We extend the results of Hyttinen and Paolini on free projective planes to free generalized n-gons. We show that the theory of free generalized n-gons corresponds to the theory of open generalized n-gons and is complete. We further characterize the elementary substructure relation and show that the theory of open generalized n-gons does not have a prime model. Hide abstract

A structure is MS-measurable if it admits a dimension-measure function on its definable sets satisfying certain definability, additivity and Fubini conditions. MS-measurable structures are necessarily supersimple of finite SU-rank and (functionally) unimodular. Elwes and Macpherson (2006) asked whether the converse is true. In that paper the authors suggested omega-categorical Hrushovski constructions as a possible source of counterexamples. We are also interested in the question of whether any omega-categorical Hrushovski construction is MS-measurable since they may provide a counterexample to the conjecture in the same paper that any omega-categorical MS-measurable structure is one-based. Results from Evans and the first year of my PhD show how in various omega-categorical Hrushovski constructions any dimension satisfying certain weak conditions must be a scalar multiple of the natural notion of dimension in a Hrushovski construction. Evans' results already gave some counterexamples to Elwes’ and Macpherson’s first question. But those methods cannot be used for showing in general that Hrushovski constructions are not MS-measurable. Hence, I've been looking at different methods for showing there is no dimension-measure function on a Hrushovski construction with the natural Hrushovski dimension. I shall discuss my proof that there is no such dimension-measure function for a class of omega-categorical Hrushovski constructions which are supersimple, finite rank and unimodular graphs. Hide abstract

On a syntactic level, the peculiarity of geometric logic can be seen from the choice of logical connectives used (in particular, we allow for infinitary disjunctions), but there are deep ramifications of this seemingly innocuous move. One, geometric logic is incomplete if we restrict ourselves to set-based models, but is complete if we also consider models in all toposes (i.e. not just Set) — as such, geometric logic can be viewed as an attempt to pull our mathematics away from a fixed set theory. Two, there is an intrinsic continuity to geometric logic, which is furnished by the definition of the classifying topos. Indeed, since every Grothendieck topos is a classifying topos of some geometric theory, this provides yet another way of viewing Grothendieck toposes as generalised spaces. Both these insights will inform the content of this talk. We shall start by giving a leisurely introduction to the theory of geometric logic and classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory. The first step of this programme is to define the geometric theory of absolute values of Q and provide a point-free account of exponentiation (which has already been completed). The next step is to construct the classifying topos of places of Q, which incidentally provides a topos-theoretic analogue of the Arakelov compactification of Spec(Z). This part is still work in progress, but some interesting observations (in particular, regarding whether we ought to view the Archimedean place as a point) have already emerged which we would like to share with the community. Fundamentally, we hope to use this framework to bring into focus certain essential aspects about how topology and algebra interact, before extending our insights to explore various connections with the number theory and homotopy theory. Hide abstract