Title: On the Garoufalidis-Kashaev State-Integral Invariant

Program: Mathematics MS

Committee Chair: Uwe Kaiser

Committee: Uwe Kaiser, Jens Harlander, Zach Teitler

Abstract: This thesis explores a construction of the family of topological invariants for certain oriented 3-manifolds based on the state-integral approach developed by Andersen, Garoufalidis, and Kashaev in the Archimedean setting. Starting from an ideal triangulation of a 3-manifold equipped with angle data, variables are assigned to the faces and tetrahedra, taking values in a so called 'Gaussian group'.
The invariant is defined by integrating a distribution defined from the combinatorics of the triangulation and a special function over a product of the Gaussian group. The special function is a quantum dilogarithm, whose valuable feature, the pentagon relation, ensures the resulting integral remains unchanged under local modifications of the triangulation, thereby reflecting the topology of the manifold.