Three-Manifolds, Singularities, and Invariants

Barnard College, New York NY

Investigators

Abstract

For hyperbolic manifolds there are postulated connections between geometric, representation-theoretic, and quantum based invariants of manifolds. The PI will work towards specific conjectures relating these invariants; in particular, he will investigate a scissors congruence approach to the volume conjecture and he will continue work on his conjecture relating the A-polynomial with variation of Bloch invariants. The research on these connections involves invariants that are related to number theory and algebraic K-theory, and machine computation has been an important component of the research. The main tool is the program Snap, and another important aspect of the project is further development of this program. The PI will also continue his investigation of commensurability properties of 3-manifolds. This proposal addresses some long-term open questions concerning 3-dimensional space forms. These questions involve invariants that span different areas of mathematics, creating interactions of low dimensional topology with algebra, number theory, and theoretical physics. Links between disparate areas provide much of the power of mathematics, and strengthening these links increases the power. The project will also lead to a major enhancement of powerful 3-manifold software, originally developed in a project led by PI and C. Hodgson, that is an important tool for the research of this proposal and is also widely used in the mathematical community.