Manifolds in Natural and Social Phenomena

A guided research project in Geometry & Dynamics.

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Experience Research Geometry & Dynamics

Question

When does complex data behave as if it lies on a simpler curved space?

A manifold can be curved overall while each small region looks like familiar Euclidean space.
A manifold can be curved overall while each small region looks like familiar Euclidean space.

Goals

  • Identify manifold-like structure in a natural or social example.
  • Compare local coordinates with an ordinary flat description.

Method

Study a complex system through small neighbourhoods that may look low-dimensional.

  • Choose data or a network whose states have many recorded variables.
  • Build local coordinates or neighbourhood graphs and estimate their dimension.
  • Compare distances, clusters, or paths before and after the manifold description.