure of compact rings
  • First we give some preliminary results, including a summary of Jacobson's Radical Theory in a form convenient for our use, some remarks on Q-rings (rings in which elements near zero are quasi-regular), and the abstract theory of boundedness. The structure theorems for compact rings are given.

• Korovin's example of a pseudocompact Tychonoff semitopological group which is not paratopological
  • At last we shall know the construction of this example.

• An unexpected obstacle for the building of close to compact semi- and para-topological groups without automatic continuity of operations
  • We show that if (G,σ) is a Hausdorff precompact topological group, (G,τ) is a paratopological or semitopological group with a base consisting of canonically open sets and τ⊂σ then (G,τ) is a topological group.

• Pseudocompactness, products and Brandt λ⁰-extensions
  • We give conditions under which the Tychonoff product of topological Brandt λ⁰-extensions of pseudocompact (countably compact) semitopological semigroups is a pseudocompact (countably compact) space.

• On H-complete topological inverse semigroups
  • We consider a category of Lawson semilattices with compact maximal chains, and give sufficient conditions to be H-compete for such semilattices. Also it will be given sufficient conditions to be H-complete for some clases of topological inverse semigroups.

• On H-complete topological inverse semigroups
  • We consider topological inverse semigroups and give sufficient conditions to be H-complete in some classes of topological inverse semigroups.

• Sequentially pseudocompact spaces
  • We reintroduce sequentially pseudocompact spaces which lie between sequentially compact and pseudocompact spaces and investigate their examples and properties with respect to standard operations with topological spaces. In particular, sequentially pseudocompact spaces unexpectedly turned out to be preserved by products, opposite to their above counterparts.

• Extrasensory perception strategy and Catalan numbers
  • The speaker, after the years of his practice of ancient mystic teachings, is going to reveal mathematical mysteries.

• On topological semilattices with compact maximal chains
  • The speaker descusses on the structure and properties of topological semilattices with compact maximal chains.

• Cone topologies of paratopological groups
  • We begin a cycle of lectures devoted to the subject which is useful for constructing counterexamples of paratopological groups. We start from an interplay between the algebraic properties of the cone semigroup S and compact-like properties of two semigroup topologies generated by S on the group.

• Cone topologies of paratopological groups
  • We continue the cycle of lectures devoted to the subject which is useful for constructing counterexamples of paratopological groups. We start from an interplay between the algebraic properties of the cone semigroup S and compact-like properties of two semigroup topologies generated by S on the group.

• Topological algebraic and categorical properties of the polycyclic monoid
  • We describe a subsemillatice of the polycyclic monoid, constract a minimal topology on the polyciclic monoid and prove that polycyclic monoid in this topology is an H-complete topological inverse semigroup. Also we study some topological, algebraic and categorical properties of the polycyclic monoid.

• On compactly factorizable topological inverse semigroups
  • We discuss on compactly factorizable topological inverse semigroups and related topics.

• On completeness of topological inverse semigroups
  • We discuss on completeness of topological inverse semigroups which is a generalization of the Raikov completeness for topological groups.