Related papers: Automorphisms of $\mathscr{P}(\lambda)/\mathscr{I}…

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…

We describe the group of continuous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.

We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of…

Given a countable abelian group $A$, we construct a row finite directed graph $\Gamma(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$ is canonically isomorphic to $A$. Moreover, each…

We show that the existence of a well-known type of ideals on a regular cardinal $\lambda$ implies a compactness property concerning the specialisability of a tree of height $\lambda$ with no cofinal branches. We also use Neeman's method of…

We present an overview of results on the question of whether the non-stationary ideal of an uncountable regular cardinal $\kappa$ can be defined by a $\Pi_1$-formula using parameters of hereditary cardinality at most $\kappa$. These results…

We determine the subgroup of automorphisms acting trivially on the second integral cohomology for hyperkaehler manifolds which are deformation equivalent to O'Grady's sporadic examples. In particular, we prove that this subgroup is trivial…

Let $P(\mathrm{sl}_2(K))$ be the Poisson enveloping algebra of the Lie algebra $\mathrm{sl}_2(K)$ over an algebraically closed field $K$ of characteristic zero. The quotient algebras $ $ $P(\mathrm{sl}_2(K))/(C_P-\lambda)$, where $C_P$ is…

It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in…

We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH…

Given two infinite cardinals $\kappa$ and $\lambda$, we introduce and study the notion of a $\kappa$-barely independent family over $\lambda.$ We provide some conditions under which these types of families exist. In particular, we relate…

For any abstract subfactor planar algebra $P$, there exists a finite index extremal subfactor $M_0 \subset M_1$ with $P$ as its standard invariant. In this paper, we classify the automorphism group of a bipartite graph planar algebra, and…

Assume that there is no quasi-measurable cardinal smaller than $2^\omega$. ($\kappa$ is quasi measurable if there exists $\kappa $-additive ideal $\ci $ of subsets of $\kappa $ such that the Boolean algebra $P(\kappa)/\ci$ satisfies c.c.c.)…

We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal ${\kappa}\in On$, (1) ${\kappa}^+\to ({\kappa},{\omega}+1)$, (2) any…

We prove that every biregular automorphism of the affine algebraic variety ${\mathbb P}^M\setminus S$, $M\geqslant 3$, where $S\subset {\mathbb P}^M$ is a hypersurface of degree $m\geqslant M+1$ with a unique singular point of multiplicity…

We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $\kappa$ is a limit of measurables then there is no wellorder of a subset of $P(\kappa)$ of length…

An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC…

Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic.

Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $\kappa$, there are no $\Sigma^1_1(\kappa)-\kappa-$mad families.

In this paper we consider invertible one-dimensional linear cellular automata (CA hereafter) defined on a finite alphabet of cardinality $p^k$, i.e. the maps $T_{f[l,r]}:\mathbb{Z}^{\mathbb{Z}}_{p^k}\to\mathbb{Z}^{\mathbb{Z}}_{p^k}$ which…