Initial steps are presented towards understanding which finitely generated groups are almost surely generated as a semigroup by the path of a random walk on the group.

This preprint is part of a longer article in which we will show that any PS–overtwisted manifold is algebraically overtwisted. In dimension 3, this has already been proved for overtwisted manifolds by Eliashberg and Yau [Yau06]. The “correct” definition of algebraically overtwisted should be the strongest contact invariant that is not able to distinguish overtwisted contact manifolds (unfortuna...

Algebraically constructible functions connect real algebra with the topology of algebraic sets. In this survey we present some history, definitions, properties, and algebraic characterizations of algebraically constructible functions, and a description of local obstructions for a topological space to be homeomorphic to a real algebraic set. More than three decades ago Sullivan proved that the l...

We extend an approach to Morse theory due to Harvey and Lawson to non compact manifolds. This provides a modified and arguably more natural version of Morse-Novikov theory, where the Novikov Ring is replaced by a new " Forward Laurent Ring ". A detailed development of standard Morse-Novikov theory is clarified in several ways. Geometrically: The Novikov complex is described as a subcomplex of t...

Stability conditions on triangulated categories were introduced by Bridgeland as a ‘continuous’ generalisation of t-structures. The set of locally-finite stability conditions on a triangulated category is a manifold which has been studied intensively. However, there are mainstream triangulated categories whose stability manifold is the empty set. One example is D ( k[X]/(X) ) , the compact deri...

We investigate a Hamiltonian model of networks. The model is a mirror formulation of the XY model (hence the name) — instead letting the XY spins vary, keeping the coupling topology static, we keep the spins conserved and sample different underlying networks. Our numerical simulations show complex scaling behaviors, but no finite-temperature critical behavior. The ground state and low-order exc...

The aim of algebraic logic is to compact series of small steps of general logical inference into larger (in)equational steps. Algebraic structures that have proved very useful in this context are modal semirings and modal Kleene algebras. We show that they can also model knowledge and belief logics as well as games without additional effort; many of the standard logical properties are theorems ...