# Ming Ng (Queen Mary)

- Date
- Wednesday 9 November 2022, 3:00 PM
- Location
- MALL
- Category
- Logic Seminar

## Adelic Geometry via Geometric Logic

**Note time change: 3pm**

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 do not allow negation), 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 $\mathrm{Set}$) — 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 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 $\mathbb{Q}$ and provide a point-free account of exponentiation. The next step is to construct the classifying topos of places of $\mathbb{Q}$, which incidentally provides a topos-theoretic analogue of the Arakelov compactification of $\operatorname{Spec}(\mathbb{Z})$. Interestingly, whereas the classical picture views the Archimedean place as a single point "at infinity", our picture reveals that the Archimedean place resembles a blurred interval living below $\operatorname{Spec}(\mathbb{Z})$. This raises challenging questions to our current understanding of the number theory, particularly in regards to reconciling the Archimedean vs. the non-Archimedean aspects.