Modal Logics of Tame Topologies
Simon Kreuzer

Abstract:
This thesis investigates the modal logics of tame topologies on the Euclidean spaces. The central tool is the theory of stratifications, for which we show that closure algebras of tame topologies are determined by the decomposition spaces of their stratifications.
These methods are applied to several natural tame structures. First, we give an algebraic proof of the Euclidean hierarchy for chequered sets. We address tensor sums of modal algebras and show that varieties generated by such tensor sums of finite modal algebras are already generated by tensor sums of subdirectly irreducible algebras.
The main focus lies on algebraic and semialgebraic geometry. We prove that the modal logic of constructible sets in the n-dimensional affine spaces is axiomatized as Grz.2⊕b_{dn+1}, thus yielding a non-trivial Euclidean hierarchy. For compact semialgebraic sets, we use the semialgebraic Hauptvermutung (German for ‘main conjecture’) to prove that their semialgebraic logic coincides with the polyhedral logic of a polyhedron uniquely determined up to PL homeomorphism. Moreover, we show that the modal logic of semialgebraic subsets of the n-dimensional Euclidean space is PL_n , the polyhedral logic of the n-simplex. These results show that polyhedral logic extends from polyhedra to a broad class of tame spaces.
Finally, we connect polyhedral logic with homotopy theory. We prove that the modal logic of all contractible polyhedra is Grz, demonstrating that contractibility is not definable in polyhedral logic. To increase expressive power, we propose Homotopy Modal Logic, extending the modal language by a reachability operator to capture homotopies between maps.