## General affine adjunctions, Nullstellensätze, and dualities

At last, we have finished and submitted our paper on “General affine adjunctions, Nullstellensätze, and dualities” co-authored with Olivia Caramello and Vincenzo Marra.

Abstract.We introduce and investigate a category-theoretic abstraction of the standard “system-solution” adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert’sNullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.

The preprint is available on arXiv. We made another preprint available some years ago(!), but the manuscript has changed in many respects. The main differences between the two versions on arXiv are the following:

- The comparison with the existing literature is now more thorough.
- The categories R and D are now taken directly without passing through the quotient categories. In our opinion, this is cleaner and, as a consequence, it is now clearer what are the minimal assumption on the triplet I: T -> S.
- There is now a section studying the issue of concreteness of the adjunction and comparing with the theory of concrete adjunction.

## Tutorial on Dualities

These are the slides of my tutorial on **Dualities** at the $16^{th}$ Latin American Symposium on Mathematical Logic. 28th July – 1st August 2014. Buenos Aires, Argentina. A shorter version can be found here.

## Dualities and geometry

Finally I wrote some slides about the long-waiting article I am writing together with Olivia Caramello and Vincenzo Marra on adjunctions, dualities, and Nullstellensätze . These slides where presented at the AILA meeting in Pisa and at the Apllied Logic seminar in Delft.