Algebraic Geometry of Exotic Embeddings in 4-Manifolds

Asselmeyer-Maluga identifies wildly embedded 3-manifolds as quantum states in exotic (\mathbb{R}^4), but their geometric properties are computationally opaque. Rocco et al.’s algorithms for computing reach/feature sizes of algebraic manifolds could quantify these embeddings. By treating the wildly embedded 3-manifold as an algebraic set, we can compute its weak feature size to measure "quantum fuzziness." This tests whether exotic embeddings violate classical smoothness assumptions (e.g., Lipschitz normal embedding). Novelty: Applies algebraic geometry to quantum gravity, bridging computational topology and theoretical physics. Could yield numerical evidence for exotic smoothness in physical systems.

References:

1. Smooth quantum gravity: Exotic smoothness and Quantum gravity. T. Asselmeyer-Maluga (2016).
2. Computing geometric feature sizes for algebraic manifolds. S. Rocco, Parker B. Edwards, David Eklund, Oliver Gäfvert, J. Hauenstein (2022). SIAM Journal on applied algebra and geometry.