A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes. As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

MODEL THEORY AND MACHINE LEARNING

About 25 years ago, it came to light that a single combinatorial property determines both an important dividing line in model theory (NIP) and machine learning (PAC-learnability). The following years saw a fruitful exchange of ideas between PAC-learning and the model theory of NIP structures. In this article, we point out a new and similar connection between model theory and machine learning, this time developing a correspondence between stability and learnability in various settings of online learning. In particular, this gives many new examples of mathematically interesting classes which are learnable in the online setting.

The Markoff Property

The Markoff property is a combinatorial property of infinite words on the alphabet {a,b}, and of bi-infinite words. Such a word has this property if whenever there is a factor xy in the word,with x,y equal to the letters a,b (in some order), then itmay be extended into a factor of the formym’xymx, wherem’ is the reversal ofm, and where the length ofmis bounded (the bound depends only on the infinite word). As discussed in this chapter, the main theorem, due toMarkoff, is that this property implies periodicity, with a periodic pattern which must be a Christoffel word. It is one of the crucial results inMarkoff’s theory.

A constructive way to compute the Tarski number of a group

The Tarski number of a group [Formula: see text] is the minimal number of the pieces of paradoxical decompositions of that group. Using configurations along with a matrix combinatorial property, we construct paradoxical decompositions. We also compute an upper bound for the Tarski number of a given non-amenable group by counting the number of paths in a diagram associated to the group.

Combinatorial Property of Sets of Boxes in Multidimensional Euclidean Spaces and Theorems in Olympiad Tasks

Theorems (in general sense) are constituents of inventing, analysing and solving olympiad tasks. Also, some theorems can be proved with computer assistance only. The main idea is (human) reducing of primary (unbounded) set to a finite one. Non-trivial immanent properties of mathematical objects are of interest because they can be considered as alternative definitions of these objects revealing their additional features. A non-formal indication of such property is only inital data (size of domain) and only output data (proven/not proven) in a corresponding algorithm. One new and two known examples of such properties are considered, some techniques to convert theorem-proving algorithms into olympiad tasks are proposed.

CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing.In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

Sphericity of a real hypersurface via projective geometry

In this work, we obtain an unexpected geometric characterization of sphericity of a real-analytic Levi-nondegenerate hypersurface [Formula: see text]. We prove that [Formula: see text] is spherical if and only if its Segre(-Webster) varieties satisfy an elementary combinatorial property, identical to a property of straight lines on the plane and known in Projective Geometry as the Desargues Theorem.