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This study establishes foundational topological results for moduli spaces of non-smooth metric measure structures using optimal transport. It demonstrates how these structures on a space relate to their universal covers, yielding new insights into geometric analysis.

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Area of Science:

  • Differential Geometry
  • Topology
  • Optimal Transport

Background:

  • Metric measure structures provide a generalization of Riemannian manifolds.
  • Optimal transport offers a synthetic approach to defining curvature.
  • Moduli spaces are crucial for classifying geometric objects.

Purpose of the Study:

  • Establish foundations for moduli spaces of non-smooth metric measure structures.
  • Prove topological results concerning these moduli spaces.
  • Investigate structures with non-negative Ricci curvature in a synthetic sense.

Main Methods:

  • Utilizing optimal transport for synthetic Ricci curvature.
  • Analyzing convergence of structures on a space and their universal covers.
  • Constructing Albanese and soul maps.

Main Results:

  • Relating convergence of -structures to equivariant convergence on the universal cover.
  • Proving continuity of Albanese and soul maps.
  • Demonstrating how structures on the universal cover split.

Conclusions:

  • The study provides a framework for understanding moduli spaces of non-smooth metric measure structures.
  • Constructed examples exhibit non-trivial rational homotopy groups.
  • The work deepens the understanding of geometric structures via optimal transport and topology.