Mathematical singularity: how swarm behaviour models turn into swarm intelligence and allow building educational trajectory of learning

The article investigates the phenomenon of mathematical singularity in the context of swarm intelligence and its application to the construction of adaptive educational trajectories. Using the example of the ant colony movement model (Ant Colony Optimisation), it is demonstrated how local interactions between agents lead to globally optimal solutions, overcoming the limitations of individual behaviour. An analogy between finding the shortest path in nature and choosing a personalised learning trajectory is analysed. The technological and ethical aspects of introducing swarm algorithms into educational systems are discussed, as well as the prospects of creating self-organising platforms capable of adapting to learners' needs. The article emphasises that singularity in swarm systems opens the way to the creation of smart educational environments where trajectories are not shaped from the top down, but through the interaction of learners, data and adaptive algorithms

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