Related papers: A Groszek-Laver pair of undistinguishable $E_0$ cl…

Let $E$ be an analytic equivalence relation on a Polish space. We introduce a framework for studying the possible "reasonable" complete classifications and the complexity of possible classifying invariants for $E$, such that: (1) the…

We establish the equality of two definitions of an Euler class in algebraic geometry: the first definition is as a "characteristic class" with values in Chow-Witt theory, while the second definition is as an "obstruction class." Along the…

We study generalized diffeomorphisms in exceptional geometry with U-duality group E_{n(n)} from an algebraic point of view. By extending the Lie algebra e_n to an infinite-dimensional Borcherds superalgebra, involving also the extension to…

Let k be a field of characteristic 2 and let L/k be a finite Galois extension with Galois group G. We show the equivalence of the following two properties: (*) The group G is generated by elements of order 2 and by elements of odd order.…

Extended geometry provides a unified framework for double geometry, exceptional geometry, etc., i.e., for the geometrisations of the string theory and M-theory dualities. In this talk, we will explain the structure of gauge transformations…

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and…

We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of…

We consider some special type extensions of an arbitrary Lie algebra ${\cal G}$, arising in the theory of Lie-Poisson structures over $({\cal G}^*)^n$, where ${\cal G}^*$ is the dual of ${\cal G}$. We show that some classes of these…

Assuming that ORD is $\omega +\omega $-Erd\"os we show that if a class forcing amenable to $L$ (an $L$-forcing) has a generic then it has one definable in a set-generic extension of $L[O^\#]$. In fact we may choose such a generic to be {\it…

Suppose $X$ and $Y$ are finite complexes, with $Y$ simply connected. Gromov conjectured that the number of mapping classes in $[X,Y]$ which can be realized by $L$-Lipschitz maps grows asymptotically as $L^\alpha$, where $\alpha$ is an…

Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a…

We characterize all pairs $(\mathcal{A}$,$\mathcal{B})$ of generalized Riemann differences for which $\mathcal{A}$-differentiability implies $\mathcal{B}$-differentiability. Two generalized Riemann derivatives $\mathcal{A}$ and…

In this paper we study two classes of $\ell$-modular standard modules of the general linear group. The first class is obtained by reducing existing standard modules over $\overline{\mathbb{Q}}_\ell$ to $\overline{\mathbb{F}}_\ell$ with…

A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero.

We show that for $G$ a simple compact Lie group, the infinitesimal subgroup $G^{00}$ is bi-intepretable with a real closed valued field. We deduce that for $G$ an infinite definably compact group definable in an o-minimal expansion of a…

Suppose $E \subseteq \mathbb{R}$ is nowhere dense. If $(\mathbb{R},<,+,(x \mapsto \lambda x)_{\lambda \in \mathbb{R} }, E)$ does not define every bounded Borel subset of every $\mathbb{R}^n$ then for every $s > 0$ we have $$ | \{ k \in…

Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.

We classify all essential extensions of the form $$0 \rightarrow \W \rightarrow \D \rightarrow A \rightarrow 0$$ where $\W$ is the unique separable simple C*-algebra with a unique tracial state, with finite nuclear dimension and with…

We prove that if $I_\ell = [a_\ell,b_\ell)$, $\ell=1, \ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \ldots, a_L, b_1, \ldots, b_L$ are linearly independent over $\mathbb{Q}$, then there exist…

Let $\mathcal{W}(b)$ be a class of free Lie conformal algebras of rank $2$ with $\mathbb{C}[\partial]$-basis ${L,H}$ and relations \begin{eqnarray*} [L_\lambda L]=(\partial+2\lambda)L,\ \ [L_\lambda H]=\big(\partial+(1-b)\lambda\big)H, \ \…