Ramsey expansions of metrically homogeneous graphs

We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and the coherent EPPA. Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's classification programme of Ramsey classes and as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion (or lift or ordering) property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a "completion algorithm" then allows us to apply several strong results in the areas that imply EPPA and respectively the Ramsey property. The main results have numerous corollaries on the automorphism groups of the Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor revisio

The pp conjecture in the theory of spaces of orderings

The notion of spaces of orderings was introduced by Murray Marshall in the 1970's and provides an abstract framework for studying orderings on fields and the reduced theory of quadratic forms over fields. The structure of a space of orderings (X, G) is completely determined by the group structure of G and the quaternary relation (a_1, a_2) = (a_3, a_4) on G -- the groups with additional structure arising in this way are called reduced special groups. The theory of reduced special groups, in turn, can be conveniently axiomatized in the first order language L_SG. Numerous important notions in this theory, such as isometry, isotropy, or being an element of a value set of a form, make an extensive use of, so called, positive primitive formulae in the language L_SG. Therefore, the following question, which can be viewed as a type of very general and highly abstract local-global principle, is of great importance:Is it true that if a positive primitive formula holds in every finite subspace of a space of orderings, then it also holds in the whole space?This problem is now known as the pp conjecture. The answer to this question is affirmative in many cases, although it has always seemed unlikely that the conjecture has a positive solution in general. In this thesis, we discuss, discovered by us, first counterexamples for which the pp conjecture fails. Namely, we classify spaces of orderings of function fields of rational conics with respect to the pp conjecture, and show for which of such spaces the conjecture fails, and then we disprove the pp conjecture for the space of orderings of the field R(x,y). Some other examples, which can be easily obtained from the developed theory, are also given. In addition, we provide a refinement of the result previously obtained by Vincent Astier and Markus Tressl, which shows that a pp formula fails on a finite subspace of a space of orderings, if and only if a certain family of formulae is verified

Abelian homotopy Dijkgraaf-Witten theory

We construct a version of Dijkgraaf-Witten theory based on a compact abelian Lie group within the formalism of Turaev's homotopy quantum field theory. As an application we show that the 2+1-dimensional theory based on U(1) classifies lens spaces up to homotopy type.Comment: 23 pages, 1 figur

Relative cellular algebras

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular. Most prominently, the restricted enveloping algebra and the small quantum group for $\mathfrak{sl}_{2}$, and an annular version of arc algebras.Comment: 39 pages, many figures, revised version, to appear in Transform. Groups, comments welcom

Three constructions of Frobenius manifolds: a comparative study

The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting with the Dolbeault complex of a Calabi-Yau manifold and conjecturally producing the B--side of the Mirror Conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of ``non-linear cohomology''. The comparison of thesestructures sheds some light on the general Mirror Problem: establishing isomorphisms between Frobenius manifolds of different classes. Another theme is the study of tensor products of Frobenius manifolds, corresponding respectively to the K\"unneth formula in Quantum Cohomology, direct sum of singularities in Saito's theory, and presumably, the tensor product of the differential Gerstenhaber-Batalin-Vilkovisky algebras. We extend the initial Gepner's construction of mirrors to the context of Frobenius manifolds and formulate the relevant mathematical conjecture.Comment: 46 pages, AMSTe