Semih Özlem -- The structure of learning: From algebra to AI скачать в хорошем качестве

Semih Özlem -- The structure of learning: From algebra to AI

We often think of mathematics as a tower of abstractions, but it begins with something deeply human: the act of telling things apart. In this talk, I'll explore how this simple idea—splitting and focusing—manifests across different fields, from linear algebra to motives to machine learning. We'll start with a basic observation: if we relax the unit axiom in a vector space, the scalar multiplication by 1 becomes an idempotent, splitting the space into what is preserved and what is annihilated. This splitting phenomenon appears in surprising places: in the theory of motives, where projectors decompose varieties; in knot theory, where Jones–Wenzl projectors filter diagram algebras; and in deep learning, where attention mechanisms focus on relevant features. I'll introduce the topos-theoretic model of neural networks (Belfiore–Bennequin) and suggest that learning difficulties like catastrophic forgetting and generalization gaps can be viewed as homotopical obstructions to achieving "nice" (fibrant) network states. Architectural tools like residual connections and attention can then be seen as learned, conditional idempotents—adaptable splitters that help networks organize information. This talk is an invitation to think structurally across disciplines. I won't present finished theorems, but a framework of connections that links motivic philosophy, categorical algebra, and the practice of machine learning. The goal is to start a conversation: can tools from pure mathematics—obstruction theory, homotopy colimits, derivators—help us design more robust, interpretable, and composable learning systems? No expertise in motives, knots, or AI is required—only curiosity about how ideas weave together.