Laurent Lafforgue

Abstract: The purpose of this talk will be to sketch a partial outline for building a new version of AI based on Grothendieck Topos Theory. We will first review some key facts which make Grothendieck toposes a natural interface between logic and topology or geometry. We will explain in particular in which sense the semantics of any first-order "geometric" theory can be incarnated by a topos, so by a mathematical object to which all intuitions of topological nature still apply. Based on that, we will consider anew the problem of designing good description languages for any type of real objects which we could want to represent mathematically, with the aim of processing their representations. This would require the choice of a vocabulary. After such a description vocabulary is chosen, basic principles of Topos Theory yield a process for deriving from instances of the type of real objects under consideration some grammar rules relating the elements of vocabulary. These grammar rules incarnate an interpretation principle for the type of objects under consideration. The way they are derived using principles of Topos Theory can be considered as a modellization of inductive reasoning. Supposing a good description language, consisting in chosen elements of vocabulary and derived grammar rules, has been elaborated, the next and most difficult step would be to construct a topos-based process for extracting information. This would be a topos-theoretic version of Deep Learning. We will propose a general form for such topos-based processes and describe an induced framework which allows at least to think about this problem in a mathematical way.

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