The Tbilisi Centre for Mathematical Sciences (TCMS) is a non-governmental and non-profit independent academic institution, continuing and developing activities in mathematical sciences that were initiated by the National Centre for Science and Technology (NCST).

We show that assuming the consistency of certain large cardinals (namely a supercompact cardinal with a measurable cardinal above it), it is possible to force and construct choiceless universes of $\mathsf{ZF}$ in which the first two uncountable cardinals $\aleph_1$ and $\aleph_2$ are both measurable and carry certain fixed numbers of normal measures. Specifically, in the models constructed, $\aleph_1$ will carry exactly one normal measure, namely $\mu_w = \{x \subseteq \aleph_1 \mid x$ contains a club set$\}$, and $\aleph_2$ will carry exactly $\tau$ normal measures, where $\tau \ge \aleph_3$ is any regular cardinal. This contrasts with the well-known facts that assuming $\mathsf{AD} + \mathsf{AC}$, $\aleph_1$ is measurable and carries exactly one normal measure, and $\aleph_2$ is measurable and carries exactly two normal measures.

We show that the first order theory of the $\Sigma^0_2\;$ $\mathrm{s}$-degrees is undecidable. Via isomorphism of the $\mathrm{s}$-degrees with the $\mathrm{Q}$-degrees, this also shows that the first order theory of the $\Pi^0_2\;$ $\mathrm{Q}$-degrees is undecidable. Together with a result of Nies, the proof of the undecidability of the $\Sigma^0_2\;$ $\mathrm{s}$-degrees yields a new proof of the known fact (due to Downey, LaForte and Nies) that the first order theory of the c.e. $\mathrm{Q}$-degrees is undecidable.

The aim of this paper is to gain explicit information about the multiplicative structure of $\ell_*\ell$, where $\ell$ is the connective Adams summand at an odd prime $p$. Our approach differs from Kane's or Lellmann's because our main technical tool is the $MU$-based Künneth spectral sequence. We prove that the algebra structure on $\ell_*\ell$ is inherited from the multiplication on a Koszul resolution of $\ell_*BP$.

We prove that the Leech dimension of any free partially commutative monoid is equal to the supremum of numbers of its mutually commuting generators. As a consequence, we confirm a conjecture that if a free partially commutative monoid does not contain more than $n$ mutually commuting generators, then it is of homological dimension $\leqslant n$. We apply this result to the homological dimension of asynchronous transition systems. We positively answer the question whether the homological dimension of an asynchronous transition system is not greater than the maximal number of its mutually independent events.

For $P$ a poset or lattice, let $\mathrm{Id}(P)$ denote the poset, respectively, lattice, of upward directed downsets in $P,$ including the empty set, and let $\mathrm{id}(P)=\mathrm{Id}(P)-\{\emptyset\}.$ This note obtains various results to the effect that $\mathrm{Id}(P)$ is always, and $\mathrm{id}(P)$ often, “essentially larger” than $P.$ In the first vein, we find that a poset $P$ admits no $\!\lt\!$-respecting map (and so in particular, no one-to-one isotone map) from $\mathrm{Id}(P)$ into $P,$ and, going the other way, that an upper semilattice $P$ admits no semilattice homomorphism from any subsemilattice of itself onto $\mathrm{Id}(P).$

The slightly smaller object $\mathrm{id}(P)$ is known to be isomorphic to $P$ if and only if $P$ has ascending chain condition. This result is strengthened to say that the only posets $P_0$ such that for every natural number $n$ there exists a poset $P_n$ with $\mathrm{id}^n(P_n)\cong P_0$ are those having ascending chain condition. On the other hand, a wide class of cases is noted where $\mathrm{id}(P)$ is embeddable in $P.$

Counterexamples are given to many variants of the statements proved.

We characterize from a functional analytic point of view almost periodicity of operators on $\ell^1$ given by infinite column-stochastic matrices. Some of the equivalent properties occur, under the name of Foster's condition, in the theory of stochastic processes. The results are applied to flows in infinite networks.

Our “long term and large scale” aim is to characterize the first order theories $T$ (at least the countable ones) such that for every ordinal $\alpha$ there are $\lambda$, $M_1$, $M_2$ such that $M_1$ and $M_2$ are non-isomorphic models of $T$ of cardinality $\lambda$ which are EF$^+_{\alpha,\lambda}$-equivalent. We expect that as in the main gap [11, XII], we get a strong dichotomy, i.e., on the non-structure side we have stronger, better examples, and on the structure side we have an analogue of [11, XIII]. We presently prove the consistency of the non-structure side for $T$ which is $\aleph_0$-independent (= not strongly dependent), even for PC$(T_1,T)$.

We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christensen.