Quantum invariants and hyperbolic manifolds in three-dimensional topology

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The project aims to broaden our understanding of three-dimensional (3-D) spaces, including spaces that arise in engineering, microbiology and physics. It is known that all 3-D spaces can be decomposed into geometric pieces. The most common type of geometry is hyperbolic. It is also known that such spaces have algebraic structures arising from quantum physics, known as quantum invariants. Several important conjectures, based on developments in physics, assert that hyperbolic geometry and quantum invariants are deeply related, but they remain unproved. The project aims to find new relationships between hyperbolic geometry and quantum invariants, advancing our understanding of both areas.