Research

Research Foundations

The mathematical framework behind adaptive learning.

Publications

Working Papers

Working papers are available on request.

Working Paper

The Riemannian Knowledge Space: Learning as Navigation on a Manifold

We model the space of learning outcomes as a Riemannian manifold where the metric tensor encodes pedagogical cost. Students navigate this space along geodesic paths. The cognitive cost of traversal is C = ρλ², linking complexity density to semantic distance.

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Working Paper

Field-Theoretic Quantities for Adaptive Learning

We derive four field-theoretic quantities — entropy, divergence, Jacobian, and Green's function — from the knowledge manifold and show how each maps to a concrete tutoring decision. Together they form a complete adaptive intelligence layer.

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Working Paper

Curriculum Alignment via Embedding Geometry

We embed multiple curricula (Cambridge, UK NC, Common Core) into a shared vector space using transformer-based embeddings, then construct the manifold from pairwise distances. Curricula become coordinate systems on a shared knowledge geometry.

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Key Concepts

Glossary

Riemannian manifold: A smooth space where distances between points are defined by a metric tensor that can vary across the surface.
Metric tensor: Encodes the "cost" of moving between learning outcomes. Different curricula parameterise the same tensor differently.
Geodesic: The optimal learning path through the knowledge space — the route that minimises total cognitive cost.
Shannon entropy: Measures uncertainty in a student's knowledge state. High entropy = we don't know what they know.
Jacobian: How mastery of one outcome propagates to connected outcomes. High Jacobian = high teaching leverage.
Lagrangian: The action functional for learning: what a student "should" do next, derived from the manifold geometry.

Active Research

What We're Working On

Optimal metric tensor construction from sparse assessment data
Multi-scale learning dynamics: micro (within-session), meso (across sessions), macro (across terms)
Publisher licensing as metric tensor parameterisation — can a textbook define the geometry?

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